diff --git a/NAMESPACE b/NAMESPACE
index 855c65b5c..5c0835022 100644
--- a/NAMESPACE
+++ b/NAMESPACE
@@ -54,6 +54,7 @@ S3method(setup_approach,timeseries)
 S3method(setup_approach,vaeac)
 export(additional_regression_setup)
 export(aicc_full_single_cpp)
+export(append_vS_list)
 export(check_convergence)
 export(cli_compute_vS)
 export(cli_iter)
diff --git a/R/approach_vaeac.R b/R/approach_vaeac.R
index da3f17555..2eff261c3 100644
--- a/R/approach_vaeac.R
+++ b/R/approach_vaeac.R
@@ -2531,7 +2531,7 @@ Last epoch:             %d. \tVLB = %.3f \tIWAE = %.3f \tIWAE_running = %.3f\n",
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = approach,
-#'   prediction_zero = p0,
+#'   phi0 = p0,
 #'   n_MC_samples = 1, # As we are only interested in the training of the vaeac
 #'   vaeac.epochs = 10, # Should be higher in applications.
 #'   vaeac.n_vaeacs_initialize = 1,
@@ -2545,7 +2545,7 @@ Last epoch:             %d. \tVLB = %.3f \tIWAE = %.3f \tIWAE_running = %.3f\n",
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = approach,
-#'   prediction_zero = p0,
+#'   phi0 = p0,
 #'   n_MC_samples = 1, # As we are only interested in the training of the vaeac
 #'   vaeac.epochs = 10, # Should be higher in applications.
 #'   vaeac.width = 16,
@@ -2766,7 +2766,7 @@ vaeac_plot_eval_crit <- function(explanation_list,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "vaeac",
-#'   prediction_zero = mean(y_train),
+#'   phi0 = mean(y_train),
 #'   n_MC_samples = 1,
 #'   vaeac.epochs = 10,
 #'   vaeac.n_vaeacs_initialize = 1
diff --git a/R/cli.R b/R/cli.R
index 5f2643add..694c7f7cd 100644
--- a/R/cli.R
+++ b/R/cli.R
@@ -80,7 +80,6 @@ cli_startup <- function(internal, model_class, verbose) {
 #' @export
 #' @keywords internal
 cli_compute_vS <- function(internal) {
-
   verbose <- internal$parameters$verbose
   approach <- internal$parameters$approach
 
diff --git a/R/compute_estimates.R b/R/compute_estimates.R
index 34c14c826..05b7fd1d8 100644
--- a/R/compute_estimates.R
+++ b/R/compute_estimates.R
@@ -76,11 +76,11 @@ compute_estimates <- function(internal, vS_list) {
 #' @keywords internal
 postprocess_vS_list <- function(vS_list, internal) {
   keep_samp_for_vS <- internal$parameters$output_args$keep_samp_for_vS
-  prediction_zero <- internal$parameters$prediction_zero
+  phi0 <- internal$parameters$phi0
   n_explain <- internal$parameters$n_explain
 
   # Appending the zero-prediction to the list
-  dt_vS0 <- as.data.table(rbind(c(1, rep(prediction_zero, n_explain))))
+  dt_vS0 <- as.data.table(rbind(c(1, rep(phi0, n_explain))))
 
   # Extracting/merging the data tables from the batch running
   # TODO: Need a memory and speed optimized way to transform the output form dt_vS_list to two different lists,
diff --git a/R/compute_vS.R b/R/compute_vS.R
index e6345e4a9..321a391d3 100644
--- a/R/compute_vS.R
+++ b/R/compute_vS.R
@@ -230,7 +230,6 @@ compute_MCint <- function(dt, pred_cols = "p_hat") {
 #' @export
 #' @keywords internal
 append_vS_list <- function(vS_list, internal) {
-
   iter <- length(internal$iter_list)
 
   # Adds v_S output above to any vS_list already computed
@@ -243,17 +242,17 @@ append_vS_list <- function(vS_list, internal) {
 
     # Creates a mapper from the last id_coalition to the new id_coalition numbering
     id_coalitions_mapper <- merge(prev_coalition_map,
-                                  current_coalition_map,
-                                  by = "coalitions_str",
-                                  suffixes = c("", "_new")
+      current_coalition_map,
+      by = "coalitions_str",
+      suffixes = c("", "_new")
     )
     prev_vS_list_new <- list()
 
     # Applies the mapper to update the prev_vS_list ot the new id_coalition numbering
     for (k in seq_along(prev_vS_list)) {
       prev_vS_list_new[[k]] <- merge(prev_vS_list[[k]],
-                                     id_coalitions_mapper[, .(id_coalition, id_coalition_new)],
-                                     by = "id_coalition"
+        id_coalitions_mapper[, .(id_coalition, id_coalition_new)],
+        by = "id_coalition"
       )
       prev_vS_list_new[[k]][, id_coalition := id_coalition_new]
       prev_vS_list_new[[k]][, id_coalition_new := NULL]
@@ -263,5 +262,4 @@ append_vS_list <- function(vS_list, internal) {
     vS_list <- c(prev_vS_list_new, vS_list)
   }
   return(vS_list)
-
 }
diff --git a/R/explain.R b/R/explain.R
index 2381eb2a8..caaaf0743 100644
--- a/R/explain.R
+++ b/R/explain.R
@@ -21,7 +21,7 @@
 #' `"categorical"`, `"timeseries"`, `"independence"`, `"regression_separate"`, or `"regression_surrogate"`.
 #' The two regression approaches can not be combined with any other approach. See details for more information.
 #'
-#' @param prediction_zero Numeric.
+#' @param phi0 Numeric.
 #' The prediction value for unseen data, i.e. an estimate of the expected prediction without conditioning on any
 #' features.
 #' Typically we set this value equal to the mean of the response variable in our training data, but other choices
@@ -211,7 +211,7 @@
 #' \describe{
 #'   \item{shapley_values_est}{data.table with the estimated Shapley values with explained observation in the rows and
 #'   features along the columns.
-#'   The column `none` is the prediction not devoted to any of the features (given by the argument `prediction_zero`)}
+#'   The column `none` is the prediction not devoted to any of the features (given by the argument `phi0`)}
 #'   \item{shapley_values_sd}{data.table with the standard deviation of the Shapley values reflecting the uncertainty.
 #'   Note that this only reflects the coalition sampling part of the kernelSHAP procedure, and is therefore by
 #'   definition 0 when all coalitions is used.
@@ -269,7 +269,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "empirical",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -279,7 +279,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "gaussian",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -289,7 +289,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "copula",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -299,7 +299,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "ctree",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -310,7 +310,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = approach,
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -332,7 +332,7 @@
 #'   x_train = x_train,
 #'   group = group_list,
 #'   approach = "empirical",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #' print(explain_groups$shapley_values_est)
@@ -347,7 +347,7 @@
 #'   model = model,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   approach = "regression_separate",
 #'   regression.model = parsnip::linear_reg()
 #' )
@@ -356,7 +356,7 @@
 #'   model = model,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   approach = "regression_surrogate",
 #'   regression.model = parsnip::linear_reg()
 #' )
@@ -370,7 +370,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "gaussian",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2,
 #'   iterative = TRUE,
 #'   iterative_args = list(initial_n_coalitions = 10)
@@ -395,7 +395,7 @@ explain <- function(model,
                     x_explain,
                     x_train,
                     approach,
-                    prediction_zero,
+                    phi0,
                     iterative = NULL,
                     max_n_coalitions = NULL,
                     group = NULL,
@@ -433,7 +433,7 @@ explain <- function(model,
     x_explain = x_explain,
     approach = approach,
     paired_shap_sampling = paired_shap_sampling,
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     max_n_coalitions = max_n_coalitions,
     group = group,
     n_MC_samples = n_MC_samples,
diff --git a/R/explain_forecast.R b/R/explain_forecast.R
index 36aaddbae..eeaff7ca3 100644
--- a/R/explain_forecast.R
+++ b/R/explain_forecast.R
@@ -79,7 +79,7 @@
 #'   explain_y_lags = 2,
 #'   horizon = 3,
 #'   approach = "empirical",
-#'   prediction_zero = p0_ar,
+#'   phi0 = p0_ar,
 #'   group_lags = FALSE
 #' )
 #'
@@ -93,7 +93,7 @@ explain_forecast <- function(model,
                              explain_xreg_lags = explain_y_lags,
                              horizon,
                              approach,
-                             prediction_zero,
+                             phi0,
                              max_n_coalitions = NULL,
                              iterative = NULL,
                              iterative_args = list(),
@@ -125,7 +125,7 @@ explain_forecast <- function(model,
   # Checks data/model compatability
   internal <- setup(
     approach = approach,
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     output_size = horizon,
     max_n_coalitions = max_n_coalitions,
     n_MC_samples = n_MC_samples,
diff --git a/R/plot.R b/R/plot.R
index 9bf19d2b7..a206d1951 100644
--- a/R/plot.R
+++ b/R/plot.R
@@ -101,7 +101,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "empirical",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -151,7 +151,7 @@
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "ctree",
-#'   prediction_zero = p,
+#'   phi0 = p,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -299,7 +299,7 @@ plot.shapr <- function(x,
     # compute start and end values for waterfall rectangles
     data.table::setorder(dt_plot, rank_waterfall)
     dt_plot[, end := cumsum(phi), by = id]
-    expected <- x$internal$parameters$prediction_zero
+    expected <- x$internal$parameters$phi0
     dt_plot[, start := c(expected, head(end, -1)), by = id]
     dt_plot[, phi_significant := format(phi, digits = digits), by = id]
 
@@ -895,7 +895,7 @@ make_waterfall_plot <- function(dt_plot,
 #' )
 #'
 #' # Specifying the phi_0, i.e. the expected prediction without any features
-#' prediction_zero <- mean(y_train)
+#' phi0 <- mean(y_train)
 #'
 #' # Independence approach
 #' explanation_independence <- explain(
@@ -903,7 +903,7 @@ make_waterfall_plot <- function(dt_plot,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "independence",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -913,7 +913,7 @@ make_waterfall_plot <- function(dt_plot,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "gaussian",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e1
 #' )
 #'
@@ -923,7 +923,7 @@ make_waterfall_plot <- function(dt_plot,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "gaussian",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -933,7 +933,7 @@ make_waterfall_plot <- function(dt_plot,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "ctree",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -943,7 +943,7 @@ make_waterfall_plot <- function(dt_plot,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = c("gaussian", "independence", "ctree"),
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -1452,7 +1452,7 @@ make_MSEv_coalition_plots <- function(MSEv_coalition_dt,
 #' )
 #'
 #' # Specifying the phi_0, i.e. the expected prediction without any features
-#' prediction_zero <- mean(y_train)
+#' phi0 <- mean(y_train)
 #'
 #' # Independence approach
 #' explanation_independence <- explain(
@@ -1460,7 +1460,7 @@ make_MSEv_coalition_plots <- function(MSEv_coalition_dt,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "independence",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -1470,7 +1470,7 @@ make_MSEv_coalition_plots <- function(MSEv_coalition_dt,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "empirical",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -1480,7 +1480,7 @@ make_MSEv_coalition_plots <- function(MSEv_coalition_dt,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "gaussian",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e1
 #' )
 #'
@@ -1490,7 +1490,7 @@ make_MSEv_coalition_plots <- function(MSEv_coalition_dt,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = "gaussian",
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
@@ -1500,7 +1500,7 @@ make_MSEv_coalition_plots <- function(MSEv_coalition_dt,
 #'   x_explain = x_explain,
 #'   x_train = x_train,
 #'   approach = c("gaussian", "ctree", "empirical"),
-#'   prediction_zero = prediction_zero,
+#'   phi0 = phi0,
 #'   n_MC_samples = 1e2
 #' )
 #'
diff --git a/R/setup.R b/R/setup.R
index be00abb95..904c7cdec 100644
--- a/R/setup.R
+++ b/R/setup.R
@@ -27,7 +27,7 @@ setup <- function(x_train,
                   x_explain,
                   approach,
                   paired_shap_sampling = TRUE,
-                  prediction_zero,
+                  phi0,
                   output_size = 1,
                   max_n_coalitions,
                   group,
@@ -76,7 +76,7 @@ setup <- function(x_train,
   internal$parameters <- get_parameters(
     approach = approach,
     paired_shap_sampling = paired_shap_sampling,
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     output_size = output_size,
     max_n_coalitions = max_n_coalitions,
     group = group,
@@ -163,7 +163,7 @@ get_prev_internal <- function(prev_shapr_object,
 #' @keywords internal
 get_parameters <- function(approach,
                            paired_shap_sampling,
-                           prediction_zero,
+                           phi0,
                            output_size = 1,
                            max_n_coalitions,
                            group,
@@ -290,12 +290,12 @@ get_parameters <- function(approach,
   if (!is.null(causal_ordering) && !is.list(causal_ordering)) stop("`causal_ordering` must be a list.\n")
 
   #### Tests combining more than one parameter ####
-  # prediction_zero vs output_size
-  if (!all((is.numeric(prediction_zero)) &&
-    all(length(prediction_zero) == output_size) &&
-    all(!is.na(prediction_zero)))) {
+  # phi0 vs output_size
+  if (!all((is.numeric(phi0)) &&
+    all(length(phi0) == output_size) &&
+    all(!is.na(phi0)))) {
     stop(paste0(
-      "`prediction_zero` (", paste0(prediction_zero, collapse = ", "),
+      "`phi0` (", paste0(phi0, collapse = ", "),
       ") must be numeric and match the output size of the model (",
       paste0(output_size, collapse = ", "), ")."
     ))
@@ -315,7 +315,7 @@ get_parameters <- function(approach,
   parameters <- list(
     approach = approach,
     paired_shap_sampling = paired_shap_sampling,
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     max_n_coalitions = max_n_coalitions,
     group = group,
     n_MC_samples = n_MC_samples,
diff --git a/README.Rmd b/README.Rmd
index d7511c54a..db17b1395 100644
--- a/README.Rmd
+++ b/README.Rmd
@@ -171,7 +171,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Printing the Shapley values for the test data.
diff --git a/README.md b/README.md
index 203526d60..d5b68201b 100644
--- a/README.md
+++ b/README.md
@@ -227,7 +227,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
diff --git a/inst/scripts/Beeswarm_illustration.R b/inst/scripts/Beeswarm_illustration.R
index 83fedf42f..72b61cce7 100644
--- a/inst/scripts/Beeswarm_illustration.R
+++ b/inst/scripts/Beeswarm_illustration.R
@@ -498,7 +498,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 10, # Do not need precise Shapley values to illustrate the behaviour of beeswarm plot
   n_MC_samples = 10 # Do not need precise Shapley values to illustrate the behaviour of beeswarm plot
 )
diff --git a/inst/scripts/Compare_Conditional_and_Causal_Categorical.R b/inst/scripts/Compare_Conditional_and_Causal_Categorical.R
index 502b16e47..f30efa475 100644
--- a/inst/scripts/Compare_Conditional_and_Causal_Categorical.R
+++ b/inst/scripts/Compare_Conditional_and_Causal_Categorical.R
@@ -33,7 +33,7 @@ causal_independence <- explain(
   x_explain = x_explain_categorical,
   x_train = x_train_categorical,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   asymmetric = FALSE,
   causal_ordering = list(3:4, 2, 1),
   confounding = c(TRUE, FALSE, FALSE),
@@ -47,7 +47,7 @@ causal_categorical <- explain(
   x_explain = x_explain_categorical,
   x_train = x_train_categorical,
   approach = "categorical",
-  prediction_zero = p0,
+  phi0 = p0,
   asymmetric = FALSE,
   causal_ordering = list(3:4, 2, 1),
   confounding = c(TRUE, FALSE, FALSE),
@@ -63,7 +63,7 @@ causal_ctree <- explain(
   x_explain = x_explain_categorical,
   x_train = x_train_categorical,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   asymmetric = FALSE,
   causal_ordering = list(3:4, 2, 1),
   confounding = c(TRUE, FALSE, FALSE),
@@ -79,7 +79,7 @@ causal_vaeac <- explain(
   x_train = x_train_categorical,
   approach = "vaeac",
   vaeac.epochs = 20,
-  prediction_zero = p0,
+  phi0 = p0,
   asymmetric = FALSE,
   causal_ordering = list(3:4, 2, 1),
   confounding = c(TRUE, FALSE, FALSE),
@@ -102,7 +102,7 @@ conditional_independence <- explain(
   x_explain = x_explain_categorical,
   x_train = x_train_categorical,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   # asymmetric = FALSE,
   # causal_ordering = list(3:4, 2, 1),
   # confounding = c(TRUE, FALSE, FALSE),
@@ -117,7 +117,7 @@ conditional_categorical <- explain(
   x_explain = x_explain_categorical,
   x_train = x_train_categorical,
   approach = "categorical",
-  prediction_zero = p0,
+  phi0 = p0,
   # asymmetric = FALSE,
   # causal_ordering = list(3:4, 2, 1),
   # confounding = c(TRUE, FALSE, FALSE),
@@ -133,7 +133,7 @@ conditional_ctree <- explain(
   x_explain = x_explain_categorical,
   x_train = x_train_categorical,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   # asymmetric = FALSE,
   # causal_ordering = list(3:4, 2, 1),
   # confounding = c(TRUE, FALSE, FALSE),
@@ -149,7 +149,7 @@ conditional_vaeac <- explain(
   x_train = x_train_categorical,
   approach = "vaeac",
   vaeac.epochs = 20,
-  prediction_zero = p0,
+  phi0 = p0,
   # asymmetric = FALSE,
   # causal_ordering = list(3:4, 2, 1),
   # confounding = c(TRUE, FALSE, FALSE),
diff --git a/inst/scripts/Compare_categorical_prepare_data.R b/inst/scripts/Compare_categorical_prepare_data.R
index c0c4ab340..dd913ee4d 100644
--- a/inst/scripts/Compare_categorical_prepare_data.R
+++ b/inst/scripts/Compare_categorical_prepare_data.R
@@ -452,7 +452,7 @@ explanation = explain(
   x_explain = x_explain_categorical,
   x_train = x_train_categorical,
   approach = "categorical",
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 1,
   timing = FALSE
 )
diff --git a/inst/scripts/Heskes_bike_rental_illustration.R b/inst/scripts/Heskes_bike_rental_illustration.R
index 9fb9d2722..e74e48ce9 100644
--- a/inst/scripts/Heskes_bike_rental_illustration.R
+++ b/inst/scripts/Heskes_bike_rental_illustration.R
@@ -51,7 +51,7 @@ save_plots <- FALSE
 #'   dat_A,
 #'   approach = "gaussian",
 #'   explainer = explainer,
-#'   prediction_zero = y_mean
+#'   phi0 = y_mean
 #' )
 #' sina_plot(explanation_classic)
 #'
@@ -59,7 +59,7 @@ save_plots <- FALSE
 #'   dat_A,
 #'   approach = "causal",
 #'   explainer = explainer,
-#'   prediction_zero = y_mean,
+#'   phi0 = y_mean,
 #'   ordering = list(1, c(2, 3))
 #' )
 #' sina_plot(explanation_causal)
@@ -195,7 +195,7 @@ model <- xgboost(
 )
 # caret::RMSE(y_explain, predict(model, x_explain))
 sqrt(mean((predict(model, x_explain) - y_explain)^2))
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 message("1. Prepared and plotted data, trained XGBoost model")
 
@@ -209,7 +209,7 @@ explanation_gaussian_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = FALSE,
         causal_ordering = list(1:7),
         confounding = FALSE,
@@ -236,7 +236,7 @@ explanation_causal_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = FALSE,
         causal_ordering = list(1, c(2, 3), c(4:7)),
         confounding = c(FALSE, TRUE, FALSE),
@@ -284,7 +284,7 @@ explanation_marginal_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "independence",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = FALSE,
         causal_ordering = list(1:7),
         confounding = FALSE,
@@ -331,7 +331,7 @@ explanation_asymmetric_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, c(2, 3), c(4:7)),
         confounding = FALSE,
@@ -376,7 +376,7 @@ explanation_asymmetric_causal_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, c(2, 3), c(4:7)),
         confounding = c(FALSE, TRUE, FALSE),
@@ -602,7 +602,7 @@ explanation_asymmetric_causal_gaussian_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, c(2, 3), c(4:7)),
         confounding = c(FALSE, TRUE, FALSE),
@@ -622,7 +622,7 @@ explanation_asymmetric_causal_copula_time = system.time({
       x_train = x_train,
       x_explain = x_explain,
       approach = "copula",
-      prediction_zero = prediction_zero,
+      phi0 = phi0,
       asymmetric = TRUE,
       causal_ordering = list(1, c(2, 3), c(4:7)),
       confounding = c(FALSE, TRUE, FALSE),
@@ -642,7 +642,7 @@ explanation_asymmetric_causal_ctree_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "ctree",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, c(2, 3), c(4:7)),
         confounding = c(FALSE, TRUE, FALSE),
@@ -663,7 +663,7 @@ explanation_asymmetric_causal_independence_time = system.time({
       x_train = x_train,
       x_explain = x_explain,
       approach = "independence",
-      prediction_zero = prediction_zero,
+      phi0 = phi0,
       asymmetric = TRUE,
       causal_ordering = list(1, c(2, 3), c(4:7)),
       confounding = c(FALSE, TRUE, FALSE),
@@ -683,7 +683,7 @@ explanation_asymmetric_causal_empirical_time = system.time({
       x_train = x_train,
       x_explain = x_explain,
       approach = "empirical",
-      prediction_zero = prediction_zero,
+      phi0 = phi0,
       asymmetric = TRUE,
       causal_ordering = list(1, c(2, 3), c(4:7)),
       confounding = c(FALSE, TRUE, FALSE),
@@ -703,7 +703,7 @@ explanation_asymmetric_causal_vaeac_time = system.time({
       x_train = x_train,
       x_explain = x_explain,
       approach = "vaeac",
-      prediction_zero = prediction_zero,
+      phi0 = phi0,
       asymmetric = TRUE,
       causal_ordering = list(1, c(2, 3), c(4:7)),
       confounding = c(FALSE, TRUE, FALSE),
@@ -744,7 +744,7 @@ explanation_asymmetric_all_gaussian2 <-
       x_train = x_train,
       x_explain = x_explain,
       approach = "gaussian",
-      prediction_zero = prediction_zero,
+      phi0 = phi0,
       asymmetric = TRUE,
       causal_ordering = list(1, c(2, 3), c(4:7)),
       confounding = FALSE,
@@ -773,7 +773,7 @@ explanation_asymmetric_gaussian <-
       x_train = x_train,
       x_explain = x_explain,
       approach = "gaussian",
-      prediction_zero = prediction_zero,
+      phi0 = phi0,
       asymmetric = TRUE,
       causal_ordering = list(1, c(2, 3), c(4:7)),
       confounding = FALSE,
@@ -800,7 +800,7 @@ explanation_causal_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, c(2, 3), c(4:7)),
         confounding = c(FALSE, TRUE, FALSE),
@@ -821,7 +821,7 @@ explanation_causal_time_sampled = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, c(2, 3), c(4:7)),
         confounding = c(FALSE, TRUE, FALSE),
@@ -869,7 +869,7 @@ explanation_group_asymmetric_causal_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, 2:3, 4:6),
         confounding = c(FALSE, TRUE, FALSE),
@@ -892,7 +892,7 @@ explanation_group_asymmetric_causal_sampled_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(1, 2:3, 4:6),
         confounding = confounding,
@@ -914,7 +914,7 @@ explanation_group_symmetric_causal_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = FALSE,
         causal_ordering = list(1, 2:3, 4:6), #FORTSETT HER MED Å ENDRE OG SE HVA SOM KRÆSJER
         confounding = confounding,
@@ -933,7 +933,7 @@ explanation_group_symmetric_causal_sampled_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = FALSE,
         causal_ordering = causal_ordering,
         confounding = confounding,
@@ -955,7 +955,7 @@ explanation_group_symmetric_conditional_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = FALSE,
         causal_ordering = NULL,
         confounding = FALSE,
@@ -974,7 +974,7 @@ explanation_group_symmetric_conditional_sampled_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = FALSE,
         causal_ordering = NULL,
         confounding = FALSE,
@@ -994,7 +994,7 @@ explanation_group_asymmetric_conditional_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = list(seq_along(group_list)),
         confounding = FALSE,
@@ -1014,7 +1014,7 @@ explanation_group_asymmetric_causal_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = causal_ordering,
         confounding = c(FALSE, TRUE, FALSE),
@@ -1039,7 +1039,7 @@ explanation_group_asymmetric_conditional_sampled_time = system.time({
         x_train = x_train,
         x_explain = x_explain,
         approach = "gaussian",
-        prediction_zero = prediction_zero,
+        phi0 = phi0,
         asymmetric = TRUE,
         causal_ordering = causal_ordering,
         confounding = FALSE,
diff --git a/inst/scripts/check_model_workflow.R b/inst/scripts/check_model_workflow.R
index ef76a36ff..296c090ae 100644
--- a/inst/scripts/check_model_workflow.R
+++ b/inst/scripts/check_model_workflow.R
@@ -50,7 +50,7 @@ explain_workflow = explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 4
 )
 
@@ -59,7 +59,7 @@ explain_xgboost = explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 4
 )
 
@@ -103,7 +103,7 @@ explain_decision_tree_ctree = explain(
   x_explain = x_explain_mixed,
   x_train = x_train_mixed,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 4
 )
 
@@ -113,7 +113,7 @@ explain_decision_tree_lm = explain(
   x_train = x_train_mixed,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 4
 )
 
@@ -149,7 +149,7 @@ explain_decision_model_rf_cv_rf = explain(
   x_train = x_train_mixed,
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression"),
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 4
 )
 
@@ -159,7 +159,7 @@ explain_decision_model_rf_cv_ctree = explain(
   x_explain = x_explain_mixed,
   x_train = x_train_mixed,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 4
 )
 
diff --git a/inst/scripts/compare_copula_in_R_and_C++.R b/inst/scripts/compare_copula_in_R_and_C++.R
index d0698dbf2..f3811c7fa 100644
--- a/inst/scripts/compare_copula_in_R_and_C++.R
+++ b/inst/scripts/compare_copula_in_R_and_C++.R
@@ -906,7 +906,7 @@ arma::cube prepare_data_copula_cpp_and_R(arma::mat MC_samples_mat,
   predictive_model <- lm(y ~ ., data = data_train_with_response)
 
   # Get the prediction zero, i.e., the phi0 Shapley value.
-  prediction_zero <- mean(response_train)
+  phi0 <- mean(response_train)
 
   model <- predictive_model
   x_explain <- data_test
@@ -925,7 +925,7 @@ arma::cube prepare_data_copula_cpp_and_R(arma::mat MC_samples_mat,
     x_train = x_train,
     x_explain = x_explain,
     approach = approach,
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_coalitions = n_coalitions,
     group = group,
     n_samples = n_samples,
@@ -1400,7 +1400,7 @@ all.equal(shapr_mat_arma_res, sourceCpp_mat_arma_res)
   predictive_model <- lm(y ~ ., data = data_train_with_response)
 
   # Get the prediction zero, i.e., the phi0 Shapley value.
-  prediction_zero <- mean(response_train)
+  phi0 <- mean(response_train)
 
   model <- predictive_model
   x_explain <- data_test
@@ -1419,7 +1419,7 @@ all.equal(shapr_mat_arma_res, sourceCpp_mat_arma_res)
     x_train = x_train,
     x_explain = x_explain,
     approach = approach,
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_coalitions = n_coalitions,
     group = group,
     n_samples = n_samples,
@@ -1511,7 +1511,7 @@ temp_shapley_value_func = function(dt, internal, model, predict_model) {
     xreg = internal$data$xreg
   )
   dt_vS2 <- compute_MCint(dt, paste0("p_hat", seq_len(internal$parameters$output_size)))
-  dt_vS <- rbind(t(as.matrix(c(1, rep(prediction_zero, n_test)))), dt_vS2, t(as.matrix(c(2^M, response_test))),
+  dt_vS <- rbind(t(as.matrix(c(1, rep(phi0, n_test)))), dt_vS2, t(as.matrix(c(2^M, response_test))),
                  use.names = FALSE)
   colnames(dt_vS) = colnames(dt_vS2)
   compute_shapley_new(internal, dt_vS)
diff --git a/inst/scripts/compare_gaussian_in_R_and_C++.R b/inst/scripts/compare_gaussian_in_R_and_C++.R
index 0106109a6..b358c9127 100644
--- a/inst/scripts/compare_gaussian_in_R_and_C++.R
+++ b/inst/scripts/compare_gaussian_in_R_and_C++.R
@@ -2289,7 +2289,7 @@ prepare_data_gaussian_new_v6 <- function(internal, index_features, ...) {
   predictive_model <- lm(y ~ ., data = data_train_with_response)
 
   # Get the prediction zero, i.e., the phi0 Shapley value.
-  prediction_zero <- mean(response_train)
+  phi0 <- mean(response_train)
 
   model <- predictive_model
   x_explain <- data_test
@@ -2308,7 +2308,7 @@ prepare_data_gaussian_new_v6 <- function(internal, index_features, ...) {
     x_train = x_train,
     x_explain = x_explain,
     approach = approach,
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_coalitions = n_coalitions,
     group = group,
     n_samples = n_samples,
diff --git a/inst/scripts/compare_shap_python.R b/inst/scripts/compare_shap_python.R
index 6a4ed7787..ebc39e2c3 100644
--- a/inst/scripts/compare_shap_python.R
+++ b/inst/scripts/compare_shap_python.R
@@ -47,12 +47,12 @@ time_R_prepare <- proc.time()
 
 # Computing the actual Shapley values with kernelSHAP accounting for feature dependence using
 # the empirical (conditional) distribution approach with bandwidth parameter sigma = 0.1 (default)
-explanation_independence <- explain(x_test, explainer, approach = "independence", prediction_zero = p0)
+explanation_independence <- explain(x_test, explainer, approach = "independence", phi0 = p0)
 
 time_R_indep0 <- proc.time()
 
 explanation_largesigma <- explain(x_test, explainer, approach = "empirical", type = "fixed_sigma",
-                                  fixed_sigma_vec = 10000, w_threshold = 1, prediction_zero = p0)
+                                  fixed_sigma_vec = 10000, w_threshold = 1, phi0 = p0)
 
 time_R_largesigma0 <- proc.time()
 
diff --git a/inst/scripts/compare_shap_python_new.R b/inst/scripts/compare_shap_python_new.R
index b8a0a2b33..5e51120f4 100644
--- a/inst/scripts/compare_shap_python_new.R
+++ b/inst/scripts/compare_shap_python_new.R
@@ -40,14 +40,14 @@ time_R_start <- proc.time()
 # Computing the actual Shapley values with kernelSHAP accounting for feature dependence using
 # the empirical (conditional) distribution approach with bandwidth parameter sigma = 0.1 (default)
 explanation_independence <- explain(model = model,x_explain = x_test,x_train=x_train,
-                                    approach = "independence", prediction_zero = p0,n_batches = 1)
+                                    approach = "independence", phi0 = p0,n_batches = 1)
 
 time_R_indep0 <- proc.time()
 
 
 explanation_largesigma <- explain(model = model,x_explain = x_test,x_train=x_train,
                                   approach = "empirical",empirical.type="fixed_sigma",empirical.fixed_sigma=10000,empirical.eta=1,
-                                  prediction_zero = p0,n_batches=1)
+                                  phi0 = p0,n_batches=1)
 
 
 time_R_largesigma0 <- proc.time()
diff --git a/inst/scripts/devel/Rscript_test_shapr.R b/inst/scripts/devel/Rscript_test_shapr.R
index 8f8b5a504..03380a6ed 100644
--- a/inst/scripts/devel/Rscript_test_shapr.R
+++ b/inst/scripts/devel/Rscript_test_shapr.R
@@ -62,7 +62,7 @@ sys_time_start_shapr <- Sys.time()
 explainer <- shapr(x_train, model)
 sys_time_end_shapr <- Sys.time()
 
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 n_batches_use <- min(nrow(explainer$S),n_batches)
 
@@ -73,7 +73,7 @@ explanation <- explain(
   x_test,
   approach = approach,
   explainer = explainer,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_batches = n_batches_use
 )
 sys_time_end_explain <- Sys.time()
diff --git a/inst/scripts/devel/compare_explain_batch.R b/inst/scripts/devel/compare_explain_batch.R
index 45473a747..48544bd80 100644
--- a/inst/scripts/devel/compare_explain_batch.R
+++ b/inst/scripts/devel/compare_explain_batch.R
@@ -25,13 +25,13 @@ model <- xgboost(
 library(shapr)
 explainer <- shapr(x_train, model,n_coalitions = 100)
 p = mean(y_train)
-gauss = explain(x_test, explainer, "gaussian", prediction_zero = p, n_samples = 10000)
-emp =  explain(x_test, explainer, "empirical", prediction_zero = p, n_samples = 10000)
-copula =  explain(x_test, explainer, "copula", prediction_zero = p, n_samples = 10000)
-indep = explain(x_test, explainer, "independence", prediction_zero = p, n_samples = 10000)
-comb = explain(x_test, explainer, c("gaussian", "gaussian", "empirical", "empirical"), prediction_zero = p, n_samples = 10000)
-ctree = explain(x_test, explainer, "ctree", mincriterion = 0.95, prediction_zero = p, n_samples = 10000)
-ctree2 = explain(x_test, explainer, "ctree", mincriterion = c(0.95, 0.95, 0.95, 0.95), prediction_zero = p, n_samples = 10000)
+gauss = explain(x_test, explainer, "gaussian", phi0 = p, n_samples = 10000)
+emp =  explain(x_test, explainer, "empirical", phi0 = p, n_samples = 10000)
+copula =  explain(x_test, explainer, "copula", phi0 = p, n_samples = 10000)
+indep = explain(x_test, explainer, "independence", phi0 = p, n_samples = 10000)
+comb = explain(x_test, explainer, c("gaussian", "gaussian", "empirical", "empirical"), phi0 = p, n_samples = 10000)
+ctree = explain(x_test, explainer, "ctree", mincriterion = 0.95, phi0 = p, n_samples = 10000)
+ctree2 = explain(x_test, explainer, "ctree", mincriterion = c(0.95, 0.95, 0.95, 0.95), phi0 = p, n_samples = 10000)
 #saveRDS(list(gauss = gauss, empirical = emp, copula = copula, indep = indep, comb = comb, ctree = ctree, ctree_comb = ctree2), file = "inst/scripts/devel/master_res2.rds")
 # saveRDS(list(ctree = ctree, ctree_comb = ctree2), file = "inst/scripts/devel/master_res_ctree.rds")
 
@@ -42,13 +42,13 @@ nobs = 6
 x_test <- as.matrix(Boston[1:nobs, x_var])
 explainer <- shapr(x_train, model,n_coalitions = 100)
 p = mean(y_train)
-gauss = explain(x_test, explainer, "gaussian", prediction_zero = p, n_samples = 10000, n_batches = 1)
-emp =  explain(x_test, explainer, "empirical", prediction_zero = p, n_samples = 10000, n_batches = 1)
-copula =  explain(x_test, explainer, "copula", prediction_zero = p, n_samples = 10000, n_batches = 1)
-indep = explain(x_test, explainer, "independence", prediction_zero = p, n_samples = 10000, n_batches = 1)
-comb = explain(x_test, explainer, c("gaussian", "gaussian", "empirical", "empirical"), prediction_zero = p, n_samples = 10000, n_batches = 1)
-ctree = explain(x_test, explainer, "ctree", mincriterion = 0.95, prediction_zero = p, n_samples = 10000, n_batches = 1)
-ctree2 = explain(x_test, explainer, "ctree", mincriterion = c(0.95, 0.95, 0.95, 0.95), prediction_zero = p, n_samples = 10000, n_batches = 1)
+gauss = explain(x_test, explainer, "gaussian", phi0 = p, n_samples = 10000, n_batches = 1)
+emp =  explain(x_test, explainer, "empirical", phi0 = p, n_samples = 10000, n_batches = 1)
+copula =  explain(x_test, explainer, "copula", phi0 = p, n_samples = 10000, n_batches = 1)
+indep = explain(x_test, explainer, "independence", phi0 = p, n_samples = 10000, n_batches = 1)
+comb = explain(x_test, explainer, c("gaussian", "gaussian", "empirical", "empirical"), phi0 = p, n_samples = 10000, n_batches = 1)
+ctree = explain(x_test, explainer, "ctree", mincriterion = 0.95, phi0 = p, n_samples = 10000, n_batches = 1)
+ctree2 = explain(x_test, explainer, "ctree", mincriterion = c(0.95, 0.95, 0.95, 0.95), phi0 = p, n_samples = 10000, n_batches = 1)
 
 res = readRDS("inst/scripts/devel/master_res2.rds")
 
@@ -60,8 +60,8 @@ res$comb$dt
 comb$dt
 
 # With batches
-gauss_b = explain(x_test, explainer, "gaussian", prediction_zero = p, n_samples = 10000, n_batches = 3)
-emp_b =  explain(x_test, explainer, "empirical", prediction_zero = p, n_samples = 10000, n_batches = 3)
+gauss_b = explain(x_test, explainer, "gaussian", phi0 = p, n_samples = 10000, n_batches = 3)
+emp_b =  explain(x_test, explainer, "empirical", phi0 = p, n_samples = 10000, n_batches = 3)
 
 gauss_b$dt
 res$gauss$dt
@@ -71,7 +71,7 @@ res$empirical$dt
 
 #### MJ stuff here:
 
-explain.independence2 <- function(x, explainer, approach, prediction_zero,
+explain.independence2 <- function(x, explainer, approach, phi0,
                                   n_samples = 1e3, n_batches = 1, seed = 1, only_return_contrib_dt = FALSE, ...) {
 
 
@@ -82,7 +82,7 @@ explain.independence2 <- function(x, explainer, approach, prediction_zero,
   explainer$approach <- approach
   explainer$n_samples <- n_samples
 
-  r <- prepare_and_predict(explainer, n_batches, prediction_zero, only_return_contrib_dt, ...)
+  r <- prepare_and_predict(explainer, n_batches, phi0, only_return_contrib_dt, ...)
 }
 
 
@@ -137,36 +137,36 @@ prepare_data.independence2 <- function(x, index_features = NULL, ...) {
 
 # Using independence with n_samples > nrow(x_train) such that no sampling is performed
 
-indep1 =  explain(x_test, explainer, "independence", prediction_zero = p, n_samples = 10000, n_batches = 1)
-indep2 =  explain(x_test, explainer, "independence2", prediction_zero = p, n_samples = 10000, n_batches = 1)
+indep1 =  explain(x_test, explainer, "independence", phi0 = p, n_samples = 10000, n_batches = 1)
+indep2 =  explain(x_test, explainer, "independence2", phi0 = p, n_samples = 10000, n_batches = 1)
 
 all.equal(indep1,indep2) # TRUE
 
-indep1_batch_2 = explain(x_test, explainer, "independence", prediction_zero = p, n_samples = 10000, n_batches = 2)
+indep1_batch_2 = explain(x_test, explainer, "independence", phi0 = p, n_samples = 10000, n_batches = 2)
 
 all.equal(indep1,indep1_batch_2) # TRUE
 
-indep1_batch_5 = explain(x_test, explainer, "independence", prediction_zero = p, n_samples = 10000, n_batches = 5)
+indep1_batch_5 = explain(x_test, explainer, "independence", phi0 = p, n_samples = 10000, n_batches = 5)
 
 all.equal(indep1,indep1_batch_5) # TRUE
 
-comb_indep_1_batch_1 = explain(x_test, explainer, c("independence", "independence", "independence", "independence"), prediction_zero = p, n_samples = 10000, n_batches = 1)
+comb_indep_1_batch_1 = explain(x_test, explainer, c("independence", "independence", "independence", "independence"), phi0 = p, n_samples = 10000, n_batches = 1)
 
 all.equal(indep1,comb_indep_1_batch_1) # TRUE
 
-comb_indep_1_batch_2 = explain(x_test, explainer, c("independence", "independence", "independence", "independence"), prediction_zero = p, n_samples = 10000, n_batches = 2)
+comb_indep_1_batch_2 = explain(x_test, explainer, c("independence", "independence", "independence", "independence"), phi0 = p, n_samples = 10000, n_batches = 2)
 
 all.equal(indep1,comb_indep_1_batch_2) # TRUE
 
-comb_indep_1_2_batch_1 = explain(x_test, explainer, c("independence", "independence", "independence2", "independence2"), prediction_zero = p, n_samples = 10000, n_batches = 1)
+comb_indep_1_2_batch_1 = explain(x_test, explainer, c("independence", "independence", "independence2", "independence2"), phi0 = p, n_samples = 10000, n_batches = 1)
 
 all.equal(indep1,comb_indep_1_2_batch_1) #TRUE
 
-comb_indep_1_2_batch_2 = explain(x_test, explainer, c("independence", "independence", "independence2", "independence2"), prediction_zero = p, n_samples = 10000, n_batches = 2)
+comb_indep_1_2_batch_2 = explain(x_test, explainer, c("independence", "independence", "independence2", "independence2"), phi0 = p, n_samples = 10000, n_batches = 2)
 
 all.equal(indep1,comb_indep_1_2_batch_2) #TRUE
 
-comb_indep_1_2_batch_5 = explain(x_test, explainer, c("independence", "independence", "independence2", "independence2"), prediction_zero = p, n_samples = 10000, n_batches = 5)
+comb_indep_1_2_batch_5 = explain(x_test, explainer, c("independence", "independence", "independence2", "independence2"), phi0 = p, n_samples = 10000, n_batches = 5)
 
 all.equal(indep1,comb_indep_1_2_batch_5) #TRUE
 
diff --git a/inst/scripts/devel/compare_indep_implementations.R b/inst/scripts/devel/compare_indep_implementations.R
index a508e2d1e..ae035b492 100644
--- a/inst/scripts/devel/compare_indep_implementations.R
+++ b/inst/scripts/devel/compare_indep_implementations.R
@@ -37,7 +37,7 @@ explanation_old <- explain(
   approach = "empirical",
   type = "independence",
   explainer = explainer,
-  prediction_zero = p, seed=111,n_samples = 100
+  phi0 = p, seed=111,n_samples = 100
 )
 print(proc.time()-t_old)
 #user  system elapsed
@@ -48,7 +48,7 @@ explanation_new <- explain(
   x_test,
   approach = "independence",
   explainer = explainer,
-  prediction_zero = p,seed = 111,n_samples = 100
+  phi0 = p,seed = 111,n_samples = 100
   )
 print(proc.time()-t_new)
 #user  system elapsed
@@ -69,7 +69,7 @@ explanation_full_old <- explain(
   approach = "empirical",
   type = "independence",
   explainer = explainer,
-  prediction_zero = p, seed=111
+  phi0 = p, seed=111
 )
 print(proc.time()-t_old)
 #user  system elapsed
@@ -80,7 +80,7 @@ explanation_full_new <- explain(
   x_test,
   approach = "independence",
   explainer = explainer,
-  prediction_zero = p,seed = 111
+  phi0 = p,seed = 111
 )
 print(proc.time()-t_new)
 #user  system elapsed
diff --git a/inst/scripts/devel/demonstrate_combined_approaches_bugs.R b/inst/scripts/devel/demonstrate_combined_approaches_bugs.R
index 57e5b9f44..bafa0bab3 100644
--- a/inst/scripts/devel/demonstrate_combined_approaches_bugs.R
+++ b/inst/scripts/devel/demonstrate_combined_approaches_bugs.R
@@ -10,7 +10,7 @@ explanation_1 = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("independence", "empirical", "gaussian", "copula", "empirical"),
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 3,
   timing = FALSE,
   seed = 1)
@@ -42,7 +42,7 @@ explanation_2 = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("independence", "ctree", "ctree", "ctree" ,"ctree"),
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 2,
   timing = FALSE,
   seed = 1)
@@ -62,7 +62,7 @@ explanation_3 = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("independence", "ctree", "ctree", "ctree" ,"ctree"),
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 15,
   timing = FALSE,
   seed = 1)
@@ -93,7 +93,7 @@ explanation_combined_1 = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("independence", "empirical", "gaussian", "copula", "empirical"),
-  prediction_zero = p0,
+  phi0 = p0,
   timing = FALSE,
   seed = 1)
 
@@ -102,7 +102,7 @@ explanation_combined_2 = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("independence", "empirical", "gaussian", "copula", "empirical"),
-  prediction_zero = p0,
+  phi0 = p0,
   timing = FALSE,
   seed = 1)
 
@@ -117,7 +117,7 @@ explanation_combined_3 = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("independence", "empirical", "gaussian", "copula", "ctree"),
-  prediction_zero = p0,
+  phi0 = p0,
   timing = FALSE,
   seed = 1)
 
@@ -126,7 +126,7 @@ explanation_combined_4 = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("independence", "empirical", "gaussian", "copula", "ctree"),
-  prediction_zero = p0,
+  phi0 = p0,
   timing = FALSE,
   seed = 1)
 
diff --git a/inst/scripts/devel/devel_batch_testing.R b/inst/scripts/devel/devel_batch_testing.R
index 8fcb2c807..20c1063f3 100644
--- a/inst/scripts/devel/devel_batch_testing.R
+++ b/inst/scripts/devel/devel_batch_testing.R
@@ -53,7 +53,7 @@ expl <- explain(model = model,
                 x_explain= x_explain,
                 x_train = x_train,
                 approach = "ctree",
-                prediction_zero = p0,
+                phi0 = p0,
                 n_batches = 100,
                 n_samples = 1000,
                 iterative = TRUE,
diff --git a/inst/scripts/devel/devel_convergence_branch.R b/inst/scripts/devel/devel_convergence_branch.R
index 9d022ba17..313a28698 100644
--- a/inst/scripts/devel/devel_convergence_branch.R
+++ b/inst/scripts/devel/devel_convergence_branch.R
@@ -43,7 +43,7 @@ explanation_iterative <- explain(
   x_train = x_train,
   approach = "gaussian",
   max_n_coalitions = 500,
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = TRUE,
   print_shapleyres = TRUE, # tmp
   print_iter_info = TRUE, # tmp
@@ -56,7 +56,7 @@ explanation_iterative <- explain(
   x_train = x_train,
   approach = "ctree",
   n_coalitions = 500,
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = TRUE,
   print_shapleyres = TRUE, # tmp
   print_iter_info = TRUE, # tmp
@@ -70,7 +70,7 @@ explanation_noniterative <- explain(
   x_train = x_train,
   approach = "gaussian",
   n_coalitions = 400,
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 
@@ -81,7 +81,7 @@ explanation_iterative <- explain(
   x_train = x_train,
   approach = "gaussian",
   n_coalitions = 500,
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = TRUE,
   iterative_args = list(initial_n_coalitions=10,convergence_tol=0.0001),
   print_shapleyres = TRUE, # tmp
@@ -142,7 +142,7 @@ explanation_regular <- explain(
   x_train = x_train,
   approach = "gaussian",
   n_coalitions = NULL,
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 
diff --git a/inst/scripts/devel/devel_non_exact_grouping.R b/inst/scripts/devel/devel_non_exact_grouping.R
index bb8cf5d3b..d5e29e3b0 100644
--- a/inst/scripts/devel/devel_non_exact_grouping.R
+++ b/inst/scripts/devel/devel_non_exact_grouping.R
@@ -38,14 +38,14 @@ explanation1 <- explain(
   x_test,
   approach = "independence",
   explainer = explainer1,
-  prediction_zero = p
+  phi0 = p
 )
 
 explanation2 <- explain(
   x_test,
   approach = "independence",
   explainer = explainer2,
-  prediction_zero = p
+  phi0 = p
 )
 
 
diff --git a/inst/scripts/devel/devel_parallelization.R b/inst/scripts/devel/devel_parallelization.R
index 6dd6d10bd..21aa964cc 100644
--- a/inst/scripts/devel/devel_parallelization.R
+++ b/inst/scripts/devel/devel_parallelization.R
@@ -35,7 +35,7 @@ explanation0 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 time0 <- stop-start
@@ -48,7 +48,7 @@ explanation1 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 time1 <- stop-start
@@ -60,7 +60,7 @@ explanation2 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 time2 <- stop-start
@@ -72,7 +72,7 @@ explanation3 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 time3 <- stop-start
@@ -84,7 +84,7 @@ explanation4 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 time4 <- stop-start
@@ -96,7 +96,7 @@ explanation5 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 time5 <- stop-start
@@ -108,7 +108,7 @@ explanation6 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 time6 <- stop-start
@@ -123,7 +123,7 @@ explanation7 <- explain(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,n_batches = 32
+  phi0 = p,n_batches = 32
 )
 stop <- proc.time()
 parallel::stopCluster(cl)
diff --git a/inst/scripts/devel/devel_tmp_new_batch.R b/inst/scripts/devel/devel_tmp_new_batch.R
index 290d5c009..37950b3a3 100644
--- a/inst/scripts/devel/devel_tmp_new_batch.R
+++ b/inst/scripts/devel/devel_tmp_new_batch.R
@@ -5,7 +5,7 @@ explainer <- explain_setup(
   x_test,
   approach = c("empirical","empirical","gaussian","copula"),
   explainer = explainer,
-  prediction_zero = p,
+  phi0 = p,
   n_batches = 4
 )
 
diff --git a/inst/scripts/devel/devel_verbose.R b/inst/scripts/devel/devel_verbose.R
index d9f0a54fc..ad4a2fb7d 100644
--- a/inst/scripts/devel/devel_verbose.R
+++ b/inst/scripts/devel/devel_verbose.R
@@ -4,7 +4,7 @@ ex <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative_args = list(
     initial_n_coalitions = 6,
@@ -21,7 +21,7 @@ ex <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "regression_separate",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative = TRUE,verbose=c("vS_details")
 )
@@ -30,7 +30,7 @@ ex <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "regression_separate",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative = TRUE,verbose=c("basic","progress","vS_details"),
   regression.model = parsnip::decision_tree(tree_depth = hardhat::tune(), engine = "rpart", mode = "regression"),
@@ -43,7 +43,7 @@ ex <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "regression_surrogate",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative = FALSE,verbose=c("basic","vS_details"),
   regression.model = parsnip::decision_tree(tree_depth = hardhat::tune(), engine = "rpart", mode = "regression"),
@@ -62,7 +62,7 @@ ex <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative = FALSE,verbose=c("basic","progress","vS_details"),
   n_MC_samples = 100,
@@ -75,7 +75,7 @@ ex2 <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative = FALSE,verbose=c("basic","progress","vS_details"),
   n_MC_samples = 100,
@@ -97,7 +97,7 @@ ex <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "regression_separate",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative = FALSE,verbose=c("basic")
 )
@@ -109,7 +109,7 @@ ex <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 30,
   iterative_args = list(
     initial_n_coalitions = 6,
@@ -127,7 +127,7 @@ explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = TRUE,
   iterative_args <- list(n_initial_)
   verbose = c("basic"),
diff --git a/inst/scripts/devel/explain_new.R b/inst/scripts/devel/explain_new.R
index b6a1e2af7..1e86d1276 100644
--- a/inst/scripts/devel/explain_new.R
+++ b/inst/scripts/devel/explain_new.R
@@ -39,7 +39,7 @@ explanation_new <- explain_new(
   x_test,
   approach = "gaussian",
   explainer = explainer1,
-  prediction_zero = p,
+  phi0 = p,
   n_samples = 5*10^5,n_batches = 1
 )
 
@@ -56,7 +56,7 @@ explanation_new <- explain_new(
   x_test,
   approach = "gaussian",
   explainer = explainer,
-  prediction_zero = p,
+  phi0 = p,
   n_samples = 10^5,n_batches = 4
 )
 
@@ -73,7 +73,7 @@ explanation_new <- explain_new(
   x_test,
   approach = "empirical",
   explainer = explainer,
-  prediction_zero = p,
+  phi0 = p,
   n_samples = 10^5,n_batches = 1
 )
 
@@ -90,7 +90,7 @@ explanation_new <- explain_new(
   x_test,
   approach = "empirical",
   explainer = explainer,
-  prediction_zero = p,
+  phi0 = p,
   n_samples = 10^5,n_batches = 4
 )
 
@@ -112,7 +112,7 @@ explanation_new$dt_shapley
 #   x_test,
 #   approach = "gaussian",
 #   explainer = explainer,
-#   prediction_zero = p
+#   phi0 = p
 # )
 #
 # str(explainer,max.level = 1)
@@ -122,7 +122,7 @@ explainer <- explain_setup(
    x_test,
    approach = "empirical",
    explainer = explainer,
-   prediction_zero = p,
+   phi0 = p,
    n_batches = 4
  )
 
@@ -130,7 +130,7 @@ explainer0 <- explain_setup(
   x_test,
   approach = c("empirical","copula","ctree","gaussian"),
   explainer = explainer,
-  prediction_zero = p,
+  phi0 = p,
   n_batches = 7
 )
 
@@ -149,7 +149,7 @@ explainer0$X
 #   x_test,
 #   approach = "gaussian",
 #   explainer = explainer,
-#   prediction_zero = p,
+#   phi0 = p,
 #   n_samples = 10^5
 # )
 
diff --git a/inst/scripts/devel/real_data_iterative_kernelshap.R b/inst/scripts/devel/real_data_iterative_kernelshap.R
index 9d6404f99..0e33ae141 100644
--- a/inst/scripts/devel/real_data_iterative_kernelshap.R
+++ b/inst/scripts/devel/real_data_iterative_kernelshap.R
@@ -107,7 +107,7 @@ expl <- shapr::explain(model = model,
                        x_explain= x_explain[inds,],
                        x_train = x_train,
                        approach = "ctree",
-                       prediction_zero = p0
+                       phi0 = p0
 )
 fwrite(expl$shapley_values_est,paste0(sim_results_saving_folder,"exact_shapley_values_", kernelSHAP_reweighting_strategy, ".csv"))
 print(Sys.time())
@@ -168,7 +168,7 @@ for (i in seq_along(testObs_computed_vec)){
                         x_explain= x_explain[testObs_computed_vec[i],],
                         x_train = x_train,
                         approach = approach,
-                        prediction_zero = p0,
+                        phi0 = p0,
                         n_coalitions = runcomps_list[[i]])
   expl_approx[i,] = unlist(expl_approx_obj$shapley_values_est)
   expl_approx_obj_list[[i]] <- expl_approx_obj
diff --git a/inst/scripts/devel/same_seed_as_master.R b/inst/scripts/devel/same_seed_as_master.R
index 977c770f7..4460b7e62 100644
--- a/inst/scripts/devel/same_seed_as_master.R
+++ b/inst/scripts/devel/same_seed_as_master.R
@@ -22,13 +22,13 @@ model <- xgboost(
 # Prepare the data for explanation
 explainer <- shapr(x_train, model,n_coalitions = 100)
 p = mean(y_train)
-gauss = explain(x_test, explainer, "gaussian", prediction_zero = p, n_samples = 10000)
-emp =  explain(x_test, explainer, "empirical", prediction_zero = p, n_samples = 10000)
-copula =  explain(x_test, explainer, "copula", prediction_zero = p, n_samples = 10000)
-indep = explain(x_test, explainer, "independence", prediction_zero = p, n_samples = 10000)
-comb = explain(x_test, explainer, c("gaussian", "gaussian", "empirical", "empirical"), prediction_zero = p, n_samples = 10000)
-ctree = explain(x_test, explainer, "ctree", mincriterion = 0.95, prediction_zero = p, n_samples = 10000)
-ctree2 = explain(x_test, explainer, "ctree", mincriterion = c(0.95, 0.95, 0.95, 0.95), prediction_zero = p, n_samples = 10000)
+gauss = explain(x_test, explainer, "gaussian", phi0 = p, n_samples = 10000)
+emp =  explain(x_test, explainer, "empirical", phi0 = p, n_samples = 10000)
+copula =  explain(x_test, explainer, "copula", phi0 = p, n_samples = 10000)
+indep = explain(x_test, explainer, "independence", phi0 = p, n_samples = 10000)
+comb = explain(x_test, explainer, c("gaussian", "gaussian", "empirical", "empirical"), phi0 = p, n_samples = 10000)
+ctree = explain(x_test, explainer, "ctree", mincriterion = 0.95, phi0 = p, n_samples = 10000)
+ctree2 = explain(x_test, explainer, "ctree", mincriterion = c(0.95, 0.95, 0.95, 0.95), phi0 = p, n_samples = 10000)
 
 
 # results from master
diff --git a/inst/scripts/devel/simtest_iterative_kernelshap_lingauss_v2.R b/inst/scripts/devel/simtest_iterative_kernelshap_lingauss_v2.R
index 1d88192c5..afbd2467f 100644
--- a/inst/scripts/devel/simtest_iterative_kernelshap_lingauss_v2.R
+++ b/inst/scripts/devel/simtest_iterative_kernelshap_lingauss_v2.R
@@ -80,7 +80,7 @@ expl <- shapr::explain(model = model,
                        x_explain= x_explain[inds,],
                        x_train = x_train,
                        approach = "gaussian",
-                       prediction_zero = p0,Sigma=Sigma,mu=mu)
+                       phi0 = p0,Sigma=Sigma,mu=mu)
 
 fwrite(expl$shapley_values_est,paste0(sim_results_saving_folder,"exact_shapley_values_",shapley_threshold_val,"_",kernelSHAP_reweighting_strategy, ".csv"))
 
@@ -149,7 +149,7 @@ for (i in testObs_computed_vec){
                                     x_explain= x_explain[inds[i],],
                                     x_train = x_train,
                                     approach = "gaussian",
-                                    prediction_zero = p0,
+                                    phi0 = p0,
                                     n_coalitions = runcomps_list[[i]],
                                     Sigma=Sigma,mu=mu)
   expl_approx[i,] = unlist(expl_approx_obj$shapley_values_est)
diff --git a/inst/scripts/devel/simtest_reweighting_strategies.R b/inst/scripts/devel/simtest_reweighting_strategies.R
index 3a5a24f0c..3f6a1e3df 100644
--- a/inst/scripts/devel/simtest_reweighting_strategies.R
+++ b/inst/scripts/devel/simtest_reweighting_strategies.R
@@ -61,7 +61,7 @@ expl <- shapr::explain(model = model,
                        x_train = x_train,
                        approach = "gaussian",
                        n_batches=100,n_samples = 10000,
-                       prediction_zero = p0,Sigma=Sigma,mu=mu)
+                       phi0 = p0,Sigma=Sigma,mu=mu)
 
 dt_vS_map <- merge(expl$internal$iter_list[[1]]$coalition_map,expl$internal$output$dt_vS,by="id_coalition")[,-"id_coalition"]
 
@@ -92,7 +92,7 @@ for(i0 in seq_along(paired_shap_sampling_vec)){
                              approach = "gaussian",
                              n_samples = 10, # Never used
                              n_batches=10,
-                             prediction_zero = p0,
+                             phi0 = p0,
                              Sigma=Sigma,
                              mu=mu,
                              seed = this_seed,
@@ -226,7 +226,7 @@ for (i in testObs_computed_vec){
                                     x_explain= x_explain[inds[i],],
                                     x_train = x_train,
                                     approach = "gaussian",
-                                    prediction_zero = p0,
+                                    phi0 = p0,
                                     n_combinations = runcomps_list[[i]],
                                     Sigma=Sigma,mu=mu)
   expl_approx[i,] = unlist(expl_approx_obj$shapley_values_est)
diff --git a/inst/scripts/devel/simtest_reweighting_strategies_nonlinear.R b/inst/scripts/devel/simtest_reweighting_strategies_nonlinear.R
index 0c1c8b521..c7ab347b7 100644
--- a/inst/scripts/devel/simtest_reweighting_strategies_nonlinear.R
+++ b/inst/scripts/devel/simtest_reweighting_strategies_nonlinear.R
@@ -78,7 +78,7 @@ expl <- shapr::explain(model = model,
                        x_train = x_train,
                        approach = "gaussian",
                        n_batches=100,n_samples = 10000,
-                       prediction_zero = p0,Sigma=Sigma,mu=mu)
+                       phi0 = p0,Sigma=Sigma,mu=mu)
 
 dt_vS_map <- merge(expl$internal$iter_list[[1]]$coalition_map,expl$internal$output$dt_vS,by="id_coalition")[,-"id_coalition"]
 
@@ -109,7 +109,7 @@ for(i0 in seq_along(paired_shap_sampling_vec)){
                              approach = "gaussian",
                              n_samples = 10, # Never used
                              n_batches=10,
-                             prediction_zero = p0,
+                             phi0 = p0,
                              Sigma=Sigma,
                              mu=mu,
                              seed = this_seed,
diff --git a/inst/scripts/devel/simtest_reweighting_strategies_nonlinear_nonunique_sampling.R b/inst/scripts/devel/simtest_reweighting_strategies_nonlinear_nonunique_sampling.R
index cc8341f92..84f9e71c4 100644
--- a/inst/scripts/devel/simtest_reweighting_strategies_nonlinear_nonunique_sampling.R
+++ b/inst/scripts/devel/simtest_reweighting_strategies_nonlinear_nonunique_sampling.R
@@ -78,7 +78,7 @@ expl <- shapr::explain(model = model,
                        x_train = x_train,
                        approach = "gaussian",
                        n_batches=100,n_samples = 10000,
-                       prediction_zero = p0,Sigma=Sigma,mu=mu)
+                       phi0 = p0,Sigma=Sigma,mu=mu)
 
 dt_vS_map <- merge(expl$internal$iter_list[[1]]$coalition_map,expl$internal$output$dt_vS,by="id_coalition")[,-"id_coalition"]
 
@@ -109,7 +109,7 @@ for(ii in seq_along(n_coalitions_vec)){
                              approach = "gaussian",
                              n_samples = 10, # Never used
                              n_batches=10,
-                             prediction_zero = p0,
+                             phi0 = p0,
                              Sigma=Sigma,
                              mu=mu,
                              seed = this_seed,
diff --git a/inst/scripts/devel/simtest_timing_to_Frida.R b/inst/scripts/devel/simtest_timing_to_Frida.R
index 93fc801bf..acc7e3e2a 100644
--- a/inst/scripts/devel/simtest_timing_to_Frida.R
+++ b/inst/scripts/devel/simtest_timing_to_Frida.R
@@ -81,7 +81,7 @@ expl <- explain(
   x_explain= x_explain[inds,],
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_coalitions = 100,
   Sigma=Sigma,
   mu=mu,
diff --git a/inst/scripts/devel/testing_explain_forevast_n_comb.R b/inst/scripts/devel/testing_explain_forevast_n_comb.R
index 43aeb6ced..03bea2181 100644
--- a/inst/scripts/devel/testing_explain_forevast_n_comb.R
+++ b/inst/scripts/devel/testing_explain_forevast_n_comb.R
@@ -9,7 +9,7 @@ h3test <- explain_forecast(model = model_arima_temp,
                                 explain_xreg_lags = 2,
                                 horizon = 3,
                                 approach = "empirical",
-                                prediction_zero = p0_ar[1:3],
+                                phi0 = p0_ar[1:3],
                                 group_lags = FALSE,
                                 n_batches = 1,
                                 timing = FALSE,
@@ -26,7 +26,7 @@ h2test <- explain_forecast(model = model_arima_temp,
                            explain_xreg_lags = 2,
                            horizon = 2,
                            approach = "empirical",
-                           prediction_zero = p0_ar[1:2],
+                           phi0 = p0_ar[1:2],
                            group_lags = FALSE,
                            n_batches = 1,
                            timing = FALSE,
@@ -43,7 +43,7 @@ h1test <- explain_forecast(model = model_arima_temp,
                            explain_xreg_lags = 2,
                            horizon = 1,
                            approach = "empirical",
-                           prediction_zero = p0_ar[1],
+                           phi0 = p0_ar[1],
                            group_lags = FALSE,
                            n_batches = 1,
                            timing = FALSE,
@@ -87,7 +87,7 @@ h3full <- explain_forecast(model = model_arima_temp,
                        explain_xreg_lags = 2,
                        horizon = 3,
                        approach = "empirical",
-                       prediction_zero = p0_ar[1:3],
+                       phi0 = p0_ar[1:3],
                        group_lags = FALSE,
                        n_batches = 1,
                        timing = FALSE,
@@ -103,7 +103,7 @@ h1full <- explain_forecast(model = model_arima_temp,
                            explain_xreg_lags = 2,
                            horizon = 1,
                            approach = "empirical",
-                           prediction_zero = p0_ar[1],
+                           phi0 = p0_ar[1],
                            group_lags = FALSE,
                            n_batches = 1,
                            timing = FALSE,
@@ -122,7 +122,7 @@ for (i in 1:reps){
                          explain_xreg_lags = 2,
                          horizon = 3,
                          approach = "empirical",
-                         prediction_zero = p0_ar[1:3],
+                         phi0 = p0_ar[1:3],
                          group_lags = FALSE,
                          n_batches = 1,
                          timing = FALSE,
@@ -139,7 +139,7 @@ for (i in 1:reps){
                          explain_xreg_lags = 2,
                          horizon = 2,
                          approach = "empirical",
-                         prediction_zero = p0_ar[1:2],
+                         phi0 = p0_ar[1:2],
                          group_lags = FALSE,
                          n_batches = 1,
                          timing = FALSE,
@@ -156,7 +156,7 @@ for (i in 1:reps){
                          explain_xreg_lags = 2,
                          horizon = 1,
                          approach = "empirical",
-                         prediction_zero = p0_ar[1],
+                         phi0 = p0_ar[1],
                          group_lags = FALSE,
                          n_batches = 1,
                          timing = FALSE,
diff --git a/inst/scripts/devel/testing_for_valid_defualt_n_batches.R b/inst/scripts/devel/testing_for_valid_defualt_n_batches.R
index a9d5739fd..a097fe73c 100644
--- a/inst/scripts/devel/testing_for_valid_defualt_n_batches.R
+++ b/inst/scripts/devel/testing_for_valid_defualt_n_batches.R
@@ -49,6 +49,6 @@ explanation <- explain(
   x_train = x_train,
   n_samples = 2, # Low value for fast computations
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_coalitions = any_number_equal_or_below_10
 )
diff --git a/inst/scripts/devel/testing_intermediate_saving.R b/inst/scripts/devel/testing_intermediate_saving.R
index 7214a2eac..85981c381 100644
--- a/inst/scripts/devel/testing_intermediate_saving.R
+++ b/inst/scripts/devel/testing_intermediate_saving.R
@@ -5,7 +5,7 @@ aa = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative_args = list(
     initial_n_coalitions = 10,
     convergence_tol = 0.01,
@@ -22,7 +22,7 @@ bb = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative_args = list(
     initial_n_coalitions = 10,
     convergence_tol = 0.001,
@@ -47,7 +47,7 @@ full = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative_args = list(
     initial_n_coalitions = 10,
     convergence_tol = 0.001,
@@ -67,7 +67,7 @@ first = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative_args = list(
     initial_n_coalitions = 10,
     convergence_tol = 0.001,
@@ -87,7 +87,7 @@ second = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative_args = list(
     initial_n_coalitions = 10,
     convergence_tol = 0.001,
@@ -110,7 +110,7 @@ second_path = explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative_args = list(
     initial_n_coalitions = 10,
     convergence_tol = 0.001,
diff --git a/inst/scripts/devel/testing_memory_monitoring.R b/inst/scripts/devel/testing_memory_monitoring.R
index 90af5c872..f161c3d2e 100644
--- a/inst/scripts/devel/testing_memory_monitoring.R
+++ b/inst/scripts/devel/testing_memory_monitoring.R
@@ -56,7 +56,7 @@ peakRAM(explain(
     x_test,
     approach = "gaussian",
     explainer = explainer,
-    prediction_zero = p,n_batches = 4)
+    phi0 = p,n_batches = 4)
     )
 
 # ,
@@ -64,28 +64,28 @@ peakRAM(explain(
 #       x_test,
 #       approach = "empirical",
 #       explainer = explainer,
-#       prediction_zero = p,n_batches = 2),
+#       phi0 = p,n_batches = 2),
 #     explain(
 #       x_test,
 #       approach = "empirical",
 #       explainer = explainer,
-#       prediction_zero = p,n_batches = 4))
+#       phi0 = p,n_batches = 4))
 
 #     explain(
 #       x_test,
 #       approach = "empirical",
 #       explainer = explainer,
-#       prediction_zero = p,n_batches = 8),
+#       phi0 = p,n_batches = 8),
 #     explain(
 #       x_test,
 #       approach = "empirical",
 #       explainer = explainer,
-#       prediction_zero = p,n_batches = 16),
+#       phi0 = p,n_batches = 16),
 #     explain(
 #       x_test,
 #       approach = "empirical",
 #       explainer = explainer,
-#       prediction_zero = p,n_batches = 32)
+#       phi0 = p,n_batches = 32)
 # )
 
 # s <- proc.time()
@@ -93,6 +93,6 @@ peakRAM(explain(
 #   x_test,
 #   approach = "empirical",
 #   explainer = explainer,
-#   prediction_zero = p,n_batches = 32)
+#   phi0 = p,n_batches = 32)
 # print(proc.time()-s)
 #
diff --git a/inst/scripts/devel/testing_n_cobinations_equal_2_power_m.R b/inst/scripts/devel/testing_n_cobinations_equal_2_power_m.R
index 980597b6f..ee8a01e3f 100644
--- a/inst/scripts/devel/testing_n_cobinations_equal_2_power_m.R
+++ b/inst/scripts/devel/testing_n_cobinations_equal_2_power_m.R
@@ -41,7 +41,7 @@ explanation_exact <- explain(
   n_samples = 2, # Low value for fast computations
   n_batches = 1, # Not related to the bug
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_coalitions = NULL
 )
 
@@ -53,7 +53,7 @@ explanation_should_also_be_exact <- explain(
   n_samples = 2, # Low value for fast computations
   n_batches = 1, # Not related to the bug
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_coalitions = 2^ncol(x_explain)
 )
 
diff --git a/inst/scripts/devel/testing_parallelization.R b/inst/scripts/devel/testing_parallelization.R
index 24cacc1a7..3f82541f2 100644
--- a/inst/scripts/devel/testing_parallelization.R
+++ b/inst/scripts/devel/testing_parallelization.R
@@ -78,7 +78,7 @@ for(i in seq_len(nrow(res_dt))){
       x_test,
       approach = approach_use,
       explainer = explainer,
-      prediction_zero = p,n_batches = n_batches_use
+      phi0 = p,n_batches = n_batches_use
     )},iterations = reps,time_unit ='s',memory = F,
     min_time = Inf
   )
diff --git a/inst/scripts/devel/testing_verification_ar_model.R b/inst/scripts/devel/testing_verification_ar_model.R
index 0271bb62f..6cf50f894 100644
--- a/inst/scripts/devel/testing_verification_ar_model.R
+++ b/inst/scripts/devel/testing_verification_ar_model.R
@@ -28,7 +28,7 @@ exp <- explain_forecast(model = model_arima_temp,
                        explain_xreg_lags = c(0,0),
                        horizon = 2,
                        approach = "empirical",
-                       prediction_zero = c(0,0),
+                       phi0 = c(0,0),
                        group_lags = FALSE,
                        n_batches = 1,
                        timing = FALSE,
diff --git a/inst/scripts/devel/time_series_annabelle.R b/inst/scripts/devel/time_series_annabelle.R
index 26e1f8b38..62fdffd7b 100644
--- a/inst/scripts/devel/time_series_annabelle.R
+++ b/inst/scripts/devel/time_series_annabelle.R
@@ -71,7 +71,7 @@ explanation_group <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "timeseries",
-  prediction_zero = p0,
+  phi0 = p0,
   group = group,
   timeseries.fixed_sigma_vec = 2
   # timeseries.bounds = c(-1, 2)
diff --git a/inst/scripts/devel/verifying_arima_model_output.R b/inst/scripts/devel/verifying_arima_model_output.R
index 4f27f3c40..47ce0641d 100644
--- a/inst/scripts/devel/verifying_arima_model_output.R
+++ b/inst/scripts/devel/verifying_arima_model_output.R
@@ -45,7 +45,7 @@ exp <- explain_forecast(model = model_arima_temp,
                         explain_xreg_lags = c(0,1),
                         horizon = 1,
                         approach = "empirical",
-                        prediction_zero = rep(mean(y),1),
+                        phi0 = rep(mean(y),1),
                         group_lags = FALSE,
                         n_batches = 1)
 
diff --git a/inst/scripts/devel/visual_bug_in_Shapley_bar_plot.R b/inst/scripts/devel/visual_bug_in_Shapley_bar_plot.R
index 0c57fe6c1..f9189e480 100644
--- a/inst/scripts/devel/visual_bug_in_Shapley_bar_plot.R
+++ b/inst/scripts/devel/visual_bug_in_Shapley_bar_plot.R
@@ -41,7 +41,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_samples = 10,
   keep_samp_for_vS = TRUE
 )
diff --git a/inst/scripts/empirical_memory_testing2.R b/inst/scripts/empirical_memory_testing2.R
index 678883b4e..84e1f863f 100644
--- a/inst/scripts/empirical_memory_testing2.R
+++ b/inst/scripts/empirical_memory_testing2.R
@@ -60,7 +60,7 @@ xy_train <- cbind(x_train,y=y_train)
 
 model <- lm(formula = y~.,data=xy_train)
 
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 n_batches_use <- min(2^p-2,n_batches)
 
@@ -71,7 +71,7 @@ explanation_many <- explain(
     x_train = x_train,
     approach = approach,
     n_batches = n_batches_use,
-    prediction_zero = prediction_zero
+    phi0 = phi0
   )
 
 
@@ -81,7 +81,7 @@ explanation_many <- explain(
 #  x_train = x_train,
 #  approach = approach,
 #  n_batches = 1,
-#  prediction_zero = prediction_zero
+#  phi0 = phi0
 #)
 
 
@@ -99,7 +99,7 @@ internal <- setup(
   x_train = x_train,
   x_explain = x_explain,
   approach = approach,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_coalitions = 2^p,
   group = NULL,
   n_samples = 1e3,
diff --git a/inst/scripts/example_annabelle.R b/inst/scripts/example_annabelle.R
index feede50bb..b2cad4031 100644
--- a/inst/scripts/example_annabelle.R
+++ b/inst/scripts/example_annabelle.R
@@ -46,7 +46,7 @@ temp = explain(
   x_explain = x_test,
   model = model,
   approach = "categorical",
-  prediction_zero = p,
+  phi0 = p,
   joint_probability_dt = joint_prob_dt
 )
 print(temp)
diff --git a/inst/scripts/example_ctree_method.R b/inst/scripts/example_ctree_method.R
index 6f0d26f12..6765a989c 100644
--- a/inst/scripts/example_ctree_method.R
+++ b/inst/scripts/example_ctree_method.R
@@ -33,7 +33,7 @@ p0 <- mean(y_train)
 # and sample = TRUE
 explanation <- explain(x_test, explainer,
                        approach = "ctree",
-                       prediction_zero = p0)
+                       phi0 = p0)
 
 # Printing the Shapley values for the test data
 explanation$dt
@@ -91,7 +91,7 @@ explanation_cat <- explain(
   dummylist$testdata_new,
   approach = "ctree",
   explainer = explainer_cat,
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Plot the resulting explanations for observations 1 and 6, excluding
diff --git a/inst/scripts/example_custom_model.R b/inst/scripts/example_custom_model.R
index 34a6377a4..c2a476a31 100644
--- a/inst/scripts/example_custom_model.R
+++ b/inst/scripts/example_custom_model.R
@@ -65,7 +65,7 @@ get_model_specs.gbm <- function(x){
 set.seed(123)
 explainer <- shapr(xy_train, model)
 p0 <- mean(xy_train[,y_var])
-explanation <- explain(x_test, explainer, approach = "empirical", prediction_zero = p0)
+explanation <- explain(x_test, explainer, approach = "empirical", phi0 = p0)
 # Plot results
 plot(explanation)
 
@@ -89,6 +89,6 @@ predict_model.gbm <- function(x, newdata) {
 set.seed(123)
 explainer <- shapr(x_train, model)
 p0 <- mean(xy_train[,y_var])
-explanation <- explain(x_test, explainer, approach = "empirical", prediction_zero = p0)
+explanation <- explain(x_test, explainer, approach = "empirical", phi0 = p0)
 # Plot results
 plot(explanation)
diff --git a/inst/scripts/example_plot_MSEv.R b/inst/scripts/example_plot_MSEv.R
index 51fd56ee2..725b1d896 100644
--- a/inst/scripts/example_plot_MSEv.R
+++ b/inst/scripts/example_plot_MSEv.R
@@ -29,7 +29,7 @@ model <- xgboost::xgboost(
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 # Independence approach
 explanation_independence <- explain(
@@ -37,7 +37,7 @@ explanation_independence <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -47,7 +47,7 @@ explanation_empirical <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -57,7 +57,7 @@ explanation_gaussian_1e1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e1
 )
 
@@ -67,7 +67,7 @@ explanation_gaussian_1e2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -77,7 +77,7 @@ explanation_ctree <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "ctree",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -87,7 +87,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("gaussian", "independence", "ctree"),
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -290,7 +290,7 @@ explanation_gaussian_seed_1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10,
   n_coalitions = 10,
   seed = 1
@@ -301,7 +301,7 @@ explanation_gaussian_seed_1_V2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10,
   n_coalitions = 10,
   seed = 1
@@ -312,7 +312,7 @@ explanation_gaussian_seed_2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10,
   n_coalitions = 10,
   seed = 2
@@ -323,7 +323,7 @@ explanation_gaussian_seed_3 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10,
   n_coalitions = 10,
   seed = 3
@@ -350,7 +350,7 @@ explanation_gaussian_all <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10
 )
 
@@ -359,7 +359,7 @@ explanation_gaussian_only_5 <- explain(
   x_explain = x_explain[1:5, ],
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10
 )
 
@@ -376,7 +376,7 @@ explanation_gaussian <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10
 )
 
@@ -397,7 +397,7 @@ explanation_gaussian <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 10
 )
 
diff --git a/inst/scripts/example_plot_SV_several_approaches.R b/inst/scripts/example_plot_SV_several_approaches.R
index a25c66b36..564e4c133 100644
--- a/inst/scripts/example_plot_SV_several_approaches.R
+++ b/inst/scripts/example_plot_SV_several_approaches.R
@@ -27,7 +27,7 @@ model = xgboost::xgboost(
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero = mean(y_train)
+phi0 = mean(y_train)
 
 # Independence approach
 explanation_independence = explain(
@@ -35,7 +35,7 @@ explanation_independence = explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -45,7 +45,7 @@ explanation_empirical = explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -55,7 +55,7 @@ explanation_gaussian_1e1 = explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e1
 )
 
@@ -65,7 +65,7 @@ explanation_gaussian_1e2 = explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
@@ -75,7 +75,7 @@ explanation_combined = explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("gaussian", "ctree", "empirical"),
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_samples = 1e2
 )
 
diff --git a/inst/scripts/example_plot_several_vaeacs_VLB_IWAE.R b/inst/scripts/example_plot_several_vaeacs_VLB_IWAE.R
index a364a9ce4..85e9e3914 100644
--- a/inst/scripts/example_plot_several_vaeacs_VLB_IWAE.R
+++ b/inst/scripts/example_plot_several_vaeacs_VLB_IWAE.R
@@ -28,7 +28,7 @@ explanation_paired_sampling_TRUE <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 2,
   n_samples = 1, #' As we are only interested in the training of the vaeac
   vaeac.epochs = 25, #' Should be higher in applications.
@@ -44,7 +44,7 @@ explanation_paired_sampling_FALSE <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 2,
   n_samples = 1, #' As we are only interested in the training of the vaeac
   vaeac.epochs = 25, #' Should be higher in applications.
@@ -61,7 +61,7 @@ explanation_paired_sampling_FALSE_small <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 2,
   n_samples = 1, #' As we are only interested in the training of the vaeac
   vaeac.epochs = 25, #' Should be higher in applications.
@@ -80,7 +80,7 @@ explanation_paired_sampling_TRUE_small <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p0,
+  phi0 = p0,
   n_batches = 2,
   n_samples = 1, #' As we are only interested in the training of the vaeac
   vaeac.epochs = 25, #' Should be higher in applications.
diff --git a/inst/scripts/explain_memory_testing.R b/inst/scripts/explain_memory_testing.R
index 7c3030ffc..d9e35e7eb 100644
--- a/inst/scripts/explain_memory_testing.R
+++ b/inst/scripts/explain_memory_testing.R
@@ -60,7 +60,7 @@ xy_train <- cbind(x_train,y=y_train)
 
 model <- lm(formula = y~.,data=xy_train)
 
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 n_batches_use <- min(2^p-2,n_batches)
 
@@ -74,7 +74,7 @@ explanation <- explain(
   x_train = x_train,
   approach = approach,
   n_batches = n_batches_use,
-  prediction_zero = prediction_zero
+  phi0 = phi0
 )
 },threshold=10^4)
 
diff --git a/inst/scripts/problematic_plots_jens.R b/inst/scripts/problematic_plots_jens.R
index 2aa26c896..176af6a9f 100644
--- a/inst/scripts/problematic_plots_jens.R
+++ b/inst/scripts/problematic_plots_jens.R
@@ -41,7 +41,7 @@ explanation_cat <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 
@@ -62,7 +62,7 @@ explanation_cat <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Works fine
@@ -85,7 +85,7 @@ explanation_cat <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Only 4 ticks in the x-axis for the factor
@@ -107,7 +107,7 @@ explanation_cat <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Duplicated labels on the x-axis
diff --git a/inst/scripts/readme_example.R b/inst/scripts/readme_example.R
index b9e8852a9..9d63bc1a1 100644
--- a/inst/scripts/readme_example.R
+++ b/inst/scripts/readme_example.R
@@ -34,7 +34,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Printing the Shapley values for the test data.
diff --git a/inst/scripts/testing_samling_ncombinations.R b/inst/scripts/testing_samling_ncombinations.R
index 1d4fdb36f..d11220a4f 100644
--- a/inst/scripts/testing_samling_ncombinations.R
+++ b/inst/scripts/testing_samling_ncombinations.R
@@ -25,7 +25,7 @@ for (i in seq_along(n)) {
       x_test,
       model = model,
       approach = "empirical",
-      prediction_zero = p_mean,
+      phi0 = p_mean,
       n_coalitions = n_coalitions[i]
     )
   )
@@ -52,7 +52,7 @@ for (i in seq_along(n)) {
       x_test,
       model = model,
       approach = "empirical",
-      prediction_zero = p_mean,
+      phi0 = p_mean,
       n_coalitions = n_coalitions[i]
     )
   )
@@ -79,7 +79,7 @@ system.time({res = explain(
   x_test,
   model = model,
   approach = "empirical",
-  prediction_zero = p_mean,
+  phi0 = p_mean,
   n_coalitions = 1000
 )})
 
@@ -89,7 +89,7 @@ system.time({res2 = explain(
   x_test,
   model = model,
   approach = "empirical",
-  prediction_zero = p_mean,
+  phi0 = p_mean,
   n_coalitions = 800
 )})
 
@@ -100,7 +100,7 @@ system.time({res3 = explain(
   x_test,
   model = model,
   approach = "empirical",
-  prediction_zero = p_mean,
+  phi0 = p_mean,
   n_coalitions = NULL
 )})
 
@@ -117,7 +117,7 @@ res = profvis({res = explain(
   x_test,
   model = model,
   approach = "empirical",
-  prediction_zero = p_mean,
+  phi0 = p_mean,
   n_coalitions = n_coalitions[i]
 )})
 res
diff --git a/inst/scripts/time_series_annabelle.R b/inst/scripts/time_series_annabelle.R
index 26e1f8b38..62fdffd7b 100644
--- a/inst/scripts/time_series_annabelle.R
+++ b/inst/scripts/time_series_annabelle.R
@@ -71,7 +71,7 @@ explanation_group <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "timeseries",
-  prediction_zero = p0,
+  phi0 = p0,
   group = group,
   timeseries.fixed_sigma_vec = 2
   # timeseries.bounds = c(-1, 2)
diff --git a/inst/scripts/timing_script_2023.R b/inst/scripts/timing_script_2023.R
index 71a25c1c7..31c258d98 100644
--- a/inst/scripts/timing_script_2023.R
+++ b/inst/scripts/timing_script_2023.R
@@ -59,7 +59,7 @@ xy_train <- cbind(x_train,y=y_train)
 
 model <- lm(formula = y~.,data=xy_train)
 
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 n_batches_use <- min(2^p-2,n_batches)
 
@@ -72,7 +72,7 @@ explanation <- explain(
   x_train = x_train,
   approach = approach,
   n_batches = n_batches_use,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_coalitions = 10^4
 )
 
diff --git a/inst/scripts/vilde/airquality_example.R b/inst/scripts/vilde/airquality_example.R
index 9c162bfe2..59d2e225a 100644
--- a/inst/scripts/vilde/airquality_example.R
+++ b/inst/scripts/vilde/airquality_example.R
@@ -15,7 +15,7 @@ x <- explain(
   test,
   model = model,
   approach = "empirical",
-  prediction_zero = p
+  phi0 = p
 )
 
 if (requireNamespace("ggplot2", quietly = TRUE)) {
diff --git a/inst/scripts/vilde/check_progress.R b/inst/scripts/vilde/check_progress.R
index aee0f765c..ec3da4887 100644
--- a/inst/scripts/vilde/check_progress.R
+++ b/inst/scripts/vilde/check_progress.R
@@ -25,34 +25,34 @@ p <- mean(y_train)
 plan(multisession, workers=3)
 
 # when we simply call explain(), no progress bar is shown
-x <- explain(x_train, x_test, model, approach="gaussian", prediction_zero=p, n_batches = 4)
+x <- explain(x_train, x_test, model, approach="gaussian", phi0=p, n_batches = 4)
 
 # the handler specifies what kind of progress bar is shown
 # Wrapping explain() in with_progress() gives a progress bar when calling explain()
 handlers("txtprogressbar")
 x <- with_progress(
-  explain(x_train, x_test, model, approach="empirical", prediction_zero=p, n_batches = 5)
+  explain(x_train, x_test, model, approach="empirical", phi0=p, n_batches = 5)
   )
 
 # with global=TRUE the progress bar is displayed whenever the explain-function is called, and there is no need to use with_progress()
 handlers(global = TRUE)
-x <- explain(x_train, x_test, model, approach="gaussian", prediction_zero=p, n_batches = 4)
+x <- explain(x_train, x_test, model, approach="gaussian", phi0=p, n_batches = 4)
 
 # there are different options for what kind of progress bar should be displayed
 handlers("txtprogressbar") #this is the default
-x <- explain(x_train, x_test, model, approach="independence", prediction_zero=p, n_batches = 4)
+x <- explain(x_train, x_test, model, approach="independence", phi0=p, n_batches = 4)
 
 handlers("progress")
-x <- explain(x_train, x_test, model, approach="independence", prediction_zero=p, n_batches = 4)
+x <- explain(x_train, x_test, model, approach="independence", phi0=p, n_batches = 4)
 
 # you can edit the symbol used to draw completed progress in the progress bar (as well as other features) with handler_progress()
 handlers(handler_progress(complete = "#"))
-x <- explain(x_train, x_test, model, approach="copula", prediction_zero=p, n_batches = 4)
+x <- explain(x_train, x_test, model, approach="copula", phi0=p, n_batches = 4)
 
 plan("sequential")
 
 handlers("progress")
-x <- explain(x_train, x_test, model, approach=c(rep("ctree",4),"independence","independence"), prediction_zero=p, n_batches = 4)
+x <- explain(x_train, x_test, model, approach=c(rep("ctree",4),"independence","independence"), phi0=p, n_batches = 4)
 
 
 
diff --git a/inst/scripts/vilde/sketch_for_waterfall_plot.R b/inst/scripts/vilde/sketch_for_waterfall_plot.R
index f8572fb55..e31971a1a 100644
--- a/inst/scripts/vilde/sketch_for_waterfall_plot.R
+++ b/inst/scripts/vilde/sketch_for_waterfall_plot.R
@@ -25,7 +25,7 @@ model <- xgboost(
 p <- mean(y_train)
 
 # Prepare the data for explanation
-res <- explain_final(x_train,x_test,model,approach="independence",prediction_zero=p,n_batches = 4)
+res <- explain_final(x_train,x_test,model,approach="independence",phi0=p,n_batches = 4)
 plot(res)
 
 i<- 1 # index for observation we want to plot
diff --git a/inst/scripts/vilde/waterfall_plot.R b/inst/scripts/vilde/waterfall_plot.R
index 531f1e4c1..5035d2528 100644
--- a/inst/scripts/vilde/waterfall_plot.R
+++ b/inst/scripts/vilde/waterfall_plot.R
@@ -19,7 +19,7 @@ model <- xgboost(
   verbose = FALSE
 )
 p <- mean(y_train)
-x <- explain_final(x_train,x_test,model,approach="independence",prediction_zero=p,n_batches = 4)
+x <- explain_final(x_train,x_test,model,approach="independence",phi0=p,n_batches = 4)
 plot.shapr(x,
            plot_type = "bar",
            digits = 3,
diff --git a/man/append_vS_list.Rd b/man/append_vS_list.Rd
new file mode 100644
index 000000000..ceb1db088
--- /dev/null
+++ b/man/append_vS_list.Rd
@@ -0,0 +1,19 @@
+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/compute_vS.R
+\name{append_vS_list}
+\alias{append_vS_list}
+\title{Appends the new vS_list to the prev vS_list}
+\usage{
+append_vS_list(vS_list, internal)
+}
+\arguments{
+\item{vS_list}{List
+Output from \code{\link[=compute_vS]{compute_vS()}}}
+
+\item{internal}{List.
+Not used directly, but passed through from \code{\link[=explain]{explain()}}.}
+}
+\description{
+Appends the new vS_list to the prev vS_list
+}
+\keyword{internal}
diff --git a/man/cli_iter.Rd b/man/cli_iter.Rd
index 7d3481899..6426af8c9 100644
--- a/man/cli_iter.Rd
+++ b/man/cli_iter.Rd
@@ -26,6 +26,9 @@ E.g. \code{verbose = c("basic", "vS_details")} will display basic information +
 
 \item{internal}{List.
 Not used directly, but passed through from \code{\link[=explain]{explain()}}.}
+
+\item{iter}{Integer.
+The iteration number. Only used internally.}
 }
 \description{
 Printing messages in iterative procedure with cli
diff --git a/man/default_doc_explain.Rd b/man/default_doc_explain.Rd
index 9fcc39a30..af0bc7014 100644
--- a/man/default_doc_explain.Rd
+++ b/man/default_doc_explain.Rd
@@ -12,6 +12,9 @@ Not used directly, but passed through from \code{\link[=explain]{explain()}}.}
 
 \item{index_features}{Positive integer vector. Specifies the id_coalition to
 apply to the present method. \code{NULL} means all coalitions. Only used internally.}
+
+\item{iter}{Integer.
+The iteration number. Only used internally.}
 }
 \description{
 Exported documentation helper function.
diff --git a/man/explain.Rd b/man/explain.Rd
index b1e3e2454..f7fd8694e 100644
--- a/man/explain.Rd
+++ b/man/explain.Rd
@@ -9,7 +9,7 @@ explain(
   x_explain,
   x_train,
   approach,
-  prediction_zero,
+  phi0,
   iterative = NULL,
   max_n_coalitions = NULL,
   group = NULL,
@@ -49,7 +49,7 @@ All elements should, either be \code{"gaussian"}, \code{"copula"}, \code{"empiri
 \code{"categorical"}, \code{"timeseries"}, \code{"independence"}, \code{"regression_separate"}, or \code{"regression_surrogate"}.
 The two regression approaches can not be combined with any other approach. See details for more information.}
 
-\item{prediction_zero}{Numeric.
+\item{phi0}{Numeric.
 The prediction value for unseen data, i.e. an estimate of the expected prediction without conditioning on any
 features.
 Typically we set this value equal to the mean of the response variable in our training data, but other choices
@@ -330,7 +330,7 @@ Object of class \code{c("shapr", "list")}. Contains the following items:
 \describe{
 \item{shapley_values_est}{data.table with the estimated Shapley values with explained observation in the rows and
 features along the columns.
-The column \code{none} is the prediction not devoted to any of the features (given by the argument \code{prediction_zero})}
+The column \code{none} is the prediction not devoted to any of the features (given by the argument \code{phi0})}
 \item{shapley_values_sd}{data.table with the standard deviation of the Shapley values reflecting the uncertainty.
 Note that this only reflects the coalition sampling part of the kernelSHAP procedure, and is therefore by
 definition 0 when all coalitions is used.
@@ -415,7 +415,7 @@ explain1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -425,7 +425,7 @@ explain2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -435,7 +435,7 @@ explain3 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "copula",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -445,7 +445,7 @@ explain4 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "ctree",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -456,7 +456,7 @@ explain5 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -478,7 +478,7 @@ explain_groups <- explain(
   x_train = x_train,
   group = group_list,
   approach = "empirical",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 print(explain_groups$shapley_values_est)
@@ -493,7 +493,7 @@ explain_separate_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p,
+  phi0 = p,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg()
 )
@@ -502,7 +502,7 @@ explain_surrogate_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p,
+  phi0 = p,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg()
 )
@@ -516,7 +516,7 @@ explain_iterative <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2,
   iterative = TRUE,
   iterative_args = list(initial_n_coalitions = 10)
diff --git a/man/explain_forecast.Rd b/man/explain_forecast.Rd
index db66aba7a..df2d3176c 100644
--- a/man/explain_forecast.Rd
+++ b/man/explain_forecast.Rd
@@ -14,7 +14,7 @@ explain_forecast(
   explain_xreg_lags = explain_y_lags,
   horizon,
   approach,
-  prediction_zero,
+  phi0,
   max_n_coalitions = NULL,
   iterative = NULL,
   iterative_args = list(),
@@ -70,7 +70,7 @@ All elements should, either be \code{"gaussian"}, \code{"copula"}, \code{"empiri
 \code{"categorical"}, \code{"timeseries"}, \code{"independence"}, \code{"regression_separate"}, or \code{"regression_surrogate"}.
 The two regression approaches can not be combined with any other approach. See details for more information.}
 
-\item{prediction_zero}{Numeric.
+\item{phi0}{Numeric.
 The prediction value for unseen data, i.e. an estimate of the expected prediction without conditioning on any
 features.
 Typically we set this value equal to the mean of the response variable in our training data, but other choices
@@ -262,7 +262,7 @@ Object of class \code{c("shapr", "list")}. Contains the following items:
 \describe{
 \item{shapley_values_est}{data.table with the estimated Shapley values with explained observation in the rows and
 features along the columns.
-The column \code{none} is the prediction not devoted to any of the features (given by the argument \code{prediction_zero})}
+The column \code{none} is the prediction not devoted to any of the features (given by the argument \code{phi0})}
 \item{shapley_values_sd}{data.table with the standard deviation of the Shapley values reflecting the uncertainty.
 Note that this only reflects the coalition sampling part of the kernelSHAP procedure, and is therefore by
 definition 0 when all coalitions is used.
@@ -316,7 +316,7 @@ explain_forecast(
   explain_y_lags = 2,
   horizon = 3,
   approach = "empirical",
-  prediction_zero = p0_ar,
+  phi0 = p0_ar,
   group_lags = FALSE
 )
 
diff --git a/man/finalize_explanation_forecast.Rd b/man/finalize_explanation_forecast.Rd
index 5ed40b0f4..6911de4a9 100644
--- a/man/finalize_explanation_forecast.Rd
+++ b/man/finalize_explanation_forecast.Rd
@@ -20,7 +20,7 @@ Object of class \code{c("shapr", "list")}. Contains the following items:
 \describe{
 \item{shapley_values_est}{data.table with the estimated Shapley values with explained observation in the rows and
 features along the columns.
-The column \code{none} is the prediction not devoted to any of the features (given by the argument \code{prediction_zero})}
+The column \code{none} is the prediction not devoted to any of the features (given by the argument \code{phi0})}
 \item{shapley_values_sd}{data.table with the standard deviation of the Shapley values reflecting the uncertainty.
 Note that this only reflects the coalition sampling part of the kernelSHAP procedure, and is therefore by
 definition 0 when all coalitions is used.
@@ -105,7 +105,7 @@ explain1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -115,7 +115,7 @@ explain2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -125,7 +125,7 @@ explain3 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "copula",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -135,7 +135,7 @@ explain4 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "ctree",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -146,7 +146,7 @@ explain5 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -168,7 +168,7 @@ explain_groups <- explain(
   x_train = x_train,
   group = group_list,
   approach = "empirical",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 print(explain_groups$shapley_values_est)
@@ -183,7 +183,7 @@ explain_separate_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p,
+  phi0 = p,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg()
 )
@@ -192,7 +192,7 @@ explain_surrogate_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p,
+  phi0 = p,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg()
 )
@@ -206,7 +206,7 @@ explain_iterative <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2,
   iterative = TRUE,
   iterative_args = list(initial_n_coalitions = 10)
diff --git a/man/plot.shapr.Rd b/man/plot.shapr.Rd
index 77df20f5a..e2e856402 100644
--- a/man/plot.shapr.Rd
+++ b/man/plot.shapr.Rd
@@ -134,7 +134,7 @@ x <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
@@ -184,7 +184,7 @@ x <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "ctree",
-  prediction_zero = p,
+  phi0 = p,
   n_MC_samples = 1e2
 )
 
diff --git a/man/plot_MSEv_eval_crit.Rd b/man/plot_MSEv_eval_crit.Rd
index cd7792070..c7d569fee 100644
--- a/man/plot_MSEv_eval_crit.Rd
+++ b/man/plot_MSEv_eval_crit.Rd
@@ -90,7 +90,7 @@ model <- xgboost::xgboost(
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 # Independence approach
 explanation_independence <- explain(
@@ -98,7 +98,7 @@ explanation_independence <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
@@ -108,7 +108,7 @@ explanation_gaussian_1e1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e1
 )
 
@@ -118,7 +118,7 @@ explanation_gaussian_1e2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
@@ -128,7 +128,7 @@ explanation_ctree <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "ctree",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
@@ -138,7 +138,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("gaussian", "independence", "ctree"),
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
diff --git a/man/plot_SV_several_approaches.Rd b/man/plot_SV_several_approaches.Rd
index cc0c046ce..2fcfd1111 100644
--- a/man/plot_SV_several_approaches.Rd
+++ b/man/plot_SV_several_approaches.Rd
@@ -114,7 +114,7 @@ model <- xgboost::xgboost(
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 # Independence approach
 explanation_independence <- explain(
@@ -122,7 +122,7 @@ explanation_independence <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
@@ -132,7 +132,7 @@ explanation_empirical <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
@@ -142,7 +142,7 @@ explanation_gaussian_1e1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e1
 )
 
@@ -152,7 +152,7 @@ explanation_gaussian_1e2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
@@ -162,7 +162,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("gaussian", "ctree", "empirical"),
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1e2
 )
 
diff --git a/man/setup.Rd b/man/setup.Rd
index 28b96bfb4..dec833a20 100644
--- a/man/setup.Rd
+++ b/man/setup.Rd
@@ -9,7 +9,7 @@ setup(
   x_explain,
   approach,
   paired_shap_sampling = TRUE,
-  prediction_zero,
+  phi0,
   output_size = 1,
   max_n_coalitions,
   group,
@@ -59,7 +59,7 @@ If \code{TRUE} (default), paired versions of all sampled coalitions are also inc
 That is, if there are 5 features and e.g. coalitions (1,3,5) are sampled, then also coalition (2,4) is used for
 computing the Shapley values. This is done to reduce the variance of the Shapley value estimates.}
 
-\item{prediction_zero}{Numeric.
+\item{phi0}{Numeric.
 The prediction value for unseen data, i.e. an estimate of the expected prediction without conditioning on any
 features.
 Typically we set this value equal to the mean of the response variable in our training data, but other choices
diff --git a/man/vaeac_plot_eval_crit.Rd b/man/vaeac_plot_eval_crit.Rd
index c7086b853..fc8e2865b 100644
--- a/man/vaeac_plot_eval_crit.Rd
+++ b/man/vaeac_plot_eval_crit.Rd
@@ -79,7 +79,7 @@ explanation_paired <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1, # As we are only interested in the training of the vaeac
   vaeac.epochs = 10, # Should be higher in applications.
   vaeac.n_vaeacs_initialize = 1,
@@ -93,7 +93,7 @@ explanation_regular <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = approach,
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1, # As we are only interested in the training of the vaeac
   vaeac.epochs = 10, # Should be higher in applications.
   vaeac.width = 16,
diff --git a/man/vaeac_plot_imputed_ggpairs.Rd b/man/vaeac_plot_imputed_ggpairs.Rd
index f95abe5b9..6b4b1a75b 100644
--- a/man/vaeac_plot_imputed_ggpairs.Rd
+++ b/man/vaeac_plot_imputed_ggpairs.Rd
@@ -108,7 +108,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = mean(y_train),
+  phi0 = mean(y_train),
   n_MC_samples = 1,
   vaeac.epochs = 10,
   vaeac.n_vaeacs_initialize = 1
diff --git a/python/README.md b/python/README.md
index b010fec77..512ce3c39 100644
--- a/python/README.md
+++ b/python/README.md
@@ -51,7 +51,7 @@ df_shapley, pred_explain, internal, timing = explain(
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'empirical',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
 )
 print(df_shapley)
 ```
diff --git a/python/examples/devel_new_explain.py b/python/examples/devel_new_explain.py
index 04088aae5..cfa4e5d52 100644
--- a/python/examples/devel_new_explain.py
+++ b/python/examples/devel_new_explain.py
@@ -28,7 +28,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'gaussian',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
     max_n_coalitions=30
 )
 
@@ -55,7 +55,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'gaussian',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
     max_n_coalitions=100,
     iterative = False
 )
@@ -65,7 +65,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = ['gaussian', 'empirical',"gaussian","empirical","gaussian","gaussian","empirical"],
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
     max_n_coalitions=100,
     iterative = True,
     verbose = ["basic", "progress"]
@@ -76,7 +76,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'vaeac',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
     max_n_coalitions=100,
     iterative = False,
     verbose = ["basic", "progress","vS_details","shapley"]
@@ -88,7 +88,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_separate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     regression_model='parsnip::linear_reg()'
 )
 
diff --git a/python/examples/keras_classifier.py b/python/examples/keras_classifier.py
index d7b31e70f..60138165f 100644
--- a/python/examples/keras_classifier.py
+++ b/python/examples/keras_classifier.py
@@ -30,7 +30,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'empirical',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
 )
 print(df_shapley)
 
@@ -57,4 +57,4 @@
 """
 	MSEv	MSEv_sd
 1	0.000312	0.00014
-"""
\ No newline at end of file
+"""
diff --git a/python/examples/pytorch_custom.py b/python/examples/pytorch_custom.py
index eac345337..d58fa5337 100644
--- a/python/examples/pytorch_custom.py
+++ b/python/examples/pytorch_custom.py
@@ -42,7 +42,7 @@ def forward(self, x):
     x_explain = dfx_test,
     approach = 'empirical',
     predict_model = lambda m, x: m(torch.from_numpy(x.values).float()).cpu().detach().numpy(),
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
 )
 print(df_shapley)
 """
@@ -65,4 +65,4 @@ def forward(self, x):
 """
 MSEv	MSEv_sd
 1	27.046126	7.253933
-"""
\ No newline at end of file
+"""
diff --git a/python/examples/regression_paradigm.py b/python/examples/regression_paradigm.py
index c5daab4c4..bf53b77fe 100644
--- a/python/examples/regression_paradigm.py
+++ b/python/examples/regression_paradigm.py
@@ -27,7 +27,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='empirical',
-    prediction_zero=dfy_train.mean().item()
+    phi0=dfy_train.mean().item()
 )
 
 # Explain the model using several separate regression methods
@@ -37,7 +37,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_separate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model='parsnip::linear_reg()'
@@ -49,7 +49,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_separate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model='parsnip::linear_reg()',
@@ -64,7 +64,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_separate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model='parsnip::linear_reg()',
@@ -79,7 +79,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_separate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model="parsnip::decision_tree(tree_depth = hardhat::tune(), engine = 'rpart', mode = 'regression')",
@@ -93,7 +93,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_separate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model="parsnip::boost_tree(engine = 'xgboost', mode = 'regression')"
@@ -105,7 +105,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_separate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model="parsnip::boost_tree(trees = hardhat::tune(), engine = 'xgboost', mode = 'regression')",
@@ -120,7 +120,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_surrogate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model='parsnip::linear_reg()'
@@ -132,7 +132,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_surrogate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model="parsnip::rand_forest(engine = 'ranger', mode = 'regression')"
@@ -144,7 +144,7 @@
     x_train=dfx_train,
     x_explain=dfx_test,
     approach='regression_surrogate',
-    prediction_zero=dfy_train.mean().item(),
+    phi0=dfy_train.mean().item(),
     verbose=2,
     n_batches=1,
     regression_model="""parsnip::rand_forest(
@@ -191,4 +191,4 @@
 3  0.276002   0.957242  
 4  0.028560   0.049815  
 5 -0.242943   0.006815  
-"""
\ No newline at end of file
+"""
diff --git a/python/examples/sklearn_classifier.py b/python/examples/sklearn_classifier.py
index 418f88016..14e7dc263 100644
--- a/python/examples/sklearn_classifier.py
+++ b/python/examples/sklearn_classifier.py
@@ -14,7 +14,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'empirical',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
 )
 print(df_shapley)
 
diff --git a/python/examples/sklearn_regressor.py b/python/examples/sklearn_regressor.py
index 6f7d59067..3c7e87ac0 100644
--- a/python/examples/sklearn_regressor.py
+++ b/python/examples/sklearn_regressor.py
@@ -14,7 +14,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'empirical',
-    prediction_zero = dfy_train.mean().item()
+    phi0 = dfy_train.mean().item()
 )
 print(df_shapley)
 
@@ -51,7 +51,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'empirical',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
     group = group
 )
 print(df_shapley_g)
diff --git a/python/examples/xgboost_booster.py b/python/examples/xgboost_booster.py
index b89044344..d000ea06b 100644
--- a/python/examples/xgboost_booster.py
+++ b/python/examples/xgboost_booster.py
@@ -14,7 +14,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'empirical',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
 )
 print(df_shapley)
 
diff --git a/python/examples/xgboost_regressor.py b/python/examples/xgboost_regressor.py
index 7183a2dd8..da9a36389 100644
--- a/python/examples/xgboost_regressor.py
+++ b/python/examples/xgboost_regressor.py
@@ -14,7 +14,7 @@
     x_train = dfx_train,
     x_explain = dfx_test,
     approach = 'empirical',
-    prediction_zero = dfy_train.mean().item(),
+    phi0 = dfy_train.mean().item(),
 )
 print(df_shapley)
 
diff --git a/python/shaprpy/explain.py b/python/shaprpy/explain.py
index b3404e30d..be2c517c1 100644
--- a/python/shaprpy/explain.py
+++ b/python/shaprpy/explain.py
@@ -24,7 +24,7 @@ def explain(
     x_explain: pd.DataFrame,
     x_train: pd.DataFrame,
     approach: str,
-    prediction_zero: float,
+    phi0: float,
     iterative: bool | None = None,
     max_n_coalitions: int | None = None,
     group: dict | None = None,
@@ -64,7 +64,7 @@ def explain(
       The method(s) to estimate the conditional expectation. All elements should,
       either be `"gaussian"`, `"copula"`, `"empirical"`, `"ctree"`, `"categorical"`, `"timeseries"`, `"independence"`, 
       `"regression_separate"`, or `"regression_surrogate"`.
-    prediction_zero: float
+    phi0: float
       The prediction value for unseen data, i.e. an estimate of the expected prediction without conditioning on any
       features. Typically we set this value equal to the mean of the response variable in our training data, but other
       choices such as the mean of the predictions in the training data are also reasonable.
@@ -181,7 +181,7 @@ def explain(
       x_explain = py2r(x_explain),
       approach = StrVector(approach),
       paired_shap_sampling = paired_shap_sampling,
-      prediction_zero = prediction_zero,
+      phi0 = phi0,
       max_n_coalitions = maybe_null(max_n_coalitions),
       group = r_group,
       n_MC_samples = n_MC_samples,
diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_gaussian_group_converges_tol.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_gaussian_group_converges_tol.rds
index 7394eaf89..ed6d05c26 100644
Binary files a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_gaussian_group_converges_tol.rds and b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_gaussian_group_converges_tol.rds differ
diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_indep_conv_max_n_coalitions.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_indep_conv_max_n_coalitions.rds
index 76286933e..0a0f7379e 100644
Binary files a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_indep_conv_max_n_coalitions.rds and b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_indep_conv_max_n_coalitions.rds differ
diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_object.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_object.rds
index 96aab6ef9..0507b05cd 100644
Binary files a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_object.rds and b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_object.rds differ
diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_path.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_path.rds
index 96aab6ef9..0507b05cd 100644
Binary files a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_path.rds and b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_cont_est_path.rds differ
diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_converges_maxit.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_converges_maxit.rds
index de01585c9..752207bf6 100644
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diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_converges_tol.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_converges_tol.rds
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diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_converges_tol_paired.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_converges_tol_paired.rds
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diff --git a/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_reach_exact.rds b/tests/testthat/_snaps/adaptive-output/output_lm_numeric_independence_reach_exact.rds
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diff --git a/tests/testthat/_snaps/adaptive-output/output_verbose_1.rds b/tests/testthat/_snaps/adaptive-output/output_verbose_1.rds
index 0697297cc..876bd1a66 100644
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diff --git a/tests/testthat/_snaps/adaptive-output/output_verbose_1_3.rds b/tests/testthat/_snaps/adaptive-output/output_verbose_1_3.rds
index 6011838ef..fd125a547 100644
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diff --git a/tests/testthat/_snaps/adaptive-output/output_verbose_1_3_4.rds b/tests/testthat/_snaps/adaptive-output/output_verbose_1_3_4.rds
index 434f78f8f..583867cac 100644
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diff --git a/tests/testthat/_snaps/adaptive-output/output_verbose_1_3_4_5.rds b/tests/testthat/_snaps/adaptive-output/output_verbose_1_3_4_5.rds
index b7ebe066f..f0b55a8cf 100644
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diff --git a/tests/testthat/_snaps/adaptive-setup.md b/tests/testthat/_snaps/adaptive-setup.md
index 1b331af40..326a03d44 100644
--- a/tests/testthat/_snaps/adaptive-setup.md
+++ b/tests/testthat/_snaps/adaptive-setup.md
@@ -3,7 +3,7 @@
     Code
       n_batches_non_numeric_1 <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_numeric_1))
     Message
       Success with message:
@@ -19,7 +19,7 @@
     Code
       n_batches_non_numeric_2 <- TRUE
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_numeric_2))
     Message
       Success with message:
@@ -35,7 +35,7 @@
     Code
       n_batches_non_integer <- 10.5
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_integer))
     Message
       Success with message:
@@ -51,7 +51,7 @@
     Code
       n_batches_too_long <- c(1, 2)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_too_long))
     Message
       Success with message:
@@ -67,7 +67,7 @@
     Code
       n_batches_is_NA <- as.numeric(NA)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_is_NA))
     Message
       Success with message:
@@ -83,7 +83,7 @@
     Code
       n_batches_non_positive <- 0
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_positive))
     Message
       Success with message:
diff --git a/tests/testthat/_snaps/asymmetric-causal-output/output_asym_caus_conf_FALSE.rds b/tests/testthat/_snaps/asymmetric-causal-output/output_asym_caus_conf_FALSE.rds
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diff --git a/tests/testthat/_snaps/asymmetric-causal-output/output_sym_caus_conf_mix_group_iterative.rds b/tests/testthat/_snaps/asymmetric-causal-output/output_sym_caus_conf_mix_group_iterative.rds
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diff --git a/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_conditional.rds b/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_conditional.rds
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diff --git a/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_marginal_gaussian.rds b/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_marginal_gaussian.rds
index 8bff4fb6e..39833ddba 100644
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diff --git a/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_marginal_independence.rds b/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_marginal_independence.rds
index 0fe9f4ca4..4f2cb1c49 100644
Binary files a/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_marginal_independence.rds and b/tests/testthat/_snaps/asymmetric-causal-output/output_symmetric_marginal_independence.rds differ
diff --git a/tests/testthat/_snaps/asymmetric-causal-setup.md b/tests/testthat/_snaps/asymmetric-causal-setup.md
index c5468f86b..984aa7b47 100644
--- a/tests/testthat/_snaps/asymmetric-causal-setup.md
+++ b/tests/testthat/_snaps/asymmetric-causal-setup.md
@@ -2,9 +2,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list(1:6), confounding = NULL, approach = "gaussian",
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          1:6), confounding = NULL, approach = "gaussian", iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all feature names or indices exactly once.
@@ -13,9 +12,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list(1:5, 5), confounding = NULL, approach = "gaussian",
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          1:5, 5), confounding = NULL, approach = "gaussian", iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all feature names or indices exactly once.
@@ -24,9 +22,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list(2:5, 5), confounding = NULL, approach = "gaussian",
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          2:5, 5), confounding = NULL, approach = "gaussian", iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all feature names or indices exactly once.
@@ -35,9 +32,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list(1:2, 4), confounding = NULL, approach = "gaussian",
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          1:2, 4), confounding = NULL, approach = "gaussian", iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all feature names or indices exactly once.
@@ -46,10 +42,9 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list("Solar.R", "Wind", "Temp", "Month", "Day",
-          "Invalid feature name"), confounding = NULL, approach = "gaussian",
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          "Solar.R", "Wind", "Temp", "Month", "Day", "Invalid feature name"),
+        confounding = NULL, approach = "gaussian", iterative = FALSE)
     Condition
       Error in `convert_feature_name_to_idx()`:
       ! `causal_ordering` contains feature names (`Invalid feature name`) that are not in the data (`Solar.R`, `Wind`, `Temp`, `Month`, `Day`).
@@ -58,9 +53,9 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list("Solar.R", "Wind", "Temp", "Month", "Day", "Day"),
-        confounding = NULL, approach = "gaussian", iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          "Solar.R", "Wind", "Temp", "Month", "Day", "Day"), confounding = NULL,
+        approach = "gaussian", iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all feature names or indices exactly once.
@@ -69,9 +64,9 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list("Solar.R", "Wind", "Temp", "Day", "Day"), confounding = NULL,
-        approach = "gaussian", iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          "Solar.R", "Wind", "Temp", "Day", "Day"), confounding = NULL, approach = "gaussian",
+        iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all feature names or indices exactly once.
@@ -80,9 +75,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list("Solar.R", "Wind"), confounding = NULL, approach = "gaussian",
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          "Solar.R", "Wind"), confounding = NULL, approach = "gaussian", iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all feature names or indices exactly once.
@@ -91,10 +85,10 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list(c("Solar.R", "Wind", "Temp", "Month"), "Day"),
-        confounding = NULL, approach = "gaussian", group = list(A = c("Solar.R",
-          "Wind"), B = "Temp", C = c("Month", "Day")), iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          c("Solar.R", "Wind", "Temp", "Month"), "Day"), confounding = NULL,
+        approach = "gaussian", group = list(A = c("Solar.R", "Wind"), B = "Temp", C = c(
+          "Month", "Day")), iterative = FALSE)
     Condition
       Error in `convert_feature_name_to_idx()`:
       ! `causal_ordering` contains group names (`Solar.R`, `Wind`, `Temp`, `Month`, `Day`) that are not in the data (`A`, `B`, `C`).
@@ -103,10 +97,10 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list(c("A", "C"), "Wrong name"), confounding = NULL,
-        approach = "gaussian", group = list(A = c("Solar.R", "Wind"), B = "Temp", C = c(
-          "Month", "Day")), iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          c("A", "C"), "Wrong name"), confounding = NULL, approach = "gaussian",
+        group = list(A = c("Solar.R", "Wind"), B = "Temp", C = c("Month", "Day")),
+        iterative = FALSE)
     Condition
       Error in `convert_feature_name_to_idx()`:
       ! `causal_ordering` contains group names (`Wrong name`) that are not in the data (`A`, `B`, `C`).
@@ -115,10 +109,9 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = TRUE,
-        causal_ordering = list(c("A"), "B"), confounding = NULL, approach = "gaussian",
-        group = list(A = c("Solar.R", "Wind"), B = "Temp", C = c("Month", "Day")),
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = TRUE, causal_ordering = list(
+          c("A"), "B"), confounding = NULL, approach = "gaussian", group = list(A = c(
+          "Solar.R", "Wind"), B = "Temp", C = c("Month", "Day")), iterative = FALSE)
     Condition
       Error in `check_and_set_causal_ordering()`:
       ! `causal_ordering` is incomplete/incorrect. It must contain all group names or indices exactly once.
@@ -127,9 +120,9 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = FALSE,
-        causal_ordering = list(1:2, 3:4, 5), confounding = TRUE, approach = c(
-          "gaussian", "independence", "empirical", "gaussian"), iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = FALSE, causal_ordering = list(
+          1:2, 3:4, 5), confounding = TRUE, approach = c("gaussian", "independence",
+          "empirical", "gaussian"), iterative = FALSE)
     Condition
       Error in `check_and_set_causal_sampling()`:
       ! Causal Shapley values is not applicable for combined approaches.
@@ -138,7 +131,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = c(FALSE, FALSE),
+        x_train = x_train_numeric, phi0 = p0, asymmetric = c(FALSE, FALSE),
         causal_ordering = list(1:2, 3:4, 5), confounding = TRUE, approach = "gaussian",
         iterative = FALSE)
     Condition
@@ -149,7 +142,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = "Must be a single logical",
+        x_train = x_train_numeric, phi0 = p0, asymmetric = "Must be a single logical",
         causal_ordering = list(1:2, 3:4, 5), confounding = TRUE, approach = "gaussian",
         iterative = FALSE)
     Condition
@@ -160,9 +153,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = 1L,
-        causal_ordering = list(1:2, 3:4, 5), confounding = TRUE, approach = "gaussian",
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = 1L, causal_ordering = list(
+          1:2, 3:4, 5), confounding = TRUE, approach = "gaussian", iterative = FALSE)
     Condition
       Error in `get_parameters()`:
       ! `asymmetric` must be a single logical.
@@ -171,9 +163,9 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = FALSE,
-        causal_ordering = list(1:2, 3:4, 5), confounding = c("A", "B", "C"),
-        approach = "gaussian", iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, asymmetric = FALSE, causal_ordering = list(
+          1:2, 3:4, 5), confounding = c("A", "B", "C"), approach = "gaussian",
+        iterative = FALSE)
     Condition
       Error in `get_parameters()`:
       ! `confounding` must be a logical (vector).
@@ -182,8 +174,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, asymmetric = FALSE,
-        causal_ordering = list(1:2, 3:4, 5), confounding = c(TRUE, FALSE), approach = "gaussian",
+        x_train = x_train_numeric, phi0 = p0, asymmetric = FALSE, causal_ordering = list(
+          1:2, 3:4, 5), confounding = c(TRUE, FALSE), approach = "gaussian",
         iterative = FALSE)
     Condition
       Error in `check_and_set_confounding()`:
diff --git a/tests/testthat/_snaps/forecast-output/forecast_output_ar_numeric.rds b/tests/testthat/_snaps/forecast-output/forecast_output_ar_numeric.rds
index 2fcf7aa9b..f7ed98834 100644
Binary files a/tests/testthat/_snaps/forecast-output/forecast_output_ar_numeric.rds and b/tests/testthat/_snaps/forecast-output/forecast_output_ar_numeric.rds differ
diff --git a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric.rds b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric.rds
index 6b5c6c5da..3b42ce6c3 100644
Binary files a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric.rds and b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric.rds differ
diff --git a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative.rds b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative.rds
index 26be2ea86..6a691f19b 100644
Binary files a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative.rds and b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative.rds differ
diff --git a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative_groups.rds b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative_groups.rds
index e7ed70acb..0061a2ab8 100644
Binary files a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative_groups.rds and b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_iterative_groups.rds differ
diff --git a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_lags.rds b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_lags.rds
index 51b75e049..c51ff8268 100644
Binary files a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_lags.rds and b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_lags.rds differ
diff --git a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_xreg.rds b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_xreg.rds
index e32fdafc5..08001992a 100644
Binary files a/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_xreg.rds and b/tests/testthat/_snaps/forecast-output/forecast_output_arima_numeric_no_xreg.rds differ
diff --git a/tests/testthat/_snaps/forecast-output/forecast_output_forecast_ARIMA_group_numeric.rds b/tests/testthat/_snaps/forecast-output/forecast_output_forecast_ARIMA_group_numeric.rds
index e5db013be..1ab8c99a1 100644
Binary files a/tests/testthat/_snaps/forecast-output/forecast_output_forecast_ARIMA_group_numeric.rds and b/tests/testthat/_snaps/forecast-output/forecast_output_forecast_ARIMA_group_numeric.rds differ
diff --git a/tests/testthat/_snaps/forecast-setup.md b/tests/testthat/_snaps/forecast-setup.md
index 6f03d5298..fdf2616a9 100644
--- a/tests/testthat/_snaps/forecast-setup.md
+++ b/tests/testthat/_snaps/forecast-setup.md
@@ -6,7 +6,7 @@
       explain_forecast(testing = TRUE, model = model_custom_arima_temp, y = data_arima[
         1:150, "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:
         150, explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3, approach = "independence",
-      prediction_zero = p0_ar)
+      phi0 = p0_ar)
     Message
       Note: You passed a model to explain() which is not natively supported, and did not supply a 'get_model_specs' function to explain().
       Consistency checks between model and data is therefore disabled.
@@ -27,7 +27,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = y_wrong_format,
         xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
         explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3, approach = "independence",
-        prediction_zero = p0_ar)
+        phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! `y` has 2 columns (Temp,Wind).
@@ -41,7 +41,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = xreg_wrong_format, train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3, approach = "independence",
-      prediction_zero = p0_ar)
+      phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! `xreg` has 2 columns (Temp,Wind).
@@ -56,7 +56,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = xreg_no_column_names, train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3, approach = "independence",
-      prediction_zero = p0_ar)
+      phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! `xreg` misses column names.
@@ -66,22 +66,22 @@
     Code
       explain_forecast(testing = TRUE, y = data_arima[1:150, "Temp"], xreg = data_arima[
         , "Wind"], train_idx = 2:148, explain_idx = 149:150, explain_y_lags = 2,
-      explain_xreg_lags = 2, horizon = 3, approach = "independence", prediction_zero = p0_ar)
+      explain_xreg_lags = 2, horizon = 3, approach = "independence", phi0 = p0_ar)
     Condition
       Error in `explain_forecast()`:
       ! argument "model" is missing, with no default
 
-# erroneous input: `prediction_zero`
+# erroneous input: `phi0`
 
     Code
       p0_wrong_length <- p0_ar[1:2]
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3, approach = "independence",
-      prediction_zero = p0_wrong_length)
+      phi0 = p0_wrong_length)
     Condition
       Error in `get_parameters()`:
-      ! `prediction_zero` (77.8823529411765, 77.8823529411765) must be numeric and match the output size of the model (3).
+      ! `phi0` (77.8823529411765, 77.8823529411765) must be numeric and match the output size of the model (3).
 
 # erroneous input: `max_n_coalitions`
 
@@ -93,8 +93,8 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = explain_y_lags, explain_xreg_lags = explain_xreg_lags,
-      horizon = horizon, approach = "independence", prediction_zero = p0_ar,
-      max_n_coalitions = n_coalitions, group_lags = FALSE)
+      horizon = horizon, approach = "independence", phi0 = p0_ar, max_n_coalitions = n_coalitions,
+      group_lags = FALSE)
     Message
       Note: Feature names extracted from the model contains NA.
       Consistency checks between model and data is therefore disabled.
@@ -117,8 +117,8 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = explain_y_lags, explain_xreg_lags = explain_xreg_lags,
-      horizon = horizon, approach = "independence", prediction_zero = p0_ar,
-      max_n_coalitions = n_coalitions, group_lags = TRUE)
+      horizon = horizon, approach = "independence", phi0 = p0_ar, max_n_coalitions = n_coalitions,
+      group_lags = TRUE)
     Message
       Note: Feature names extracted from the model contains NA.
       Consistency checks between model and data is therefore disabled.
@@ -153,7 +153,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = train_idx_too_short,
       explain_idx = 149:150, explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `train_idx` must be a vector of positive finite integers and length > 1.
@@ -165,7 +165,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = train_idx_not_integer,
       explain_idx = 149:150, explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `train_idx` must be a vector of positive finite integers and length > 1.
@@ -177,7 +177,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = train_idx_out_of_range,
       explain_idx = 149:150, explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! The train (`train_idx`) and explain (`explain_idx`) indices must fit in the lagged data.
@@ -190,7 +190,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = explain_idx_not_integer,
       explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3, approach = "independence",
-      prediction_zero = p0_ar)
+      phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `explain_idx` must be a vector of positive finite integers.
@@ -202,7 +202,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = explain_idx_out_of_range,
       explain_y_lags = 2, explain_xreg_lags = 2, horizon = 3, approach = "independence",
-      prediction_zero = p0_ar)
+      phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! The train (`train_idx`) and explain (`explain_idx`) indices must fit in the lagged data.
@@ -215,7 +215,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = explain_y_lags_negative, explain_xreg_lags = 2, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `explain_y_lags` must be a vector of positive finite integers.
@@ -227,7 +227,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = explain_y_lags_not_integer, explain_xreg_lags = 2, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `explain_y_lags` must be a vector of positive finite integers.
@@ -239,7 +239,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = explain_y_lags_more_than_one, explain_xreg_lags = 2, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! `y` has 1 columns (Temp).
@@ -252,7 +252,7 @@
       explain_y_lags_zero <- 0
       explain_forecast(testing = TRUE, model = model_arima_temp_noxreg, y = data_arima[
         1:150, "Temp"], train_idx = 2:148, explain_idx = 149:150, explain_y_lags = 0,
-      horizon = 3, approach = "independence", prediction_zero = p0_ar)
+      horizon = 3, approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! `explain_y_lags=0` is not allowed for models without exogeneous variables
@@ -264,7 +264,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = explain_xreg_lags_negative, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `explain_xreg_lags` must be a vector of positive finite integers.
@@ -276,7 +276,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = explain_xreg_lags_not_integer, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `explain_xreg_lags` must be a vector of positive finite integers.
@@ -288,7 +288,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = explain_x_lags_wrong_length, horizon = 3,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_data_forecast()`:
       ! `xreg` has 1 columns (Wind).
@@ -302,7 +302,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = 2, horizon = horizon_negative,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `horizon` must be a vector (or scalar) of positive integers.
@@ -314,7 +314,7 @@
       explain_forecast(testing = TRUE, model = model_arima_temp, y = data_arima[1:150,
       "Temp"], xreg = data_arima[, "Wind"], train_idx = 2:148, explain_idx = 149:150,
       explain_y_lags = 2, explain_xreg_lags = 2, horizon = horizon_not_integer,
-      approach = "independence", prediction_zero = p0_ar)
+      approach = "independence", phi0 = p0_ar)
     Condition
       Error in `get_parameters()`:
       ! `horizon` must be a vector (or scalar) of positive integers.
diff --git a/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_separate.rds b/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_separate.rds
index 51b4a330b..d5bc1b7ef 100644
Binary files a/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_separate.rds and b/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_separate.rds differ
diff --git a/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_surrogate.rds b/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_surrogate.rds
index 9cc5a5799..5287a9c1f 100644
Binary files a/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_surrogate.rds and b/tests/testthat/_snaps/regression-output/output_lm_categorical_lm_surrogate.rds differ
diff --git a/tests/testthat/_snaps/regression-output/output_lm_mixed_decision_tree_cv_separate.rds b/tests/testthat/_snaps/regression-output/output_lm_mixed_decision_tree_cv_separate.rds
index b6dd532ee..374301c6f 100644
Binary files a/tests/testthat/_snaps/regression-output/output_lm_mixed_decision_tree_cv_separate.rds and b/tests/testthat/_snaps/regression-output/output_lm_mixed_decision_tree_cv_separate.rds differ
diff --git a/tests/testthat/_snaps/regression-output/output_lm_mixed_decision_tree_cv_separate_parallel.rds b/tests/testthat/_snaps/regression-output/output_lm_mixed_decision_tree_cv_separate_parallel.rds
index 67c49f255..f0b5651a3 100644
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diff --git a/tests/testthat/_snaps/regression-output/output_lm_numeric_lm_surrogate_reg_surr_n_comb.rds b/tests/testthat/_snaps/regression-output/output_lm_numeric_lm_surrogate_reg_surr_n_comb.rds
index 2ec897d61..114fa6707 100644
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diff --git a/tests/testthat/_snaps/regression-setup.md b/tests/testthat/_snaps/regression-setup.md
index e63719891..754236c2e 100644
--- a/tests/testthat/_snaps/regression-setup.md
+++ b/tests/testthat/_snaps/regression-setup.md
@@ -2,9 +2,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = c(
-          "regression_surrogate", "gaussian", "independence", "empirical"),
-        iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, approach = c("regression_surrogate",
+          "gaussian", "independence", "empirical"), iterative = FALSE)
     Condition
       Error in `check_approach()`:
       ! The `regression_separate` and `regression_surrogate` approaches cannot be combined with other approaches.
@@ -13,8 +12,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = c(
-          "regression_separate", "gaussian", "independence", "empirical"), iterative = FALSE)
+        x_train = x_train_numeric, phi0 = p0, approach = c("regression_separate",
+          "gaussian", "independence", "empirical"), iterative = FALSE)
     Condition
       Error in `check_approach()`:
       ! The `regression_separate` and `regression_surrogate` approaches cannot be combined with other approaches.
@@ -23,7 +22,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = NULL)
     Message
       Success with message:
@@ -38,7 +37,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = lm)
     Message
       Success with message:
@@ -53,7 +52,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"))
     Message
@@ -69,7 +68,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"), regression.tune_values = data.frame(num_terms = c(1, 2,
           3)))
@@ -86,7 +85,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"), regression.tune_values = data.frame(tree_depth = c(1, 2,
           3), num_terms = c(1, 2, 3)))
@@ -103,7 +102,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = 2, engine = "rpart",
           mode = "regression"), regression.tune_values = data.frame(tree_depth = c(1,
           2, 3)))
@@ -120,7 +119,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_surrogate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_surrogate",
         regression.tune_values = data.frame(tree_depth = c(1, 2, 3)), iterative = FALSE)
     Message
       Success with message:
@@ -144,7 +143,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = 2, engine = "rpart",
           mode = "regression"), regression.tune_values = as.matrix(data.frame(
           tree_depth = c(1, 2, 3))))
@@ -161,7 +160,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"), regression.tune_values = function(x) c(1, 2, 3))
     Message
@@ -177,7 +176,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"), regression.tune_values = function(x) data.frame(
           wrong_name = c(1, 2, 3)))
@@ -194,7 +193,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"), regression.tune_values = data.frame(tree_depth = c(1, 2,
           3)), regression.vfold_cv_para = 10)
@@ -211,7 +210,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"), regression.tune_values = data.frame(tree_depth = c(1, 2,
           3)), regression.vfold_cv_para = list(10))
@@ -228,7 +227,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart",
         mode = "regression"), regression.tune_values = data.frame(tree_depth = c(1, 2,
           3)), regression.vfold_cv_para = list(hey = 10))
@@ -245,7 +244,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_separate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_separate",
         regression.recipe_func = 3)
     Message
       Success with message:
@@ -260,7 +259,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_surrogate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_surrogate",
         regression.recipe_func = function(x) {
           return(2)
         }, iterative = FALSE)
@@ -286,7 +285,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_surrogate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_surrogate",
         regression.surrogate_n_comb = 2^ncol(x_explain_numeric) - 1, iterative = FALSE)
     Message
       Success with message:
@@ -310,7 +309,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "regression_surrogate",
+        x_train = x_train_numeric, phi0 = p0, approach = "regression_surrogate",
         regression.surrogate_n_comb = 0, iterative = FALSE)
     Message
       Success with message:
diff --git a/tests/testthat/_snaps/regular-output/output_custom_lm_numeric_independence_1.rds b/tests/testthat/_snaps/regular-output/output_custom_lm_numeric_independence_1.rds
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index a005d77b7..c21420f31 100644
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diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_n_coalitions.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_n_coalitions.rds
index e9bd604f4..4b240bd44 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_n_coalitions.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_n_coalitions.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_progress.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_progress.rds
index 2316a3acc..aaf9e052f 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_progress.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_empirical_progress.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_gaussian.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_gaussian.rds
index e2fba82a2..9a197ced8 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_gaussian.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_gaussian.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_independence.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_independence.rds
index dfca2b741..b23b244eb 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_independence.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_independence.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_MSEv_Shapley_weights.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_MSEv_Shapley_weights.rds
index b4b1b1231..379359169 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_MSEv_Shapley_weights.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_MSEv_Shapley_weights.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_keep_samp_for_vS.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_keep_samp_for_vS.rds
index 8eed1a51f..f9f4575a1 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_keep_samp_for_vS.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_independence_keep_samp_for_vS.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_interaction.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_interaction.rds
index 507e17dfa..e7a21d736 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_interaction.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_interaction.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_numeric_vaeac.rds b/tests/testthat/_snaps/regular-output/output_lm_numeric_vaeac.rds
index 5b5ac43ce..edea23233 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_numeric_vaeac.rds and b/tests/testthat/_snaps/regular-output/output_lm_numeric_vaeac.rds differ
diff --git a/tests/testthat/_snaps/regular-output/output_lm_timeseries_method.rds b/tests/testthat/_snaps/regular-output/output_lm_timeseries_method.rds
index 6ba6c2942..5aa38d8f8 100644
Binary files a/tests/testthat/_snaps/regular-output/output_lm_timeseries_method.rds and b/tests/testthat/_snaps/regular-output/output_lm_timeseries_method.rds differ
diff --git a/tests/testthat/_snaps/regular-setup.md b/tests/testthat/_snaps/regular-setup.md
index e689c5b8c..12ca26adf 100644
--- a/tests/testthat/_snaps/regular-setup.md
+++ b/tests/testthat/_snaps/regular-setup.md
@@ -4,7 +4,7 @@
       model_custom_lm_mixed <- model_lm_mixed
       class(model_custom_lm_mixed) <- "whatever"
       explain(testing = TRUE, model = model_custom_lm_mixed, x_train = x_train_mixed,
-        x_explain = x_explain_mixed, approach = "independence", prediction_zero = p0)
+        x_explain = x_explain_mixed, approach = "independence", phi0 = p0)
     Message
       Note: You passed a model to explain() which is not natively supported, and did not supply a 'get_model_specs' function to explain().
       Consistency checks between model and data is therefore disabled.
@@ -22,7 +22,7 @@
 
     Code
       explain(testing = TRUE, model = model_custom_lm_mixed, x_train = x_train_mixed,
-        x_explain = x_explain_mixed, approach = "independence", prediction_zero = p0,
+        x_explain = x_explain_mixed, approach = "independence", phi0 = p0,
         predict_model = custom_predict_model, get_model_specs = NA)
     Message
       Note: You passed a model to explain() which is not natively supported, and did not supply a 'get_model_specs' function to explain().
@@ -55,7 +55,7 @@
         feature_specs <- list(labels = NA, classes = NA, factor_levels = NA)
       })
       explain(testing = TRUE, model = model_custom_lm_mixed, x_train = x_train_mixed,
-        x_explain = x_explain_mixed, approach = "independence", prediction_zero = p0,
+        x_explain = x_explain_mixed, approach = "independence", phi0 = p0,
         predict_model = custom_predict_model, get_model_specs = custom_get_model_specs_no_lab)
     Message
       Note: Feature names extracted from the model contains NA.
@@ -88,7 +88,7 @@
         feature_specs <- list(labels = labels(x$terms), classes = NA, factor_levels = NA)
       })
       explain(testing = TRUE, model = model_custom_lm_mixed, x_train = x_train_mixed,
-        x_explain = x_explain_mixed, approach = "independence", prediction_zero = p0,
+        x_explain = x_explain_mixed, approach = "independence", phi0 = p0,
         predict_model = custom_predict_model, get_model_specs = custom_gms_no_classes)
     Message
       Note: Feature classes extracted from the model contains NA.
@@ -122,7 +122,7 @@
         "dataClasses")[-1], factor_levels = NA)
       })
       explain(testing = TRUE, model = model_custom_lm_mixed, x_train = x_train_mixed,
-        x_explain = x_explain_mixed, approach = "independence", prediction_zero = p0,
+        x_explain = x_explain_mixed, approach = "independence", phi0 = p0,
         predict_model = custom_predict_model, get_model_specs = custom_gms_no_factor_levels)
     Message
       Note: Feature factor levels extracted from the model contains NA.
@@ -153,7 +153,7 @@
     Code
       x_train_wrong_format <- c(a = 1, b = 2)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_wrong_format, approach = "independence", prediction_zero = p0)
+        x_train = x_train_wrong_format, approach = "independence", phi0 = p0)
     Condition
       Error in `get_data()`:
       ! x_train should be a matrix or a data.frame/data.table.
@@ -163,7 +163,7 @@
     Code
       x_explain_wrong_format <- c(a = 1, b = 2)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_wrong_format,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0)
     Condition
       Error in `get_data()`:
       ! x_explain should be a matrix or a data.frame/data.table.
@@ -174,7 +174,7 @@
       x_train_wrong_format <- c(a = 1, b = 2)
       x_explain_wrong_format <- c(a = 3, b = 4)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_wrong_format,
-        x_train = x_train_wrong_format, approach = "independence", prediction_zero = p0)
+        x_train = x_train_wrong_format, approach = "independence", phi0 = p0)
     Condition
       Error in `get_data()`:
       ! x_train should be a matrix or a data.frame/data.table.
@@ -186,8 +186,7 @@
       x_train_no_column_names <- as.data.frame(x_train_numeric)
       names(x_train_no_column_names) <- NULL
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_no_column_names, approach = "independence",
-        prediction_zero = p0)
+        x_train = x_train_no_column_names, approach = "independence", phi0 = p0)
     Condition
       Error in `get_data()`:
       ! x_train misses column names.
@@ -198,7 +197,7 @@
       x_explain_no_column_names <- as.data.frame(x_explain_numeric)
       names(x_explain_no_column_names) <- NULL
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_no_column_names,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0)
     Condition
       Error in `get_data()`:
       ! x_explain misses column names.
@@ -210,8 +209,7 @@
       x_explain_no_column_names <- as.data.frame(x_explain_numeric)
       names(x_explain_no_column_names) <- NULL
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_no_column_names,
-        x_train = x_train_no_column_names, approach = "independence",
-        prediction_zero = p0)
+        x_train = x_train_no_column_names, approach = "independence", phi0 = p0)
     Condition
       Error in `get_data()`:
       ! x_explain misses column names.
@@ -220,7 +218,7 @@
 
     Code
       explain(testing = TRUE, x_explain = x_explain_numeric, x_train = x_train_numeric,
-        approach = "independence", prediction_zero = p0)
+        approach = "independence", phi0 = p0)
     Condition
       Error in `explain()`:
       ! argument "model" is missing, with no default
@@ -230,8 +228,7 @@
     Code
       approach_non_character <- 1
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = approach_non_character,
-        prediction_zero = p0)
+        x_train = x_train_numeric, approach = approach_non_character, phi0 = p0)
     Condition
       Error in `check_approach()`:
       ! `approach` must be one of the following: 'categorical', 'copula', 'ctree', 'empirical', 'gaussian', 'independence', 'regression_separate', 'regression_surrogate', 'timeseries', 'vaeac'.
@@ -242,8 +239,7 @@
     Code
       approach_incorrect_length <- c("empirical", "gaussian")
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = approach_incorrect_length,
-        prediction_zero = p0)
+        x_train = x_train_numeric, approach = approach_incorrect_length, phi0 = p0)
     Condition
       Error in `check_approach()`:
       ! `approach` must be one of the following: 'categorical', 'copula', 'ctree', 'empirical', 'gaussian', 'independence', 'regression_separate', 'regression_surrogate', 'timeseries', 'vaeac'.
@@ -254,59 +250,58 @@
     Code
       approach_incorrect_character <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = approach_incorrect_character,
-        prediction_zero = p0)
+        x_train = x_train_numeric, approach = approach_incorrect_character, phi0 = p0)
     Condition
       Error in `check_approach()`:
       ! `approach` must be one of the following: 'categorical', 'copula', 'ctree', 'empirical', 'gaussian', 'independence', 'regression_separate', 'regression_surrogate', 'timeseries', 'vaeac'.
       These can also be combined (except 'regression_surrogate' and 'regression_separate') by passing a vector of length one less than the number of features (4).
 
-# erroneous input: `prediction_zero`
+# erroneous input: `phi0`
 
     Code
       p0_non_numeric_1 <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0_non_numeric_1)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0_non_numeric_1)
     Condition
       Error in `get_parameters()`:
-      ! `prediction_zero` (bla) must be numeric and match the output size of the model (1).
+      ! `phi0` (bla) must be numeric and match the output size of the model (1).
 
 ---
 
     Code
       p0_non_numeric_2 <- NULL
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0_non_numeric_2)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0_non_numeric_2)
     Condition
       Error in `get_parameters()`:
-      ! `prediction_zero` () must be numeric and match the output size of the model (1).
+      ! `phi0` () must be numeric and match the output size of the model (1).
 
 ---
 
     Code
       p0_too_long <- c(1, 2)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0_too_long)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0_too_long)
     Condition
       Error in `get_parameters()`:
-      ! `prediction_zero` (1, 2) must be numeric and match the output size of the model (1).
+      ! `phi0` (1, 2) must be numeric and match the output size of the model (1).
 
 ---
 
     Code
       p0_is_NA <- as.numeric(NA)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0_is_NA)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0_is_NA)
     Condition
       Error in `get_parameters()`:
-      ! `prediction_zero` (NA) must be numeric and match the output size of the model (1).
+      ! `phi0` (NA) must be numeric and match the output size of the model (1).
 
 # erroneous input: `max_n_coalitions`
 
     Code
       max_n_comb_non_numeric_1 <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         max_n_coalitions = max_n_comb_non_numeric_1)
     Condition
       Error in `get_parameters()`:
@@ -317,7 +312,7 @@
     Code
       max_n_comb_non_numeric_2 <- TRUE
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         max_n_coalitions = max_n_comb_non_numeric_2)
     Condition
       Error in `get_parameters()`:
@@ -328,7 +323,7 @@
     Code
       max_n_coalitions_non_integer <- 10.5
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         max_n_coalitions = max_n_coalitions_non_integer)
     Condition
       Error in `get_parameters()`:
@@ -339,7 +334,7 @@
     Code
       max_n_coalitions_too_long <- c(1, 2)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         max_n_coalitions = max_n_coalitions_too_long)
     Condition
       Error in `get_parameters()`:
@@ -350,7 +345,7 @@
     Code
       max_n_coalitions_is_NA <- as.numeric(NA)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         max_n_coalitions = max_n_coalitions_is_NA)
     Condition
       Error in `get_parameters()`:
@@ -361,7 +356,7 @@
     Code
       max_n_comb_non_positive <- 0
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         max_n_coalitions = max_n_comb_non_positive)
     Condition
       Error in `get_parameters()`:
@@ -372,7 +367,7 @@
     Code
       max_n_coalitions <- ncol(x_explain_numeric) - 1
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "gaussian",
+        x_train = x_train_numeric, phi0 = p0, approach = "gaussian",
         max_n_coalitions = max_n_coalitions)
     Message
       Success with message:
@@ -401,8 +396,8 @@
       groups <- list(A = c("Solar.R", "Wind"), B = c("Temp", "Month"), C = "Day")
       max_n_coalitions <- length(groups) - 1
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, prediction_zero = p0, approach = "gaussian",
-        group = groups, max_n_coalitions = max_n_coalitions)
+        x_train = x_train_numeric, phi0 = p0, approach = "gaussian", group = groups,
+        max_n_coalitions = max_n_coalitions)
     Message
       Success with message:
       n_groups is smaller than or equal to 3, meaning there are so few unique coalitions (8) that we should use all to get reliable results.
@@ -429,8 +424,7 @@
     Code
       group_non_list <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        group = group_non_list)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, group = group_non_list)
     Condition
       Error in `get_parameters()`:
       ! `group` must be NULL or a list
@@ -440,8 +434,7 @@
     Code
       group_with_non_characters <- list(A = 1, B = 2)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        group = group_with_non_characters)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, group = group_with_non_characters)
     Condition
       Error in `check_groups()`:
       ! All components of group should be a character.
@@ -452,8 +445,7 @@
       group_with_non_data_features <- list(A = c("Solar.R", "Wind",
         "not_a_data_feature"), B = c("Temp", "Month", "Day"))
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        group = group_with_non_data_features)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, group = group_with_non_data_features)
     Condition
       Error in `check_groups()`:
       ! The group feature(s) not_a_data_feature are not
@@ -465,8 +457,7 @@
       group_missing_data_features <- list(A = c("Solar.R"), B = c("Temp", "Month",
         "Day"))
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        group = group_missing_data_features)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, group = group_missing_data_features)
     Condition
       Error in `check_groups()`:
       ! The data feature(s) Wind do not
@@ -478,8 +469,7 @@
       group_dup_data_features <- list(A = c("Solar.R", "Solar.R", "Wind"), B = c(
         "Temp", "Month", "Day"))
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        group = group_dup_data_features)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, group = group_dup_data_features)
     Condition
       Error in `check_groups()`:
       ! Feature(s) Solar.R are found in more than one group or multiple times per group.
@@ -490,8 +480,7 @@
     Code
       single_group <- list(A = c("Solar.R", "Wind", "Temp", "Month", "Day"))
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        group = single_group)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, group = single_group)
     Condition
       Error in `check_groups()`:
       ! You have specified only a single group named A, containing the features: Solar.R, Wind, Temp, Month, Day.
@@ -502,7 +491,7 @@
     Code
       n_samples_non_numeric_1 <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         n_MC_samples = n_samples_non_numeric_1)
     Condition
       Error in `get_parameters()`:
@@ -513,7 +502,7 @@
     Code
       n_samples_non_numeric_2 <- TRUE
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         n_MC_samples = n_samples_non_numeric_2)
     Condition
       Error in `get_parameters()`:
@@ -524,7 +513,7 @@
     Code
       n_samples_non_integer <- 10.5
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         n_MC_samples = n_samples_non_integer)
     Condition
       Error in `get_parameters()`:
@@ -535,7 +524,7 @@
     Code
       n_samples_too_long <- c(1, 2)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         n_MC_samples = n_samples_too_long)
     Condition
       Error in `get_parameters()`:
@@ -546,7 +535,7 @@
     Code
       n_samples_is_NA <- as.numeric(NA)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         n_MC_samples = n_samples_is_NA)
     Condition
       Error in `get_parameters()`:
@@ -557,7 +546,7 @@
     Code
       n_samples_non_positive <- 0
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         n_MC_samples = n_samples_non_positive)
     Condition
       Error in `get_parameters()`:
@@ -568,8 +557,7 @@
     Code
       seed_not_integer_interpretable <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        seed = seed_not_integer_interpretable)
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, seed = seed_not_integer_interpretable)
     Condition
       Warning in `set.seed()`:
       NAs introduced by coercion
@@ -581,8 +569,8 @@
     Code
       keep_samp_for_vS_non_logical_1 <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        output_args = list(keep_samp_for_vS = keep_samp_for_vS_non_logical_1))
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, output_args = list(
+          keep_samp_for_vS = keep_samp_for_vS_non_logical_1))
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -597,8 +585,8 @@
     Code
       keep_samp_for_vS_non_logical_2 <- NULL
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        output_args = list(keep_samp_for_vS = keep_samp_for_vS_non_logical_2))
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, output_args = list(
+          keep_samp_for_vS = keep_samp_for_vS_non_logical_2))
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -613,8 +601,8 @@
     Code
       keep_samp_for_vS_too_long <- c(TRUE, FALSE)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        output_args = list(keep_samp_for_vS = keep_samp_for_vS_too_long))
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, output_args = list(
+          keep_samp_for_vS = keep_samp_for_vS_too_long))
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -629,8 +617,8 @@
     Code
       MSEv_uniform_comb_weights_nl_1 <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        output_args = list(MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_nl_1))
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, output_args = list(
+          MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_nl_1))
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -645,8 +633,8 @@
     Code
       MSEv_uniform_comb_weights_nl_2 <- NULL
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        output_args = list(MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_nl_2))
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, output_args = list(
+          MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_nl_2))
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -661,8 +649,8 @@
     Code
       MSEv_uniform_comb_weights_long <- c(TRUE, FALSE)
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
-        output_args = list(MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_long))
+        x_train = x_train_numeric, approach = "independence", phi0 = p0, output_args = list(
+          MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_long))
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -677,7 +665,7 @@
     Code
       predict_model_nonfunction <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         predict_model = predict_model_nonfunction)
     Message
       Success with message:
@@ -695,7 +683,7 @@
         "bla"
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         predict_model = predict_model_non_num_output)
     Message
       Success with message:
@@ -719,7 +707,7 @@
         rep(1, nrow(x) + 1)
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         predict_model = predict_model_wrong_output_len)
     Message
       Success with message:
@@ -743,7 +731,7 @@
         rep(1, nrow(x))
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         predict_model = predict_model_invalid_argument)
     Message
       Success with message:
@@ -766,7 +754,7 @@
         1 + "bla"
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         predict_model = predict_model_error)
     Message
       Success with message:
@@ -787,7 +775,7 @@
     Code
       get_model_specs_nonfunction <- "bla"
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         get_model_specs = get_model_specs_nonfunction)
     Condition
       Error in `get_feature_specs()`:
@@ -800,7 +788,7 @@
         "bla"
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         get_model_specs = get_ms_output_not_list)
     Condition
       Error in `get_feature_specs()`:
@@ -816,7 +804,7 @@
         list(1, 2, 3, 4)
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         get_model_specs = get_ms_output_too_long)
     Condition
       Error in `get_feature_specs()`:
@@ -832,7 +820,7 @@
         list(labels = 1, classes = 2, not_a_name = 3)
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         get_model_specs = get_ms_output_wrong_names)
     Condition
       Error in `get_feature_specs()`:
@@ -848,7 +836,7 @@
         1 + "bla"
       })
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_train_numeric, approach = "independence", prediction_zero = p0,
+        x_train = x_train_numeric, approach = "independence", phi0 = p0,
         get_model_specs = get_model_specs_error)
     Condition
       Error in `get_feature_specs()`:
@@ -866,7 +854,7 @@
     Code
       non_factor_approach_1 <- "gaussian"
       explain(testing = TRUE, model = model_lm_mixed, x_explain = x_explain_mixed,
-        x_train = x_explain_mixed, approach = non_factor_approach_1, prediction_zero = p0)
+        x_train = x_explain_mixed, approach = non_factor_approach_1, phi0 = p0)
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -883,7 +871,7 @@
     Code
       non_factor_approach_2 <- "empirical"
       explain(testing = TRUE, model = model_lm_mixed, x_explain = x_explain_mixed,
-        x_train = x_explain_mixed, approach = non_factor_approach_2, prediction_zero = p0)
+        x_train = x_explain_mixed, approach = non_factor_approach_2, phi0 = p0)
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -900,7 +888,7 @@
     Code
       non_factor_approach_3 <- "copula"
       explain(testing = TRUE, model = model_lm_mixed, x_explain = x_explain_mixed,
-        x_train = x_explain_mixed, approach = non_factor_approach_3, prediction_zero = p0)
+        x_train = x_explain_mixed, approach = non_factor_approach_3, phi0 = p0)
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -916,7 +904,7 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_explain_numeric, prediction_zero = p0, approach = "gaussian",
+        x_train = x_explain_numeric, phi0 = p0, approach = "gaussian",
         max_n_coalitions = max_n_coalitions)
     Message
       Success with message:
@@ -943,8 +931,8 @@
 
     Code
       explain(testing = TRUE, model = model_lm_numeric, x_explain = x_explain_numeric,
-        x_train = x_explain_numeric, prediction_zero = p0, approach = "gaussian",
-        group = groups, max_n_coalitions = max_n_coalitions)
+        x_train = x_explain_numeric, phi0 = p0, approach = "gaussian", group = groups,
+        max_n_coalitions = max_n_coalitions)
     Message
       Success with message:
       n_groups is smaller than or equal to 3, meaning there are so few unique coalitions (8) that we should use all to get reliable results.
@@ -971,8 +959,7 @@
     Code
       explanation_exact <- explain(testing = TRUE, model = model_lm_numeric,
         x_explain = x_explain_numeric, x_train = x_train_numeric, approach = "gaussian",
-        prediction_zero = p0, n_MC_samples = 2, seed = 123, max_n_coalitions = NULL,
-        iterative = FALSE)
+        phi0 = p0, n_MC_samples = 2, seed = 123, max_n_coalitions = NULL, iterative = FALSE)
     Message
       Success with message:
       max_n_coalitions is NULL or larger than or 2^n_features = 32, 
@@ -993,7 +980,7 @@
     Code
       explanation_equal <- explain(testing = TRUE, model = model_lm_numeric,
         x_explain = x_explain_numeric, x_train = x_train_numeric, approach = "gaussian",
-        prediction_zero = p0, n_MC_samples = 2, seed = 123, extra_computation_args = list(
+        phi0 = p0, n_MC_samples = 2, seed = 123, extra_computation_args = list(
           compute_sd = FALSE), max_n_coalitions = 2^ncol(x_explain_numeric),
         iterative = FALSE)
     Message
@@ -1012,7 +999,7 @@
     Code
       explanation_larger <- explain(testing = TRUE, model = model_lm_numeric,
         x_explain = x_explain_numeric, x_train = x_train_numeric, approach = "gaussian",
-        prediction_zero = p0, n_MC_samples = 2, seed = 123, extra_computation_args = list(
+        phi0 = p0, n_MC_samples = 2, seed = 123, extra_computation_args = list(
           compute_sd = FALSE), max_n_coalitions = 2^ncol(x_explain_numeric) + 1,
         iterative = FALSE)
     Message
diff --git a/tests/testthat/test-adaptive-output.R b/tests/testthat/test-adaptive-output.R
index 20b132075..0be8006c0 100644
--- a/tests/testthat/test-adaptive-output.R
+++ b/tests/testthat/test-adaptive-output.R
@@ -8,7 +8,7 @@ test_that("output_lm_numeric_independence_reach_exact", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = TRUE,
       verbose = c("basic", "convergence", "shapley"),
       paired_shap_sampling = TRUE
@@ -25,7 +25,7 @@ test_that("output_lm_numeric_independence_converges_tol", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative_args = list(
         initial_n_coalitions = 10,
         convergence_tol = 0.1
@@ -45,7 +45,7 @@ test_that("output_lm_numeric_independence_converges_maxit", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative_args = list(
         initial_n_coalitions = 10,
         convergence_tol = 0.001,
@@ -67,7 +67,7 @@ test_that("output_lm_numeric_indep_conv_max_n_coalitions", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       max_n_coalitions = 20,
       iterative = TRUE,
       verbose = c("convergence", "shapley")
@@ -92,7 +92,7 @@ test_that("output_lm_numeric_gaussian_group_converges_tol", {
       x_train = x_train_numeric,
       approach = "gaussian",
       group = groups,
-      prediction_zero = p0,
+      phi0 = p0,
       iterative_args = list(
         initial_n_coalitions = 5,
         convergence_tol = 0.1
@@ -112,7 +112,7 @@ test_that("output_lm_numeric_independence_converges_tol_paired", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative_args = list(
         initial_n_coalitions = 10,
         convergence_tol = 0.1
@@ -135,7 +135,7 @@ test_that("output_lm_numeric_independence_saving_and_cont_est", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "independence",
-    prediction_zero = p0,
+    phi0 = p0,
     paired_shap_sampling = FALSE,
     iterative_args = list(
       initial_n_coalitions = 10,
@@ -157,7 +157,7 @@ test_that("output_lm_numeric_independence_saving_and_cont_est", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "independence",
-    prediction_zero = p0,
+    phi0 = p0,
     paired_shap_sampling = FALSE,
     iterative_args = list(
       initial_n_coalitions = 10,
@@ -178,7 +178,7 @@ test_that("output_lm_numeric_independence_saving_and_cont_est", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       paired_shap_sampling = FALSE,
       iterative_args = list(
         initial_n_coalitions = 10,
@@ -206,7 +206,7 @@ test_that("output_lm_numeric_independence_saving_and_cont_est", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "independence",
-    prediction_zero = p0,
+    phi0 = p0,
     paired_shap_sampling = FALSE,
     iterative_args = list(
       initial_n_coalitions = 10,
@@ -227,7 +227,7 @@ test_that("output_lm_numeric_independence_saving_and_cont_est", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       paired_shap_sampling = FALSE,
       iterative_args = list(
         initial_n_coalitions = 10,
@@ -255,7 +255,7 @@ test_that("output_verbose_1", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = TRUE,
       verbose = c("basic")
     ),
@@ -271,7 +271,7 @@ test_that("output_verbose_1_3", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = TRUE,
       verbose = c("basic", "convergence")
     ),
@@ -287,7 +287,7 @@ test_that("output_verbose_1_3_4", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = TRUE,
       verbose = c("basic", "convergence", "shapley")
     ),
@@ -303,7 +303,7 @@ test_that("output_verbose_1_3_4_5", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = TRUE,
       verbose = c("basic", "convergence", "shapley", "vS_details")
     ),
diff --git a/tests/testthat/test-adaptive-setup.R b/tests/testthat/test-adaptive-setup.R
index fbc85a9f2..45f132e6b 100644
--- a/tests/testthat/test-adaptive-setup.R
+++ b/tests/testthat/test-adaptive-setup.R
@@ -5,7 +5,7 @@ test_that("iterative_args are respected", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "independence",
-    prediction_zero = p0,
+    phi0 = p0,
     max_n_coalitions = 30,
     iterative_args = list(
       initial_n_coalitions = 6,
@@ -40,7 +40,7 @@ test_that("iterative feature wise and groupwise computations identical", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "gaussian",
-    prediction_zero = p0,
+    phi0 = p0,
     iterative_args = list(
       initial_n_coalitions = 5,
       convergence_tol = 0.1
@@ -56,7 +56,7 @@ test_that("iterative feature wise and groupwise computations identical", {
     x_train = x_train_numeric,
     approach = "gaussian",
     group = groups,
-    prediction_zero = p0,
+    phi0 = p0,
     iterative_args = list(
       initial_n_coalitions = 5,
       convergence_tol = 0.1
@@ -82,7 +82,7 @@ test_that("erroneous input: `min_n_batches`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_numeric_1)
       )
     },
@@ -99,7 +99,7 @@ test_that("erroneous input: `min_n_batches`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_numeric_2)
       )
     },
@@ -116,7 +116,7 @@ test_that("erroneous input: `min_n_batches`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_integer)
       )
     },
@@ -133,7 +133,7 @@ test_that("erroneous input: `min_n_batches`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_too_long)
       )
     },
@@ -150,7 +150,7 @@ test_that("erroneous input: `min_n_batches`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_is_NA)
       )
     },
@@ -167,7 +167,7 @@ test_that("erroneous input: `min_n_batches`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         extra_computation_args = list(min_n_batches = n_batches_non_positive)
       )
     },
@@ -183,7 +183,7 @@ test_that("different n_batches gives same/different shapley values for different
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "empirical",
-    prediction_zero = p0,
+    phi0 = p0,
     extra_computation_args = list(min_n_batches = 5, max_batch_size = 10)
   )
 
@@ -193,7 +193,7 @@ test_that("different n_batches gives same/different shapley values for different
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "empirical",
-    prediction_zero = p0,
+    phi0 = p0,
     extra_computation_args = list(min_n_batches = 10, max_batch_size = 10)
   )
 
@@ -215,7 +215,7 @@ test_that("different n_batches gives same/different shapley values for different
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "ctree",
-    prediction_zero = p0,
+    phi0 = p0,
     extra_computation_args = list(min_n_batches = 5, max_batch_size = 10)
   )
 
@@ -225,7 +225,7 @@ test_that("different n_batches gives same/different shapley values for different
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "ctree",
-    prediction_zero = p0,
+    phi0 = p0,
     extra_computation_args = list(min_n_batches = 10, max_batch_size = 10)
   )
 
diff --git a/tests/testthat/test-asymmetric-causal-output.R b/tests/testthat/test-asymmetric-causal-output.R
index 9fb4668c8..bc8f0f017 100644
--- a/tests/testthat/test-asymmetric-causal-output.R
+++ b/tests/testthat/test-asymmetric-causal-output.R
@@ -7,7 +7,7 @@ test_that("output_asymmetric_conditional", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = NULL,
@@ -27,7 +27,7 @@ test_that("output_asym_cond_reg", {
       x_train = x_train_numeric,
       approach = "regression_separate",
       regression.model = parsnip::linear_reg(),
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = NULL,
@@ -46,7 +46,7 @@ test_that("output_asym_cond_reg_iterative", {
       x_train = x_train_numeric,
       approach = "regression_separate",
       regression.model = parsnip::linear_reg(),
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = NULL,
@@ -65,7 +65,7 @@ test_that("output_symmetric_conditional", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:2, 3, 4:5), # Does not matter when asymmetric = TRUE and confounding = NULL
       confounding = NULL,
@@ -83,7 +83,7 @@ test_that("output_symmetric_marginal_independence", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:5),
       confounding = TRUE,
@@ -101,7 +101,7 @@ test_that("output_symmetric_marginal_gaussian", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:5),
       confounding = TRUE,
@@ -119,7 +119,7 @@ test_that("output_asym_caus_conf_TRUE", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = TRUE,
@@ -140,7 +140,7 @@ test_that("output_asym_caus_conf_FALSE", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = FALSE,
@@ -159,7 +159,7 @@ test_that("output_asym_caus_conf_mix", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = c(TRUE, FALSE, FALSE),
@@ -178,7 +178,7 @@ test_that("output_asym_caus_conf_mix_n_coal", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = c(TRUE, FALSE, FALSE),
@@ -198,7 +198,7 @@ test_that("output_asym_caus_conf_mix_empirical", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "empirical",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = c(TRUE, FALSE, FALSE),
@@ -217,7 +217,7 @@ test_that("output_asym_caus_conf_mix_ctree", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = c(TRUE, FALSE, FALSE),
@@ -236,7 +236,7 @@ test_that("output_sym_caus_conf_TRUE", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = TRUE,
@@ -254,7 +254,7 @@ test_that("output_sym_caus_conf_FALSE", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = FALSE,
@@ -272,7 +272,7 @@ test_that("output_sym_caus_conf_mix", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = c(TRUE, FALSE, FALSE),
@@ -292,7 +292,7 @@ test_that("output_sym_caus_conf_TRUE_group", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:2, 3),
       confounding = TRUE,
@@ -312,7 +312,7 @@ test_that("output_sym_caus_conf_mix_group", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1, 2, 3),
       confounding = c(TRUE, TRUE, FALSE),
@@ -331,7 +331,7 @@ test_that("output_sym_caus_conf_mix_group_iterative", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1, 2, 3),
       confounding = c(TRUE, TRUE, FALSE),
@@ -357,7 +357,7 @@ test_that("output_mixed_sym_caus_conf_TRUE", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = TRUE,
@@ -375,7 +375,7 @@ test_that("output_mixed_sym_caus_conf_TRUE_iterative", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = TRUE,
@@ -394,7 +394,7 @@ test_that("output_mixed_asym_caus_conf_mixed", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = c(TRUE, FALSE, FALSE),
@@ -413,7 +413,7 @@ test_that("output_mixed_asym_caus_conf_mixed_2", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       confounding = c(FALSE, TRUE, TRUE),
@@ -434,7 +434,7 @@ test_that("output_mixed_asym_cond_reg", {
       x_train = x_train_mixed,
       approach = "regression_separate",
       regression.model = parsnip::linear_reg(),
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = TRUE,
       causal_ordering = list(1:2, 3, 4:5),
       paired_shap_sampling = FALSE,
@@ -456,7 +456,7 @@ test_that("output_categorical_asym_causal_mixed_cat", {
       x_explain = x_explain_categorical[1:2], # Temp [1:2] as [1:3] give different sample on GHA-macOS (unknown reason)
       x_train = x_train_categorical,
       approach = "categorical",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(3:4, 2, 1),
       confounding = c(TRUE, FALSE, FALSE),
@@ -477,7 +477,7 @@ test_that("output_cat_asym_causal_mixed_cat_ad", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "categorical",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(3:4, 2, 1),
       confounding = c(TRUE, FALSE, FALSE),
@@ -496,7 +496,7 @@ test_that("output_categorical_asym_causal_mixed_ctree", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       asymmetric = FALSE,
       causal_ordering = list(3:4, 2, 1),
       confounding = c(TRUE, FALSE, FALSE),
diff --git a/tests/testthat/test-asymmetric-causal-setup.R b/tests/testthat/test-asymmetric-causal-setup.R
index 3a17d9d78..75f03cb98 100644
--- a/tests/testthat/test-asymmetric-causal-setup.R
+++ b/tests/testthat/test-asymmetric-causal-setup.R
@@ -9,7 +9,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list(1:6),
         confounding = NULL,
@@ -28,7 +28,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list(1:5, 5),
         confounding = NULL,
@@ -47,7 +47,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list(2:5, 5),
         confounding = NULL,
@@ -66,7 +66,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list(1:2, 4),
         confounding = NULL,
@@ -85,7 +85,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list("Solar.R", "Wind", "Temp", "Month", "Day", "Invalid feature name"),
         confounding = NULL,
@@ -104,7 +104,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list("Solar.R", "Wind", "Temp", "Month", "Day", "Day"),
         confounding = NULL,
@@ -123,7 +123,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list("Solar.R", "Wind", "Temp", "Day", "Day"),
         confounding = NULL,
@@ -142,7 +142,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list("Solar.R", "Wind"),
         confounding = NULL,
@@ -161,7 +161,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list(c("Solar.R", "Wind", "Temp", "Month"), "Day"),
         confounding = NULL,
@@ -181,7 +181,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list(c("A", "C"), "Wrong name"),
         confounding = NULL,
@@ -201,7 +201,7 @@ test_that("asymmetric erroneous input: `causal_ordering`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = TRUE,
         causal_ordering = list(c("A"), "B"),
         confounding = NULL,
@@ -226,7 +226,7 @@ test_that("asymmetric erroneous input: `approach`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = FALSE,
         causal_ordering = list(1:2, 3:4, 5),
         confounding = TRUE,
@@ -249,7 +249,7 @@ test_that("asymmetric erroneous input: `asymmetric`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = c(FALSE, FALSE),
         causal_ordering = list(1:2, 3:4, 5),
         confounding = TRUE,
@@ -268,7 +268,7 @@ test_that("asymmetric erroneous input: `asymmetric`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = "Must be a single logical",
         causal_ordering = list(1:2, 3:4, 5),
         confounding = TRUE,
@@ -287,7 +287,7 @@ test_that("asymmetric erroneous input: `asymmetric`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = 1L,
         causal_ordering = list(1:2, 3:4, 5),
         confounding = TRUE,
@@ -311,7 +311,7 @@ test_that("asymmetric erroneous input: `confounding`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = FALSE,
         causal_ordering = list(1:2, 3:4, 5),
         confounding = c("A", "B", "C"),
@@ -330,7 +330,7 @@ test_that("asymmetric erroneous input: `confounding`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         asymmetric = FALSE,
         causal_ordering = list(1:2, 3:4, 5),
         confounding = c(TRUE, FALSE),
diff --git a/tests/testthat/test-forecast-output.R b/tests/testthat/test-forecast-output.R
index 1f90c9b3c..803f028e4 100644
--- a/tests/testthat/test-forecast-output.R
+++ b/tests/testthat/test-forecast-output.R
@@ -9,7 +9,7 @@ test_that("forecast_output_ar_numeric", {
       explain_y_lags = 2,
       horizon = 3,
       approach = "empirical",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       group_lags = FALSE,
       n_batches = 1
     ),
@@ -30,7 +30,7 @@ test_that("forecast_output_arima_numeric", {
       explain_xreg_lags = 2,
       horizon = 3,
       approach = "empirical",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       group_lags = FALSE,
       max_n_coalitions = 150,
       iterative = FALSE
@@ -52,7 +52,7 @@ test_that("forecast_output_arima_numeric_iterative", {
       explain_xreg_lags = 3,
       horizon = 3,
       approach = "empirical",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       group_lags = FALSE,
       max_n_coalitions = 150,
       iterative = TRUE,
@@ -75,7 +75,7 @@ test_that("forecast_output_arima_numeric_iterative_groups", {
       explain_xreg_lags = c(3, 3, 3),
       horizon = 3,
       approach = "empirical",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       group_lags = TRUE,
       max_n_coalitions = 150,
       iterative = TRUE,
@@ -96,7 +96,7 @@ test_that("forecast_output_arima_numeric_no_xreg", {
       explain_y_lags = 2,
       horizon = 3,
       approach = "empirical",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       group_lags = FALSE,
       n_batches = 1
     ),
@@ -118,7 +118,7 @@ test_that("forecast_output_forecast_ARIMA_group_numeric", {
       explain_xreg_lags = 2,
       horizon = 3,
       approach = "empirical",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       group_lags = TRUE,
       n_batches = 1
     ),
@@ -139,7 +139,7 @@ test_that("forecast_output_arima_numeric_no_lags", {
       explain_xreg_lags = 0,
       horizon = 3,
       approach = "independence",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       group_lags = FALSE,
       n_batches = 1
     ),
@@ -159,7 +159,7 @@ test_that("ARIMA gives the same output with different horizons", {
     explain_xreg_lags = 2,
     horizon = 3,
     approach = "empirical",
-    prediction_zero = p0_ar[1:3],
+    phi0 = p0_ar[1:3],
     group_lags = FALSE,
     n_batches = 1,
     max_n_coalitions = 200,
@@ -178,7 +178,7 @@ test_that("ARIMA gives the same output with different horizons", {
     explain_xreg_lags = 2,
     horizon = 2,
     approach = "empirical",
-    prediction_zero = p0_ar[1:2],
+    phi0 = p0_ar[1:2],
     group_lags = FALSE,
     n_batches = 1,
     max_n_coalitions = 100,
@@ -196,7 +196,7 @@ test_that("ARIMA gives the same output with different horizons", {
     explain_xreg_lags = 2,
     horizon = 1,
     approach = "empirical",
-    prediction_zero = p0_ar[1],
+    phi0 = p0_ar[1],
     group_lags = FALSE,
     n_batches = 1,
     max_n_coalitions = 50,
@@ -233,7 +233,7 @@ test_that("ARIMA gives the same output with different horizons with grouping", {
     explain_xreg_lags = 2,
     horizon = 3,
     approach = "empirical",
-    prediction_zero = p0_ar[1:3],
+    phi0 = p0_ar[1:3],
     group_lags = TRUE,
     n_batches = 1,
     max_n_coalitions = 50,
@@ -252,7 +252,7 @@ test_that("ARIMA gives the same output with different horizons with grouping", {
     explain_xreg_lags = 2,
     horizon = 2,
     approach = "empirical",
-    prediction_zero = p0_ar[1:2],
+    phi0 = p0_ar[1:2],
     group_lags = TRUE,
     n_batches = 1,
     max_n_coalitions = 50,
@@ -270,7 +270,7 @@ test_that("ARIMA gives the same output with different horizons with grouping", {
     explain_xreg_lags = 2,
     horizon = 1,
     approach = "empirical",
-    prediction_zero = p0_ar[1],
+    phi0 = p0_ar[1],
     group_lags = TRUE,
     n_batches = 1,
     max_n_coalitions = 50,
diff --git a/tests/testthat/test-forecast-setup.R b/tests/testthat/test-forecast-setup.R
index 00da0fe0a..cd211d392 100644
--- a/tests/testthat/test-forecast-setup.R
+++ b/tests/testthat/test-forecast-setup.R
@@ -19,7 +19,7 @@ test_that("error with custom model without providing predict_model", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -46,7 +46,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -68,7 +68,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -91,7 +91,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -114,7 +114,7 @@ test_that("erroneous input: `model`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -122,7 +122,7 @@ test_that("erroneous input: `model`", {
 })
 
 
-test_that("erroneous input: `prediction_zero`", {
+test_that("erroneous input: `phi0`", {
   set.seed(123)
 
   expect_snapshot(
@@ -141,7 +141,7 @@ test_that("erroneous input: `prediction_zero`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_wrong_length
+        phi0 = p0_wrong_length
       )
     },
     error = TRUE
@@ -171,7 +171,7 @@ test_that("erroneous input: `max_n_coalitions`", {
         explain_xreg_lags = explain_xreg_lags,
         horizon = horizon,
         approach = "independence",
-        prediction_zero = p0_ar,
+        phi0 = p0_ar,
         max_n_coalitions = n_coalitions,
         group_lags = FALSE
       )
@@ -199,7 +199,7 @@ test_that("erroneous input: `max_n_coalitions`", {
       explain_xreg_lags = explain_xreg_lags,
       horizon = horizon,
       approach = "independence",
-      prediction_zero = p0_ar,
+      phi0 = p0_ar,
       max_n_coalitions = n_coalitions,
       group_lags = TRUE
     )
@@ -226,7 +226,7 @@ test_that("erroneous input: `train_idx`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -249,7 +249,7 @@ test_that("erroneous input: `train_idx`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -271,7 +271,7 @@ test_that("erroneous input: `train_idx`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -297,7 +297,7 @@ test_that("erroneous input: `explain_idx`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -319,7 +319,7 @@ test_that("erroneous input: `explain_idx`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -345,7 +345,7 @@ test_that("erroneous input: `explain_y_lags`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -367,7 +367,7 @@ test_that("erroneous input: `explain_y_lags`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -389,7 +389,7 @@ test_that("erroneous input: `explain_y_lags`", {
         explain_xreg_lags = 2,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -410,7 +410,7 @@ test_that("erroneous input: `explain_y_lags`", {
         explain_y_lags = 0,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -437,7 +437,7 @@ test_that("erroneous input: `explain_x_lags`", {
         explain_xreg_lags = explain_xreg_lags_negative,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -459,7 +459,7 @@ test_that("erroneous input: `explain_x_lags`", {
         explain_xreg_lags = explain_xreg_lags_not_integer,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -481,7 +481,7 @@ test_that("erroneous input: `explain_x_lags`", {
         explain_xreg_lags = explain_x_lags_wrong_length,
         horizon = 3,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -507,7 +507,7 @@ test_that("erroneous input: `horizon`", {
         explain_xreg_lags = 2,
         horizon = horizon_negative,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
@@ -529,7 +529,7 @@ test_that("erroneous input: `horizon`", {
         explain_xreg_lags = 2,
         horizon = horizon_not_integer,
         approach = "independence",
-        prediction_zero = p0_ar
+        phi0 = p0_ar
       )
     },
     error = TRUE
diff --git a/tests/testthat/test-plot.R b/tests/testthat/test-plot.R
index 18598ea2b..6fe9d69de 100644
--- a/tests/testthat/test-plot.R
+++ b/tests/testthat/test-plot.R
@@ -6,7 +6,7 @@ explain_mixed <- explain(
   x_explain = x_explain_mixed,
   x_train = x_train_mixed,
   approach = "independence",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 explain_numeric_empirical <- explain(
@@ -15,7 +15,7 @@ explain_numeric_empirical <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 explain_numeric_gaussian <- explain(
@@ -24,7 +24,7 @@ explain_numeric_gaussian <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "gaussian",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 explain_numeric_ctree <- explain(
@@ -33,7 +33,7 @@ explain_numeric_ctree <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = "ctree",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 explain_numeric_combined <- explain(
@@ -42,7 +42,7 @@ explain_numeric_combined <- explain(
   x_explain = x_explain_numeric,
   x_train = x_train_numeric,
   approach = c("empirical", "ctree", "gaussian", "ctree"),
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Create a list of explanations with names
diff --git a/tests/testthat/test-regression-output.R b/tests/testthat/test-regression-output.R
index d730fdb60..d43acc701 100644
--- a/tests/testthat/test-regression-output.R
+++ b/tests/testthat/test-regression-output.R
@@ -7,7 +7,7 @@ test_that("output_lm_numeric_lm_separate_iterative", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "regression_separate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = TRUE
     ),
@@ -24,7 +24,7 @@ test_that("output_lm_numeric_lm_separate", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "regression_separate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
     ),
@@ -40,7 +40,7 @@ test_that("output_lm_numeric_lm_separate_n_comb", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "regression_separate",
-      prediction_zero = p0,
+      phi0 = p0,
       max_n_coalitions = 10,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
@@ -57,7 +57,7 @@ test_that("output_lm_categorical_lm_separate", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "regression_separate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
     ),
@@ -73,7 +73,7 @@ test_that("output_lm_mixed_lm_separate", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "regression_separate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
     ),
@@ -89,7 +89,7 @@ test_that("output_lm_mixed_splines_separate", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "regression_separate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       regression.recipe_func = function(regression.recipe) {
         recipes::step_ns(regression.recipe, recipes::all_numeric_predictors(), deg_free = 2)
@@ -107,7 +107,7 @@ test_that("output_lm_mixed_decision_tree_cv_separate", {
       model = model_lm_mixed,
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "regression_separate",
       regression.model = parsnip::decision_tree(tree_depth = hardhat::tune(), engine = "rpart", mode = "regression"),
       regression.tune_values = data.frame(tree_depth = c(1, 2)),
@@ -126,7 +126,7 @@ test_that("output_lm_mixed_decision_tree_cv_separate_parallel", {
       model = model_lm_mixed,
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "regression_separate",
       regression.model = parsnip::decision_tree(tree_depth = hardhat::tune(), engine = "rpart", mode = "regression"),
       regression.tune_values = data.frame(tree_depth = c(1, 2)),
@@ -145,7 +145,7 @@ test_that("output_lm_mixed_xgboost_separate", {
       model = model_lm_mixed,
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "regression_separate",
       regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
       regression.recipe_func = function(regression.recipe) {
@@ -166,7 +166,7 @@ test_that("output_lm_numeric_lm_surrogate_iterative", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "regression_surrogate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = TRUE
     ),
@@ -183,7 +183,7 @@ test_that("output_lm_numeric_lm_surrogate", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "regression_surrogate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
     ),
@@ -199,7 +199,7 @@ test_that("output_lm_numeric_lm_surrogate_n_comb", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "regression_surrogate",
-      prediction_zero = p0,
+      phi0 = p0,
       max_n_coalitions = 10,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
@@ -216,7 +216,7 @@ test_that("output_lm_numeric_lm_surrogate_reg_surr_n_comb", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "regression_surrogate",
-      prediction_zero = p0,
+      phi0 = p0,
       max_n_coalitions = 10,
       regression.model = parsnip::linear_reg(),
       regression.surrogate_n_comb = 8,
@@ -234,7 +234,7 @@ test_that("output_lm_categorical_lm_surrogate", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "regression_surrogate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
     ),
@@ -250,7 +250,7 @@ test_that("output_lm_mixed_lm_surrogate", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "regression_surrogate",
-      prediction_zero = p0,
+      phi0 = p0,
       regression.model = parsnip::linear_reg(),
       iterative = FALSE
     ),
@@ -265,7 +265,7 @@ test_that("output_lm_mixed_decision_tree_cv_surrogate", {
       model = model_lm_mixed,
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "regression_surrogate",
       regression.model = parsnip::decision_tree(tree_depth = hardhat::tune(), engine = "rpart", mode = "regression"),
       regression.tune_values = data.frame(tree_depth = c(1, 2)),
@@ -283,7 +283,7 @@ test_that("output_lm_mixed_xgboost_surrogate", {
       model = model_lm_mixed,
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "regression_surrogate",
       regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
       regression.recipe_func = function(regression.recipe) {
diff --git a/tests/testthat/test-regression-setup.R b/tests/testthat/test-regression-setup.R
index 9a8998ae0..f88c3692f 100644
--- a/tests/testthat/test-regression-setup.R
+++ b/tests/testthat/test-regression-setup.R
@@ -9,7 +9,7 @@ test_that("regression erroneous input: `approach`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = c("regression_surrogate", "gaussian", "independence", "empirical"),
         iterative = FALSE
       )
@@ -25,7 +25,7 @@ test_that("regression erroneous input: `approach`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = c("regression_separate", "gaussian", "independence", "empirical"),
         iterative = FALSE
       )
@@ -45,7 +45,7 @@ test_that("regression erroneous input: `regression.model`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = NULL
       )
@@ -61,7 +61,7 @@ test_that("regression erroneous input: `regression.model`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = lm
       )
@@ -77,7 +77,7 @@ test_that("regression erroneous input: `regression.model`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression")
       )
@@ -93,7 +93,7 @@ test_that("regression erroneous input: `regression.model`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression"),
         regression.tune_values = data.frame(num_terms = c(1, 2, 3))
@@ -110,7 +110,7 @@ test_that("regression erroneous input: `regression.model`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression"),
         regression.tune_values = data.frame(tree_depth = c(1, 2, 3), num_terms = c(1, 2, 3))
@@ -127,7 +127,7 @@ test_that("regression erroneous input: `regression.model`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = 2, engine = "rpart", mode = "regression"),
         regression.tune_values = data.frame(tree_depth = c(1, 2, 3))
@@ -144,7 +144,7 @@ test_that("regression erroneous input: `regression.model`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_surrogate",
         regression.tune_values = data.frame(tree_depth = c(1, 2, 3)),
         iterative = FALSE
@@ -166,7 +166,7 @@ test_that("regression erroneous input: `regression.tune_values`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = 2, engine = "rpart", mode = "regression"),
         regression.tune_values = as.matrix(data.frame(tree_depth = c(1, 2, 3)))
@@ -183,7 +183,7 @@ test_that("regression erroneous input: `regression.tune_values`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression"),
         regression.tune_values = function(x) c(1, 2, 3)
@@ -200,7 +200,7 @@ test_that("regression erroneous input: `regression.tune_values`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression"),
         regression.tune_values = function(x) data.frame(wrong_name = c(1, 2, 3))
@@ -221,7 +221,7 @@ test_that("regression erroneous input: `regression.vfold_cv_para`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression"),
         regression.tune_values = data.frame(tree_depth = c(1, 2, 3)),
@@ -239,7 +239,7 @@ test_that("regression erroneous input: `regression.vfold_cv_para`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression"),
         regression.tune_values = data.frame(tree_depth = c(1, 2, 3)),
@@ -257,7 +257,7 @@ test_that("regression erroneous input: `regression.vfold_cv_para`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.model = parsnip::decision_tree(tree_depth = tune(), engine = "rpart", mode = "regression"),
         regression.tune_values = data.frame(tree_depth = c(1, 2, 3)),
@@ -280,7 +280,7 @@ test_that("regression erroneous input: `regression.recipe_func`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_separate",
         regression.recipe_func = 3
       )
@@ -296,7 +296,7 @@ test_that("regression erroneous input: `regression.recipe_func`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_surrogate",
         regression.recipe_func = function(x) {
           return(2)
@@ -319,7 +319,7 @@ test_that("regression erroneous input: `regression.surrogate_n_comb`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_surrogate",
         regression.surrogate_n_comb = 2^ncol(x_explain_numeric) - 1,
         iterative = FALSE
@@ -336,7 +336,7 @@ test_that("regression erroneous input: `regression.surrogate_n_comb`", {
         model = model_lm_numeric,
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
-        prediction_zero = p0,
+        phi0 = p0,
         approach = "regression_surrogate",
         regression.surrogate_n_comb = 0,
         iterative = FALSE
diff --git a/tests/testthat/test-regular-output.R b/tests/testthat/test-regular-output.R
index d9c5cba9f..747080607 100644
--- a/tests/testthat/test-regular-output.R
+++ b/tests/testthat/test-regular-output.R
@@ -8,7 +8,7 @@ test_that("output_lm_numeric_independence", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_independence"
@@ -23,7 +23,7 @@ test_that("output_lm_numeric_independence_MSEv_Shapley_weights", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       output_args = list(MSEv_uniform_comb_weights = FALSE),
       iterative = FALSE
     ),
@@ -39,7 +39,7 @@ test_that("output_lm_numeric_empirical", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "empirical",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_empirical"
@@ -54,7 +54,7 @@ test_that("output_lm_numeric_empirical_n_coalitions", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "empirical",
-      prediction_zero = p0,
+      phi0 = p0,
       max_n_coalitions = 20,
       iterative = FALSE
     ),
@@ -71,7 +71,7 @@ test_that("output_lm_numeric_empirical_independence", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "empirical",
-      prediction_zero = p0,
+      phi0 = p0,
       empirical.type = "independence",
       iterative = FALSE
     ),
@@ -88,7 +88,7 @@ test_that("output_lm_numeric_empirical_AICc_each", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "empirical",
-      prediction_zero = p0,
+      phi0 = p0,
       max_n_coalitions = 8,
       empirical.type = "AICc_each_k",
       iterative = FALSE
@@ -106,7 +106,7 @@ test_that("output_lm_numeric_empirical_AICc_full", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "empirical",
-      prediction_zero = p0,
+      phi0 = p0,
       max_n_coalitions = 8,
       empirical.type = "AICc_full",
       iterative = FALSE
@@ -123,7 +123,7 @@ test_that("output_lm_numeric_gaussian", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_gaussian"
@@ -138,7 +138,7 @@ test_that("output_lm_numeric_copula", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "copula",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_copula"
@@ -153,7 +153,7 @@ test_that("output_lm_numeric_ctree", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_ctree"
@@ -169,7 +169,7 @@ test_that("output_lm_numeric_vaeac", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "vaeac",
-      prediction_zero = p0,
+      phi0 = p0,
       n_MC_samples = 10, # Low value here to speed up the time
       vaeac.epochs = 4, # Low value here to speed up the time
       vaeac.n_vaeacs_initialize = 2, # Low value here to speed up the time
@@ -191,7 +191,7 @@ test_that("output_lm_categorical_ctree", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_categorical_ctree"
@@ -207,7 +207,7 @@ test_that("output_lm_categorical_vaeac", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "vaeac",
-      prediction_zero = p0,
+      phi0 = p0,
       n_MC_samples = 10, # Low value here to speed up the time
       vaeac.epochs = 4, # Low value here to speed up the time
       vaeac.n_vaeacs_initialize = 2, # Low value here to speed up the time
@@ -229,7 +229,7 @@ test_that("output_lm_categorical_categorical", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "categorical",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_categorical_method"
@@ -244,7 +244,7 @@ test_that("output_lm_categorical_independence", {
       x_explain = x_explain_categorical,
       x_train = x_train_categorical,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_categorical_independence"
@@ -259,7 +259,7 @@ test_that("output_lm_ts_timeseries", {
       x_explain = x_explain_ts,
       x_train = x_train_ts,
       approach = "timeseries",
-      prediction_zero = p0_ts,
+      phi0 = p0_ts,
       group = group_ts,
       iterative = FALSE
     ),
@@ -275,7 +275,7 @@ test_that("output_lm_numeric_comb1", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = c("gaussian", "empirical", "ctree", "independence"),
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_comb1"
@@ -290,7 +290,7 @@ test_that("output_lm_numeric_comb2", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = c("ctree", "copula", "independence", "copula"),
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_comb2"
@@ -305,7 +305,7 @@ test_that("output_lm_numeric_comb3", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = c("independence", "empirical", "gaussian", "empirical"),
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_comb3"
@@ -323,7 +323,7 @@ test_that("output_lm_mixed_independence", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_mixed_independence"
@@ -338,7 +338,7 @@ test_that("output_lm_mixed_ctree", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "ctree",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_mixed_ctree"
@@ -354,7 +354,7 @@ test_that("output_lm_mixed_vaeac", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = "vaeac",
-      prediction_zero = p0,
+      phi0 = p0,
       n_MC_samples = 10, # Low value here to speed up the time
       vaeac.epochs = 4, # Low value here to speed up the time
       vaeac.n_vaeacs_initialize = 2, # Low value here to speed up the time
@@ -377,7 +377,7 @@ test_that("output_lm_mixed_comb", {
       x_explain = x_explain_mixed,
       x_train = x_train_mixed,
       approach = c("ctree", "independence", "ctree", "independence"),
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_mixed_comb"
@@ -404,7 +404,7 @@ test_that("output_custom_lm_numeric_independence_1", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       predict_model = custom_pred_func,
       iterative = FALSE
     ),
@@ -431,7 +431,7 @@ test_that("output_custom_lm_numeric_independence_2", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       predict_model = custom_pred_func,
       iterative = FALSE
     )),
@@ -444,7 +444,7 @@ test_that("output_custom_lm_numeric_independence_2", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "independence",
-    prediction_zero = p0,
+    phi0 = p0,
     iterative = FALSE
   )
 
@@ -494,7 +494,7 @@ test_that("output_custom_xgboost_mixed_dummy_ctree", {
           x_train = x_train_mixed,
           x_explain = x_explain_mixed,
           approach = "ctree",
-          prediction_zero = p0,
+          phi0 = p0,
           predict_model = predict_model.xgboost_dummy,
           get_model_specs = NA,
           iterative = FALSE
@@ -517,7 +517,7 @@ test_that("output_lm_numeric_interaction", {
       x_explain = x_explain_interaction,
       x_train = x_train_interaction,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       iterative = FALSE
     ),
     "output_lm_numeric_interaction"
@@ -534,7 +534,7 @@ test_that("output_lm_numeric_ctree_parallelized", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "ctree",
-        prediction_zero = p0,
+        phi0 = p0,
         iterative = FALSE
       )
     },
@@ -556,7 +556,7 @@ test_that("output_lm_numeric_empirical_progress", {
           x_explain = x_explain_numeric,
           x_train = x_train_numeric,
           approach = "empirical",
-          prediction_zero = p0,
+          phi0 = p0,
           iterative = FALSE
         )
       })
@@ -575,7 +575,7 @@ test_that("output_lm_numeric_independence_keep_samp_for_vS", {
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       output_args = list(keep_samp_for_vS = TRUE),
       iterative = FALSE
     )),
diff --git a/tests/testthat/test-regular-setup.R b/tests/testthat/test-regular-setup.R
index a42314511..ba610ad33 100644
--- a/tests/testthat/test-regular-setup.R
+++ b/tests/testthat/test-regular-setup.R
@@ -15,7 +15,7 @@ test_that("error with custom model without providing predict_model", {
         x_train = x_train_mixed,
         x_explain = x_explain_mixed,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -42,7 +42,7 @@ test_that("messages with missing detail in get_model_specs", {
       x_train = x_train_mixed,
       x_explain = x_explain_mixed,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       predict_model = custom_predict_model,
       get_model_specs = NA
     )
@@ -61,7 +61,7 @@ test_that("messages with missing detail in get_model_specs", {
       x_train = x_train_mixed,
       x_explain = x_explain_mixed,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       predict_model = custom_predict_model,
       get_model_specs = custom_get_model_specs_no_lab
     )
@@ -80,7 +80,7 @@ test_that("messages with missing detail in get_model_specs", {
       x_train = x_train_mixed,
       x_explain = x_explain_mixed,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       predict_model = custom_predict_model,
       get_model_specs = custom_gms_no_classes
     )
@@ -103,7 +103,7 @@ test_that("messages with missing detail in get_model_specs", {
       x_train = x_train_mixed,
       x_explain = x_explain_mixed,
       approach = "independence",
-      prediction_zero = p0,
+      phi0 = p0,
       predict_model = custom_predict_model,
       get_model_specs = custom_gms_no_factor_levels
     )
@@ -124,7 +124,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         x_explain = x_explain_numeric,
         x_train = x_train_wrong_format,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -141,7 +141,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         x_explain = x_explain_wrong_format,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -159,7 +159,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         x_explain = x_explain_wrong_format,
         x_train = x_train_wrong_format,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -178,7 +178,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         x_explain = x_explain_numeric,
         x_train = x_train_no_column_names,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -196,7 +196,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         x_explain = x_explain_no_column_names,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -215,7 +215,7 @@ test_that("erroneous input: `x_train/x_explain`", {
         x_explain = x_explain_no_column_names,
         x_train = x_train_no_column_names,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -233,7 +233,7 @@ test_that("erroneous input: `model`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -254,7 +254,7 @@ test_that("erroneous input: `approach`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = approach_non_character,
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -271,7 +271,7 @@ test_that("erroneous input: `approach`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = approach_incorrect_length,
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -288,14 +288,14 @@ test_that("erroneous input: `approach`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = approach_incorrect_character,
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
   )
 })
 
-test_that("erroneous input: `prediction_zero`", {
+test_that("erroneous input: `phi0`", {
   set.seed(123)
 
   expect_snapshot(
@@ -309,7 +309,7 @@ test_that("erroneous input: `prediction_zero`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0_non_numeric_1
+        phi0 = p0_non_numeric_1
       )
     },
     error = TRUE
@@ -326,7 +326,7 @@ test_that("erroneous input: `prediction_zero`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0_non_numeric_2
+        phi0 = p0_non_numeric_2
       )
     },
     error = TRUE
@@ -344,7 +344,7 @@ test_that("erroneous input: `prediction_zero`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0_too_long
+        phi0 = p0_too_long
       )
     },
     error = TRUE
@@ -361,7 +361,7 @@ test_that("erroneous input: `prediction_zero`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0_is_NA
+        phi0 = p0_is_NA
       )
     },
     error = TRUE
@@ -382,7 +382,7 @@ test_that("erroneous input: `max_n_coalitions`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         max_n_coalitions = max_n_comb_non_numeric_1
       )
     },
@@ -400,7 +400,7 @@ test_that("erroneous input: `max_n_coalitions`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         max_n_coalitions = max_n_comb_non_numeric_2
       )
     },
@@ -419,7 +419,7 @@ test_that("erroneous input: `max_n_coalitions`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         max_n_coalitions = max_n_coalitions_non_integer
       )
     },
@@ -439,7 +439,7 @@ test_that("erroneous input: `max_n_coalitions`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         max_n_coalitions = max_n_coalitions_too_long
       )
     },
@@ -457,7 +457,7 @@ test_that("erroneous input: `max_n_coalitions`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         max_n_coalitions = max_n_coalitions_is_NA
       )
     },
@@ -475,7 +475,7 @@ test_that("erroneous input: `max_n_coalitions`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         max_n_coalitions = max_n_comb_non_positive
       )
     },
@@ -491,7 +491,7 @@ test_that("erroneous input: `max_n_coalitions`", {
       model = model_lm_numeric,
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "gaussian",
       max_n_coalitions = max_n_coalitions
     )
@@ -513,7 +513,7 @@ test_that("erroneous input: `max_n_coalitions`", {
       model = model_lm_numeric,
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "gaussian",
       group = groups,
       max_n_coalitions = max_n_coalitions
@@ -535,7 +535,7 @@ test_that("erroneous input: `group`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         group = group_non_list
       )
     },
@@ -553,7 +553,7 @@ test_that("erroneous input: `group`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         group = group_with_non_characters
       )
     },
@@ -573,7 +573,7 @@ test_that("erroneous input: `group`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         group = group_with_non_data_features
       )
     },
@@ -593,7 +593,7 @@ test_that("erroneous input: `group`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         group = group_missing_data_features
       )
     },
@@ -613,7 +613,7 @@ test_that("erroneous input: `group`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         group = group_dup_data_features
       )
     },
@@ -630,7 +630,7 @@ test_that("erroneous input: `group`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         group = single_group
       )
     },
@@ -652,7 +652,7 @@ test_that("erroneous input: `n_MC_samples`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         n_MC_samples = n_samples_non_numeric_1
       )
     },
@@ -670,7 +670,7 @@ test_that("erroneous input: `n_MC_samples`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         n_MC_samples = n_samples_non_numeric_2
       )
     },
@@ -687,7 +687,7 @@ test_that("erroneous input: `n_MC_samples`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         n_MC_samples = n_samples_non_integer
       )
     },
@@ -704,7 +704,7 @@ test_that("erroneous input: `n_MC_samples`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         n_MC_samples = n_samples_too_long
       )
     },
@@ -721,7 +721,7 @@ test_that("erroneous input: `n_MC_samples`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         n_MC_samples = n_samples_is_NA
       )
     },
@@ -738,7 +738,7 @@ test_that("erroneous input: `n_MC_samples`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         n_MC_samples = n_samples_non_positive
       )
     },
@@ -760,7 +760,7 @@ test_that("erroneous input: `seed`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         seed = seed_not_integer_interpretable
       )
     },
@@ -781,7 +781,7 @@ test_that("erroneous input: `keep_samp_for_vS`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         output_args = list(keep_samp_for_vS = keep_samp_for_vS_non_logical_1)
       )
     },
@@ -798,7 +798,7 @@ test_that("erroneous input: `keep_samp_for_vS`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         output_args = list(keep_samp_for_vS = keep_samp_for_vS_non_logical_2)
       )
     },
@@ -815,7 +815,7 @@ test_that("erroneous input: `keep_samp_for_vS`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         output_args = list(keep_samp_for_vS = keep_samp_for_vS_too_long)
       )
     },
@@ -836,7 +836,7 @@ test_that("erroneous input: `MSEv_uniform_comb_weights`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         output_args = list(MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_nl_1)
       )
     },
@@ -853,7 +853,7 @@ test_that("erroneous input: `MSEv_uniform_comb_weights`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         output_args = list(MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_nl_2)
       )
     },
@@ -870,7 +870,7 @@ test_that("erroneous input: `MSEv_uniform_comb_weights`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         output_args = list(MSEv_uniform_comb_weights = MSEv_uniform_comb_weights_long)
       )
     },
@@ -892,7 +892,7 @@ test_that("erroneous input: `predict_model`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         predict_model = predict_model_nonfunction
       )
     },
@@ -912,7 +912,7 @@ test_that("erroneous input: `predict_model`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         predict_model = predict_model_non_num_output
       )
     },
@@ -932,7 +932,7 @@ test_that("erroneous input: `predict_model`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         predict_model = predict_model_wrong_output_len
       )
     },
@@ -952,7 +952,7 @@ test_that("erroneous input: `predict_model`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         predict_model = predict_model_invalid_argument
       )
     },
@@ -972,7 +972,7 @@ test_that("erroneous input: `predict_model`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         predict_model = predict_model_error
       )
     },
@@ -994,7 +994,7 @@ test_that("erroneous input: `get_model_specs`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         get_model_specs = get_model_specs_nonfunction
       )
     },
@@ -1015,7 +1015,7 @@ test_that("erroneous input: `get_model_specs`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         get_model_specs = get_ms_output_not_list
       )
     },
@@ -1035,7 +1035,7 @@ test_that("erroneous input: `get_model_specs`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         get_model_specs = get_ms_output_too_long
       )
     },
@@ -1059,7 +1059,7 @@ test_that("erroneous input: `get_model_specs`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         get_model_specs = get_ms_output_wrong_names
       )
     },
@@ -1079,7 +1079,7 @@ test_that("erroneous input: `get_model_specs`", {
         x_explain = x_explain_numeric,
         x_train = x_train_numeric,
         approach = "independence",
-        prediction_zero = p0,
+        phi0 = p0,
         get_model_specs = get_model_specs_error
       )
     },
@@ -1100,7 +1100,7 @@ test_that("incompatible input: `data/approach`", {
         x_explain = x_explain_mixed,
         x_train = x_explain_mixed,
         approach = non_factor_approach_1,
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -1116,7 +1116,7 @@ test_that("incompatible input: `data/approach`", {
         x_explain = x_explain_mixed,
         x_train = x_explain_mixed,
         approach = non_factor_approach_2,
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -1132,7 +1132,7 @@ test_that("incompatible input: `data/approach`", {
         x_explain = x_explain_mixed,
         x_train = x_explain_mixed,
         approach = non_factor_approach_3,
-        prediction_zero = p0
+        phi0 = p0
       )
     },
     error = TRUE
@@ -1147,7 +1147,7 @@ test_that("Correct dimension of S when sampling combinations", {
     model = model_lm_mixed,
     x_explain = x_explain_mixed,
     x_train = x_explain_mixed,
-    prediction_zero = p0,
+    phi0 = p0,
     approach = "ctree",
     max_n_coalitions = max_n_coalitions
   )
@@ -1164,7 +1164,7 @@ test_that("Message with too low `max_n_coalitions`", {
       model = model_lm_numeric,
       x_explain = x_explain_numeric,
       x_train = x_explain_numeric,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "gaussian",
       max_n_coalitions = max_n_coalitions
     )
@@ -1185,7 +1185,7 @@ test_that("Message with too low `max_n_coalitions`", {
       model = model_lm_numeric,
       x_explain = x_explain_numeric,
       x_train = x_explain_numeric,
-      prediction_zero = p0,
+      phi0 = p0,
       approach = "gaussian",
       group = groups,
       max_n_coalitions = max_n_coalitions
@@ -1207,7 +1207,7 @@ test_that("Shapr with `max_n_coalitions` >= 2^m uses exact Shapley kernel weight
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       n_MC_samples = 2, # Low value for fast computations
       seed = 123,
       max_n_coalitions = NULL,
@@ -1222,7 +1222,7 @@ test_that("Shapr with `max_n_coalitions` >= 2^m uses exact Shapley kernel weight
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       n_MC_samples = 2, # Low value for fast computations
       seed = 123,
       extra_computation_args = list(compute_sd = FALSE),
@@ -1240,7 +1240,7 @@ test_that("Shapr with `max_n_coalitions` >= 2^m uses exact Shapley kernel weight
       x_explain = x_explain_numeric,
       x_train = x_train_numeric,
       approach = "gaussian",
-      prediction_zero = p0,
+      phi0 = p0,
       n_MC_samples = 2, # Low value for fast computations
       seed = 123,
       extra_computation_args = list(compute_sd = FALSE),
@@ -1284,7 +1284,7 @@ test_that("Correct dimension of S when sampling combinations with groups", {
     model = model_lm_mixed,
     x_explain = x_explain_mixed,
     x_train = x_explain_mixed,
-    prediction_zero = p0,
+    phi0 = p0,
     approach = "ctree",
     group = groups,
     max_n_coalitions = max_n_coalitions
@@ -1300,7 +1300,7 @@ test_that("data feature ordering is output_lm_numeric_column_order", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "empirical",
-    prediction_zero = p0
+    phi0 = p0
   )
 
   ex.new_data_feature_order <- explain(
@@ -1309,7 +1309,7 @@ test_that("data feature ordering is output_lm_numeric_column_order", {
     x_explain = rev(x_explain_numeric),
     x_train = rev(x_train_numeric),
     approach = "empirical",
-    prediction_zero = p0
+    phi0 = p0
   )
 
   explain.new_model_feat_order <- explain(
@@ -1318,7 +1318,7 @@ test_that("data feature ordering is output_lm_numeric_column_order", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "empirical",
-    prediction_zero = p0
+    phi0 = p0
   )
 
   # Same Shapley values, but different order
@@ -1343,7 +1343,7 @@ test_that("parallelization gives same output for any approach", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "empirical",
-    prediction_zero = p0
+    phi0 = p0
   )
 
   future::plan("multisession", workers = 2) # Parallelized with 2 cores
@@ -1353,7 +1353,7 @@ test_that("parallelization gives same output for any approach", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "empirical",
-    prediction_zero = p0
+    phi0 = p0
   )
 
   future::plan("sequential") # Resetting to sequential computation
@@ -1372,7 +1372,7 @@ test_that("parallelization gives same output for any approach", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "ctree",
-    prediction_zero = p0
+    phi0 = p0
   )
 
   future::plan("multisession", workers = 2) # Parallelized with 2 cores
@@ -1382,7 +1382,7 @@ test_that("parallelization gives same output for any approach", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "ctree",
-    prediction_zero = p0
+    phi0 = p0
   )
 
   future::plan("sequential") # Resetting to sequential computation
@@ -1403,7 +1403,7 @@ test_that("gaussian approach use the user provided parameters", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "gaussian",
-    prediction_zero = p0,
+    phi0 = p0,
   )
 
   # Expect that gaussian.mu is the sample mean when no values are provided
@@ -1428,7 +1428,7 @@ test_that("gaussian approach use the user provided parameters", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "gaussian",
-    prediction_zero = p0,
+    phi0 = p0,
     gaussian.mu = gaussian.provided_mu,
     gaussian.cov_mat = gaussian.provided_cov_mat
   )
@@ -1455,7 +1455,7 @@ test_that("setting the seed for combined approaches works", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = c("independence", "empirical", "gaussian", "copula"),
-    prediction_zero = p0,
+    phi0 = p0,
     seed = 1
   )
 
@@ -1465,7 +1465,7 @@ test_that("setting the seed for combined approaches works", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = c("independence", "empirical", "gaussian", "copula"),
-    prediction_zero = p0,
+    phi0 = p0,
     seed = 1
   )
 
@@ -1484,7 +1484,7 @@ test_that("counting the number of unique approaches", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = c("independence", "empirical", "gaussian", "copula"),
-    prediction_zero = p0,
+    phi0 = p0,
     seed = 1
   )
   expect_equal(explanation_combined_1$internal$parameters$n_approaches, 4)
@@ -1496,7 +1496,7 @@ test_that("counting the number of unique approaches", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = c("empirical"),
-    prediction_zero = p0,
+    phi0 = p0,
     seed = 1
   )
   expect_equal(explanation_combined_2$internal$parameters$n_approaches, 1)
@@ -1508,7 +1508,7 @@ test_that("counting the number of unique approaches", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = c("gaussian", "gaussian", "gaussian", "gaussian"),
-    prediction_zero = p0,
+    phi0 = p0,
     seed = 1
   )
   expect_equal(explanation_combined_3$internal$parameters$n_approaches, 4)
@@ -1520,7 +1520,7 @@ test_that("counting the number of unique approaches", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = c("independence", "empirical", "independence", "empirical"),
-    prediction_zero = p0,
+    phi0 = p0,
     seed = 1
   )
   expect_equal(explanation_combined_4$internal$parameters$n_approaches, 4)
@@ -1533,7 +1533,7 @@ test_that("counting the number of unique approaches", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = c("independence", "empirical", "independence", "empirical"),
-    prediction_zero = p0,
+    phi0 = p0,
     seed = 1
   )
   expect_equal(explanation_combined_5$internal$parameters$n_approaches, 4)
@@ -1550,7 +1550,7 @@ test_that("vaeac_set_seed_works", {
     x_explain = x_explain_mixed,
     x_train = x_train_mixed,
     approach = "vaeac",
-    prediction_zero = p0,
+    phi0 = p0,
     n_MC_samples = 10,
     seed = 1,
     vaeac.epochs = 4,
@@ -1567,7 +1567,7 @@ test_that("vaeac_set_seed_works", {
     x_explain = x_explain_mixed,
     x_train = x_train_mixed,
     approach = "vaeac",
-    prediction_zero = p0,
+    phi0 = p0,
     n_MC_samples = 10,
     seed = 1,
     vaeac.epochs = 4,
@@ -1592,7 +1592,7 @@ test_that("vaeac_pretreained_vaeac_model", {
     x_explain = x_explain_mixed,
     x_train = x_train_mixed,
     approach = "vaeac",
-    prediction_zero = p0,
+    phi0 = p0,
     n_MC_samples = 10,
     seed = 1,
     vaeac.epochs = 4,
@@ -1614,7 +1614,7 @@ test_that("vaeac_pretreained_vaeac_model", {
     x_explain = x_explain_mixed,
     x_train = x_train_mixed,
     approach = "vaeac",
-    prediction_zero = p0,
+    phi0 = p0,
     n_MC_samples = 10,
     seed = 1,
     vaeac.extra_parameters = list(
@@ -1637,7 +1637,7 @@ test_that("vaeac_pretreained_vaeac_model", {
     x_explain = x_explain_mixed,
     x_train = x_train_mixed,
     approach = "vaeac",
-    prediction_zero = p0,
+    phi0 = p0,
     n_MC_samples = 10,
     seed = 1,
     vaeac.extra_parameters = list(
@@ -1666,7 +1666,7 @@ test_that("feature wise and groupwise computations are identical", {
     x_explain = x_explain_numeric,
     x_train = x_train_numeric,
     approach = "gaussian",
-    prediction_zero = p0
+    phi0 = p0
   )
 
 
@@ -1677,7 +1677,7 @@ test_that("feature wise and groupwise computations are identical", {
     x_train = x_train_numeric,
     approach = "gaussian",
     group = groups,
-    prediction_zero = p0
+    phi0 = p0
   )
 
 
diff --git a/vignettes/understanding_shapr.Rmd b/vignettes/understanding_shapr.Rmd
index f556d6298..6eedd3824 100644
--- a/vignettes/understanding_shapr.Rmd
+++ b/vignettes/understanding_shapr.Rmd
@@ -471,7 +471,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 #> Note: Feature classes extracted from the model contains NA.
@@ -541,7 +541,7 @@ explanation_plot <- explain(
   x_explain = x_explain_many,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 #> Note: Feature classes extracted from the model contains NA.
@@ -605,7 +605,7 @@ explanation_lm_cat <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 #> Success with message:
@@ -645,7 +645,7 @@ explanation_ctree <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   ctree.mincriterion = 0.80,
   ctree.minsplit = 20,
   ctree.minbucket = 20,
@@ -702,7 +702,7 @@ explanation_cat_method <- explain(
   x_explain = x_explain_all_cat,
   x_train = x_train_all_cat,
   approach = "categorical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 #> Success with message:
@@ -779,7 +779,7 @@ explanation_timeseries <- explain(
   x_explain = x_explain_ts,
   x_train = x_train_ts,
   approach = "timeseries",
-  prediction_zero = p0_ts,
+  phi0 = p0_ts,
   group = group_ts,
   iterative = FALSE
 )
@@ -905,7 +905,7 @@ explanation_independence <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -933,7 +933,7 @@ explanation_empirical <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -961,7 +961,7 @@ explanation_gaussian_1e1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e1,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -989,7 +989,7 @@ explanation_gaussian_1e2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -1017,7 +1017,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("gaussian", "empirical", "independence"),
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -1213,7 +1213,7 @@ ex <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = TRUE,
   iterative_args = list(convergence_tol = 0.1)
 )
@@ -1299,7 +1299,7 @@ explanation_par <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -1378,7 +1378,7 @@ ex_progress <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 handlers("progress")
@@ -1463,7 +1463,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("empirical", "copula", "gaussian"),
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -1500,7 +1500,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("ctree", "ctree", "empirical"),
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -1543,7 +1543,7 @@ explanation_group <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   group = group_list,
   iterative = FALSE
 )
@@ -1680,7 +1680,7 @@ explanation_custom <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   predict_model = MY_predict_model,
   get_model_specs = MY_get_model_specs
 )
@@ -1724,7 +1724,7 @@ explanation_custom_minimal <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   predict_model = MY_MINIMAL_predict_model
 )
 #> Note: You passed a model to explain() which is not natively supported, and did not supply a 'get_model_specs' function to explain().
@@ -1794,7 +1794,7 @@ explanation_tidymodels <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
   )
 #> Success with message:
@@ -1842,7 +1842,7 @@ explanation_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 100,
   vaeac.width = 16,
   vaeac.depth = 2,
@@ -1887,7 +1887,7 @@ explanation_vaeac_early_stop <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 100,
   vaeac.width = 16,
   vaeac.depth = 2,
@@ -1971,7 +1971,7 @@ ex_init <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 20,
   iterative = TRUE
 )
@@ -2009,7 +2009,7 @@ ex_further <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 25,
   iterative_args = list(convergence_tol = 0.005), # Decrease the convergence threshold
   prev_shapr_object = ex_init
@@ -2039,7 +2039,7 @@ ex_even_further <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = NULL,
   prev_shapr_object = ex_further$saving_path
 )
@@ -2098,7 +2098,7 @@ must also be 2, because at time $t = 1$ there was only a single
 observation available.
 
 Since the data is stationary, the mean of the data is used as value of
-`prediction_zero` (i.e. $\phi_0$). This can however be chosen
+`phi0` (i.e. $\phi_0$). This can however be chosen
 differently depending on the data and application.
 
 For a multivariate model such as a VAR (Vector AutoRegressive model), it
@@ -2130,7 +2130,7 @@ explanation_forecast <- explain_forecast(
   explain_y_lags = 2,
   horizon = 3,
   approach = "empirical",
-  prediction_zero = p0_ar,
+  phi0 = p0_ar,
   group_lags = FALSE
 )
 #> Note: Feature names extracted from the model contains NA.
@@ -2200,8 +2200,8 @@ that we want to explain a forecast made from time point 153 in the data.
 The argument `horizon` is set to 2 in order to explain a forecast of
 length 2.
 
-The argument `prediction_zero` is set to the mean of the time series,
-and is repeated two times. Each value of `prediction_zero` is the
+The argument `phi0` is set to the mean of the time series,
+and is repeated two times. Each value of `phi0` is the
 baseline for each forecast horizon. In our example, we assume that given
 no effect from the two lags, the temperature would just be the average
 during the observed period. Finally, we opt to not group the lags by
@@ -2220,7 +2220,7 @@ explanation_forecast <- explain_forecast(
   explain_y_lags = 2,
   horizon = 2,
   approach = "empirical",
-  prediction_zero = rep(mean(data$Temp), 2),
+  phi0 = rep(mean(data$Temp), 2),
   group_lags = FALSE
 )
 #> Note: Feature names extracted from the model contains NA.
@@ -2289,7 +2289,7 @@ explanation_forecast <- explain_forecast(
   explain_xreg_lags = 1,
   horizon = 2,
   approach = "empirical",
-  prediction_zero = rep(mean(data_fit$Temp), 2),
+  phi0 = rep(mean(data_fit$Temp), 2),
   group_lags = FALSE
 )
 #> Note: Feature names extracted from the model contains NA.
diff --git a/vignettes/understanding_shapr.Rmd.orig b/vignettes/understanding_shapr.Rmd.orig
index 30ce83a7c..9eb39feb7 100644
--- a/vignettes/understanding_shapr.Rmd.orig
+++ b/vignettes/understanding_shapr.Rmd.orig
@@ -485,7 +485,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 
@@ -524,7 +524,7 @@ explanation_plot <- explain(
   x_explain = x_explain_many,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 plot(explanation_plot, plot_type = "beeswarm")
@@ -565,7 +565,7 @@ explanation_lm_cat <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 
@@ -587,7 +587,7 @@ explanation_ctree <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = p0,
+  phi0 = p0,
   ctree.mincriterion = 0.80,
   ctree.minsplit = 20,
   ctree.minbucket = 20,
@@ -628,7 +628,7 @@ explanation_cat_method <- explain(
   x_explain = x_explain_all_cat,
   x_train = x_train_all_cat,
   approach = "categorical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
 )
 ```
@@ -689,7 +689,7 @@ explanation_timeseries <- explain(
   x_explain = x_explain_ts,
   x_train = x_train_ts,
   approach = "timeseries",
-  prediction_zero = p0_ts,
+  phi0 = p0_ts,
   group = group_ts,
   iterative = FALSE
 )
@@ -799,7 +799,7 @@ explanation_independence <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -810,7 +810,7 @@ explanation_empirical <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -821,7 +821,7 @@ explanation_gaussian_1e1 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e1,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -832,7 +832,7 @@ explanation_gaussian_1e2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -843,7 +843,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("gaussian", "empirical", "independence"),
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 1e2,
   MSEv_uniform_comb_weights = TRUE
 )
@@ -995,7 +995,7 @@ ex <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = TRUE,
   iterative_args = list(convergence_tol = 0.1)
 )
@@ -1057,7 +1057,7 @@ explanation_par <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 future::plan(sequential) # To return to non-parallel computation
@@ -1118,7 +1118,7 @@ ex_progress <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 handlers("progress")
@@ -1202,7 +1202,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("empirical", "copula", "gaussian"),
-  prediction_zero = p0
+  phi0 = p0
 )
 # Plot the resulting explanations for observations 1 and 6, excluding
 # the no-covariate effect
@@ -1219,7 +1219,7 @@ explanation_combined <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = c("ctree", "ctree", "empirical"),
-  prediction_zero = p0
+  phi0 = p0
 )
 ```
 
@@ -1244,7 +1244,7 @@ explanation_group <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   group = group_list,
   iterative = FALSE
 )
@@ -1351,7 +1351,7 @@ explanation_custom <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   predict_model = MY_predict_model,
   get_model_specs = MY_get_model_specs
 )
@@ -1375,7 +1375,7 @@ explanation_custom_minimal <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   predict_model = MY_MINIMAL_predict_model
 )
 
@@ -1424,7 +1424,7 @@ explanation_tidymodels <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0,
+  phi0 = p0,
   iterative = FALSE
   )
 
@@ -1455,7 +1455,7 @@ explanation_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 100,
   vaeac.width = 16,
   vaeac.depth = 2,
@@ -1491,7 +1491,7 @@ explanation_vaeac_early_stop <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   n_MC_samples = 100,
   vaeac.width = 16,
   vaeac.depth = 2,
@@ -1563,7 +1563,7 @@ ex_init <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 20,
   iterative = TRUE
 )
@@ -1574,7 +1574,7 @@ ex_further <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = 25,
   iterative_args = list(convergence_tol = 0.005), # Decrease the convergence threshold
   prev_shapr_object = ex_init
@@ -1589,7 +1589,7 @@ ex_even_further <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0,
+  phi0 = p0,
   max_n_coalitions = NULL,
   prev_shapr_object = ex_further$saving_path
 )
@@ -1633,7 +1633,7 @@ must also be 2, because at time $t = 1$ there was only a single
 observation available.
 
 Since the data is stationary, the mean of the data is used as value of
-`prediction_zero` (i.e. $\phi_0$). This can however be chosen
+`phi0` (i.e. $\phi_0$). This can however be chosen
 differently depending on the data and application.
 
 For a multivariate model such as a VAR (Vector AutoRegressive model), it
@@ -1664,7 +1664,7 @@ explanation_forecast <- explain_forecast(
   explain_y_lags = 2,
   horizon = 3,
   approach = "empirical",
-  prediction_zero = p0_ar,
+  phi0 = p0_ar,
   group_lags = FALSE
 )
 explanation_forecast
@@ -1712,8 +1712,8 @@ that we want to explain a forecast made from time point 153 in the data.
 The argument `horizon` is set to 2 in order to explain a forecast of
 length 2.
 
-The argument `prediction_zero` is set to the mean of the time series,
-and is repeated two times. Each value of `prediction_zero` is the
+The argument `phi0` is set to the mean of the time series,
+and is repeated two times. Each value of `phi0` is the
 baseline for each forecast horizon. In our example, we assume that given
 no effect from the two lags, the temperature would just be the average
 during the observed period. Finally, we opt to not group the lags by
@@ -1731,7 +1731,7 @@ explanation_forecast <- explain_forecast(
   explain_y_lags = 2,
   horizon = 2,
   approach = "empirical",
-  prediction_zero = rep(mean(data$Temp), 2),
+  phi0 = rep(mean(data$Temp), 2),
   group_lags = FALSE
 )
 
@@ -1784,7 +1784,7 @@ explanation_forecast <- explain_forecast(
   explain_xreg_lags = 1,
   horizon = 2,
   approach = "empirical",
-  prediction_zero = rep(mean(data_fit$Temp), 2),
+  phi0 = rep(mean(data_fit$Temp), 2),
   group_lags = FALSE
 )
 
diff --git a/vignettes/understanding_shapr_asymmetric_causal.Rmd b/vignettes/understanding_shapr_asymmetric_causal.Rmd
index 711654140..cd1e3e55c 100644
--- a/vignettes/understanding_shapr_asymmetric_causal.Rmd
+++ b/vignettes/understanding_shapr_asymmetric_causal.Rmd
@@ -309,7 +309,7 @@ model <- xgboost::xgboost(
 )
 
 # Save the phi0
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 # Look at the root mean squared error
 sqrt(mean((predict(model, x_explain) - y_explain)^2))
@@ -413,7 +413,7 @@ explanation_sym_con[["gaussian"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   asymmetric = FALSE, # Default value (TRUE will give the same since `causal_ordering = NULL`)
   causal_ordering = NULL, # Default value
@@ -452,7 +452,7 @@ explanation_sym_con[["ctree"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "ctree",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   asymmetric = FALSE, # Default value (TRUE will give the same since `causal_ordering = NULL`)
   causal_ordering = NULL, # Default value
@@ -493,7 +493,7 @@ explanation_sym_con[["xgboost"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   asymmetric = FALSE, # Default value (TRUE will give the same as `causal_ordering = NULL`)
@@ -584,7 +584,7 @@ explanation_asym_con[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   paired_shap_sampling = FALSE,
@@ -617,7 +617,7 @@ explanation_asym_con[["gaussian_non_iterative"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   paired_shap_sampling = FALSE,
@@ -650,7 +650,7 @@ explanation_asym_con[["ctree"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "ctree",
   paired_shap_sampling = FALSE,
@@ -683,7 +683,7 @@ explanation_asym_con[["xgboost"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   paired_shap_sampling = FALSE,
@@ -855,7 +855,7 @@ explanation_sym_marg[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   asymmetric = FALSE,
@@ -894,7 +894,7 @@ explanation_sym_marg[["independence_marg"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "independence",
   asymmetric = FALSE,
@@ -933,7 +933,7 @@ explanation_sym_marg[["independence_con"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "independence"
 )
@@ -1009,7 +1009,7 @@ explanation_sym_cau[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   asymmetric = FALSE,
@@ -1047,7 +1047,7 @@ explanation_sym_cau[["copula"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "copula",
   asymmetric = FALSE,
@@ -1110,7 +1110,7 @@ explanation_asym_cau[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   paired_shap_sampling = FALSE,
@@ -1145,7 +1145,7 @@ explanation_asym_cau[["copula"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "copula",
   paired_shap_sampling = FALSE,
@@ -1183,7 +1183,7 @@ explanation_asym_cau[["ctree"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "ctree",
   paired_shap_sampling = FALSE,
@@ -1218,7 +1218,7 @@ explanation_asym_cau[["vaeac"]] <- explain(
    model = model,
    x_train = x_train,
    x_explain = x_explain,
-   prediction_zero = prediction_zero,
+   phi0 = phi0,
    n_MC_samples = 1000,
    approach = "vaeac",
    vaeac.epochs = 20,
@@ -1590,7 +1590,7 @@ explanation_n_coal[["sym_cau_gaussian_64"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   asymmetric = FALSE,
   causal_ordering = list(1, 2:3, 4:7),
   confounding = c(FALSE, TRUE, FALSE),
@@ -1625,7 +1625,7 @@ explanation_n_coal[["asym_cau_gaussian_10"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   asymmetric = TRUE,
   causal_ordering = list(1, 2:3, 4:7),
   confounding = c(FALSE, TRUE, FALSE),
@@ -1761,7 +1761,7 @@ explanation_group_gaussian[["symmetric_marginal"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = FALSE,
     causal_ordering = list(seq(length(group_list))), # or `NULL`
     confounding = TRUE,
@@ -1801,7 +1801,7 @@ explanation_group_gaussian[["symmetric_conditional"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = FALSE,
     causal_ordering = list(seq(length(group_list))), # or `NULL`
     confounding = NULL,
@@ -1839,7 +1839,7 @@ explanation_group_gaussian[["asymmetric_conditional"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = TRUE,
     causal_ordering = causal_ordering_group,
     confounding = NULL,
@@ -1874,7 +1874,7 @@ explanation_group_gaussian[["symmetric_causal"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = FALSE,
     causal_ordering = causal_ordering_group,
     confounding = confounding,
@@ -1914,7 +1914,7 @@ explanation_group_gaussian[["asymmetric_causal"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = TRUE,
     causal_ordering = causal_ordering_group,
     confounding = confounding,
diff --git a/vignettes/understanding_shapr_asymmetric_causal.Rmd.orig b/vignettes/understanding_shapr_asymmetric_causal.Rmd.orig
index 66f19c49d..4f97fb010 100644
--- a/vignettes/understanding_shapr_asymmetric_causal.Rmd.orig
+++ b/vignettes/understanding_shapr_asymmetric_causal.Rmd.orig
@@ -311,7 +311,7 @@ model <- xgboost::xgboost(
 )
 
 # Save the phi0
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 # Look at the root mean squared error
 sqrt(mean((predict(model, x_explain) - y_explain)^2))
@@ -409,7 +409,7 @@ explanation_sym_con[["gaussian"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   asymmetric = FALSE, # Default value (TRUE will give the same since `causal_ordering = NULL`)
   causal_ordering = NULL, # Default value
@@ -421,7 +421,7 @@ explanation_sym_con[["ctree"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "ctree",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   asymmetric = FALSE, # Default value (TRUE will give the same since `causal_ordering = NULL`)
   causal_ordering = NULL, # Default value
@@ -432,7 +432,7 @@ explanation_sym_con[["xgboost"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   asymmetric = FALSE, # Default value (TRUE will give the same as `causal_ordering = NULL`)
@@ -481,7 +481,7 @@ explanation_asym_con[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   paired_shap_sampling = FALSE,
@@ -494,7 +494,7 @@ explanation_asym_con[["gaussian_non_iterative"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   paired_shap_sampling = FALSE,
@@ -508,7 +508,7 @@ explanation_asym_con[["ctree"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "ctree",
   paired_shap_sampling = FALSE,
@@ -521,7 +521,7 @@ explanation_asym_con[["xgboost"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   paired_shap_sampling = FALSE,
@@ -599,7 +599,7 @@ explanation_sym_marg[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   asymmetric = FALSE,
@@ -612,7 +612,7 @@ explanation_sym_marg[["independence_marg"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "independence",
   asymmetric = FALSE,
@@ -625,7 +625,7 @@ explanation_sym_marg[["independence_con"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "independence"
 )
@@ -669,7 +669,7 @@ explanation_sym_cau[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   asymmetric = FALSE,
@@ -687,7 +687,7 @@ explanation_sym_cau[["copula"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "copula",
   asymmetric = FALSE,
@@ -717,7 +717,7 @@ explanation_asym_cau[["gaussian"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "gaussian",
   paired_shap_sampling = FALSE,
@@ -731,7 +731,7 @@ explanation_asym_cau[["copula"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "copula",
   paired_shap_sampling = FALSE,
@@ -745,7 +745,7 @@ explanation_asym_cau[["ctree"]] <- explain(
   model = model,
   x_train = x_train,
   x_explain = x_explain,
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 1000,
   approach = "ctree",
   paired_shap_sampling = FALSE,
@@ -759,7 +759,7 @@ explanation_asym_cau[["vaeac"]] <- explain(
    model = model,
    x_train = x_train,
    x_explain = x_explain,
-   prediction_zero = prediction_zero,
+   phi0 = phi0,
    n_MC_samples = 1000,
    approach = "vaeac",
    vaeac.epochs = 20,
@@ -1086,7 +1086,7 @@ explanation_n_coal[["sym_cau_gaussian_64"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   asymmetric = FALSE,
   causal_ordering = list(1, 2:3, 4:7),
   confounding = c(FALSE, TRUE, FALSE),
@@ -1098,7 +1098,7 @@ explanation_n_coal[["asym_cau_gaussian_10"]] <- explain(
   x_train = x_train,
   x_explain = x_explain,
   approach = "gaussian",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   asymmetric = TRUE,
   causal_ordering = list(1, 2:3, 4:7),
   confounding = c(FALSE, TRUE, FALSE),
@@ -1171,7 +1171,7 @@ explanation_group_gaussian[["symmetric_marginal"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = FALSE,
     causal_ordering = list(seq(length(group_list))), # or `NULL`
     confounding = TRUE,
@@ -1185,7 +1185,7 @@ explanation_group_gaussian[["symmetric_conditional"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = FALSE,
     causal_ordering = list(seq(length(group_list))), # or `NULL`
     confounding = NULL,
@@ -1199,7 +1199,7 @@ explanation_group_gaussian[["asymmetric_conditional"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = TRUE,
     causal_ordering = causal_ordering_group,
     confounding = NULL,
@@ -1214,7 +1214,7 @@ explanation_group_gaussian[["symmetric_causal"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = FALSE,
     causal_ordering = causal_ordering_group,
     confounding = confounding,
@@ -1228,7 +1228,7 @@ explanation_group_gaussian[["asymmetric_causal"]] <-
     x_train = x_train,
     x_explain = x_explain,
     approach = "gaussian",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     asymmetric = TRUE,
     causal_ordering = causal_ordering_group,
     confounding = confounding,
diff --git a/vignettes/understanding_shapr_regression.Rmd b/vignettes/understanding_shapr_regression.Rmd
index 788f33526..84cd223d7 100644
--- a/vignettes/understanding_shapr_regression.Rmd
+++ b/vignettes/understanding_shapr_regression.Rmd
@@ -263,7 +263,7 @@ explanation_list$MC_empirical <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -295,7 +295,7 @@ explanation_list$sep_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg()
 )
@@ -368,7 +368,7 @@ explanation_list$sep_pcr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(regression_recipe) {
@@ -404,7 +404,7 @@ explanation_list$sep_splines <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(regression_recipe) {
@@ -475,7 +475,7 @@ explanation_list$sep_reicpe_example <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = regression.recipe_func
@@ -548,7 +548,7 @@ explanation_list$sep_tree_stump <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = 1,
@@ -581,7 +581,7 @@ explanation_list$sep_tree_default <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(engine = "rpart", mode = "regression")
 )
@@ -698,7 +698,7 @@ explanation_list$sep_tree_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(), engine = "rpart", mode = "regression"
@@ -729,7 +729,7 @@ explanation_list$sep_tree_cv_2 <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(),
@@ -777,7 +777,7 @@ explanation_list$sep_rf <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -804,7 +804,7 @@ explanation_list$sep_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   verbose = c("basic","vS_details"), # To get printouts
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(
@@ -1033,7 +1033,7 @@ explanation_list$sep_xgboost <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression")
 )
@@ -1060,7 +1060,7 @@ explanation_list$sep_xgboost_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model =
     parsnip::boost_tree(trees = hardhat::tune(), engine = "xgboost", mode = "regression"),
@@ -1091,7 +1091,7 @@ explanation_list$sep_xgboost_cv_par <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model =
     parsnip::boost_tree(trees = hardhat::tune(), engine = "xgboost", mode = "regression"),
@@ -1122,7 +1122,7 @@ explanation_list$sep_xgboost_cv_2_par <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -1229,7 +1229,7 @@ explanation_list$sur_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg()
 )
@@ -1256,7 +1256,7 @@ explanation_list$sur_xgboost <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression")
 )
@@ -1283,7 +1283,7 @@ explanation_list$sur_xgboost_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -1317,7 +1317,7 @@ explanation_list$sur_rf <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -1344,7 +1344,7 @@ explanation_list$sur_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -1401,7 +1401,7 @@ explanation_list$sur_rf_cv_par <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -1625,7 +1625,7 @@ explanation_list$sep_ppr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = ppr_reg(num_terms = 2)
 )
@@ -1652,7 +1652,7 @@ explanation_list$sep_ppr_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = ppr_reg(num_terms = hardhat::tune()),
   regression.tune_values = dials::grid_regular(dials::num_terms(c(1, 4)), levels = 3),
@@ -1681,7 +1681,7 @@ explanation_list$sur_ppr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = ppr_reg(num_terms = 3)
 )
@@ -1708,7 +1708,7 @@ explanation_list$sur_ppr_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = ppr_reg(num_terms = hardhat::tune()),
   regression.tune_values = dials::grid_regular(dials::num_terms(c(1, 8)), levels = 4),
@@ -1796,7 +1796,7 @@ explanation_list_MC$MC_independence <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -1825,7 +1825,7 @@ explanation_list_MC$MC_gaussian <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -1851,7 +1851,7 @@ explanation_list_MC$MC_copula <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "copula",
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -1877,7 +1877,7 @@ explanation_list_MC$MC_ctree <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "ctree",
-  prediction_zero = p0
+  phi0 = p0
 )
 #> Note: Feature classes extracted from the model contains NA.
 #> Assuming feature classes from the data are correct.
@@ -1903,7 +1903,7 @@ explanation_list_MC$MC_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   vaeac.epochs = 10
 )
 #> Note: Feature classes extracted from the model contains NA.
@@ -2089,7 +2089,7 @@ explanation_list_mixed$MC_independence <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "independence"
 )
 #> Success with message:
@@ -2113,7 +2113,7 @@ explanation_list_mixed$MC_ctree <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "ctree"
 )
 #> Success with message:
@@ -2137,7 +2137,7 @@ explanation_list_mixed$MC_vaeac <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "vaeac"
 )
 #> Success with message:
@@ -2172,7 +2172,7 @@ explanation_list_mixed$sep_lm <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg()
 )
@@ -2197,7 +2197,7 @@ explanation_list_mixed$sep_splines <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(regression_recipe) {
@@ -2225,7 +2225,7 @@ explanation_list_mixed$sep_tree <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(engine = "rpart", mode = "regression")
 )
@@ -2250,7 +2250,7 @@ explanation_list_mixed$sep_tree_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(),
@@ -2283,7 +2283,7 @@ explanation_list_mixed$sep_rf <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -2308,7 +2308,7 @@ explanation_list_mixed$sep_rf_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -2340,7 +2340,7 @@ explanation_list_mixed$sep_xgboost <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   regression.recipe_func = function(regression_recipe) {
@@ -2368,7 +2368,7 @@ explanation_list_mixed$sep_xgboost_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -2413,7 +2413,7 @@ explanation_list_mixed$sur_lm <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg()
 )
@@ -2439,7 +2439,7 @@ explanation_list_mixed$sur_splines <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(recipe) {
@@ -2467,7 +2467,7 @@ explanation_list_mixed$sur_tree <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::decision_tree(engine = "rpart", mode = "regression")
 )
@@ -2492,7 +2492,7 @@ explanation_list_mixed$sur_tree_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(),
@@ -2525,7 +2525,7 @@ explanation_list_mixed$sur_rf <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -2550,7 +2550,7 @@ explanation_list_mixed$sur_rf_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -2579,7 +2579,7 @@ explanation_list_mixed$sur_xgboost <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   regression.recipe_func = function(regression_recipe) {
@@ -2607,7 +2607,7 @@ explanation_list_mixed$sur_xgboost_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -2739,7 +2739,7 @@ explanation_list_str$sep_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::linear_reg()"
 )
@@ -2765,7 +2765,7 @@ explanation_list_str$sep_pcr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::linear_reg()",
   regression.recipe_func = "function(regression_recipe) {
@@ -2794,7 +2794,7 @@ explanation_list_str$sep_splines <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = "function(regression_recipe) {
@@ -2823,7 +2823,7 @@ explanation_list_str$sep_tree_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::decision_tree(
     tree_depth = hardhat::tune(), engine = 'rpart', mode = 'regression'
@@ -2854,7 +2854,7 @@ explanation_list_str$sep_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = 'ranger', mode = 'regression'
@@ -2888,7 +2888,7 @@ explanation_list_str$sur_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = "parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = 'ranger', mode = 'regression'
diff --git a/vignettes/understanding_shapr_regression.Rmd.orig b/vignettes/understanding_shapr_regression.Rmd.orig
index c9f5f53b7..5c170fcd5 100644
--- a/vignettes/understanding_shapr_regression.Rmd.orig
+++ b/vignettes/understanding_shapr_regression.Rmd.orig
@@ -272,7 +272,7 @@ explanation_list$MC_empirical <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "empirical",
-  prediction_zero = p0
+  phi0 = p0
 )
 ```
 
@@ -286,7 +286,7 @@ explanation_list$sep_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg()
 )
@@ -338,7 +338,7 @@ explanation_list$sep_pcr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(regression_recipe) {
@@ -356,7 +356,7 @@ explanation_list$sep_splines <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(regression_recipe) {
@@ -409,7 +409,7 @@ explanation_list$sep_reicpe_example <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = regression.recipe_func
@@ -452,7 +452,7 @@ explanation_list$sep_tree_stump <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = 1,
@@ -468,7 +468,7 @@ explanation_list$sep_tree_default <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(engine = "rpart", mode = "regression")
 )
@@ -552,7 +552,7 @@ explanation_list$sep_tree_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(), engine = "rpart", mode = "regression"
@@ -566,7 +566,7 @@ explanation_list$sep_tree_cv_2 <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(),
@@ -596,7 +596,7 @@ explanation_list$sep_rf <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -606,7 +606,7 @@ explanation_list$sep_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   verbose = c("basic","vS_details"), # To get printouts
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(
@@ -680,7 +680,7 @@ explanation_list$sep_xgboost <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression")
 )
@@ -690,7 +690,7 @@ explanation_list$sep_xgboost_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model =
     parsnip::boost_tree(trees = hardhat::tune(), engine = "xgboost", mode = "regression"),
@@ -704,7 +704,7 @@ explanation_list$sep_xgboost_cv_par <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model =
     parsnip::boost_tree(trees = hardhat::tune(), engine = "xgboost", mode = "regression"),
@@ -718,7 +718,7 @@ explanation_list$sep_xgboost_cv_2_par <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -790,7 +790,7 @@ explanation_list$sur_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg()
 )
@@ -800,7 +800,7 @@ explanation_list$sur_xgboost <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression")
 )
@@ -810,7 +810,7 @@ explanation_list$sur_xgboost_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -827,7 +827,7 @@ explanation_list$sur_rf <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -837,7 +837,7 @@ explanation_list$sur_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -876,7 +876,7 @@ explanation_list$sur_rf_cv_par <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -1055,7 +1055,7 @@ explanation_list$sep_ppr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = ppr_reg(num_terms = 2)
 )
@@ -1065,7 +1065,7 @@ explanation_list$sep_ppr_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = ppr_reg(num_terms = hardhat::tune()),
   regression.tune_values = dials::grid_regular(dials::num_terms(c(1, 4)), levels = 3),
@@ -1077,7 +1077,7 @@ explanation_list$sur_ppr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = ppr_reg(num_terms = 3)
 )
@@ -1087,7 +1087,7 @@ explanation_list$sur_ppr_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = ppr_reg(num_terms = hardhat::tune()),
   regression.tune_values = dials::grid_regular(dials::num_terms(c(1, 8)), levels = 4),
@@ -1128,7 +1128,7 @@ explanation_list_MC$MC_independence <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "independence",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Copy the Shapley value explanations for the empirical method
@@ -1140,7 +1140,7 @@ explanation_list_MC$MC_gaussian <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "gaussian",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Compute the Shapley value explanations using the copula method
@@ -1149,7 +1149,7 @@ explanation_list_MC$MC_copula <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "copula",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Compute the Shapley value explanations using the ctree method
@@ -1158,7 +1158,7 @@ explanation_list_MC$MC_ctree <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "ctree",
-  prediction_zero = p0
+  phi0 = p0
 )
 
 # Compute the Shapley value explanations using the vaeac method
@@ -1167,7 +1167,7 @@ explanation_list_MC$MC_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = p0,
+  phi0 = p0,
   vaeac.epochs = 10
 )
 
@@ -1281,7 +1281,7 @@ explanation_list_mixed$MC_independence <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "independence"
 )
 
@@ -1289,7 +1289,7 @@ explanation_list_mixed$MC_ctree <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "ctree"
 )
 
@@ -1297,7 +1297,7 @@ explanation_list_mixed$MC_vaeac <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "vaeac"
 )
 ```
@@ -1315,7 +1315,7 @@ explanation_list_mixed$sep_lm <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg()
 )
@@ -1325,7 +1325,7 @@ explanation_list_mixed$sep_splines <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(regression_recipe) {
@@ -1338,7 +1338,7 @@ explanation_list_mixed$sep_tree <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(engine = "rpart", mode = "regression")
 )
@@ -1348,7 +1348,7 @@ explanation_list_mixed$sep_tree_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(),
@@ -1366,7 +1366,7 @@ explanation_list_mixed$sep_rf <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -1376,7 +1376,7 @@ explanation_list_mixed$sep_rf_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -1393,7 +1393,7 @@ explanation_list_mixed$sep_xgboost <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   regression.recipe_func = function(regression_recipe) {
@@ -1406,7 +1406,7 @@ explanation_list_mixed$sep_xgboost_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_separate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -1435,7 +1435,7 @@ explanation_list_mixed$sur_lm <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg()
 )
@@ -1446,7 +1446,7 @@ explanation_list_mixed$sur_splines <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = function(recipe) {
@@ -1459,7 +1459,7 @@ explanation_list_mixed$sur_tree <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::decision_tree(engine = "rpart", mode = "regression")
 )
@@ -1469,7 +1469,7 @@ explanation_list_mixed$sur_tree_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::decision_tree(
     tree_depth = hardhat::tune(),
@@ -1487,7 +1487,7 @@ explanation_list_mixed$sur_rf <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(engine = "ranger", mode = "regression")
 )
@@ -1497,7 +1497,7 @@ explanation_list_mixed$sur_rf_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = "ranger", mode = "regression"
@@ -1511,7 +1511,7 @@ explanation_list_mixed$sur_xgboost <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(engine = "xgboost", mode = "regression"),
   regression.recipe_func = function(regression_recipe) {
@@ -1524,7 +1524,7 @@ explanation_list_mixed$sur_xgboost_cv <- explain(
   model = model_cat,
   x_explain = x_explain_cat,
   x_train = x_train_cat,
-  prediction_zero = p0_cat,
+  phi0 = p0_cat,
   approach = "regression_surrogate",
   regression.model = parsnip::boost_tree(
     trees = hardhat::tune(),
@@ -1608,7 +1608,7 @@ explanation_list_str$sep_lm <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::linear_reg()"
 )
@@ -1617,7 +1617,7 @@ explanation_list_str$sep_pcr <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::linear_reg()",
   regression.recipe_func = "function(regression_recipe) {
@@ -1629,7 +1629,7 @@ explanation_list_str$sep_splines <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = parsnip::linear_reg(),
   regression.recipe_func = "function(regression_recipe) {
@@ -1641,7 +1641,7 @@ explanation_list_str$sep_tree_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::decision_tree(
     tree_depth = hardhat::tune(), engine = 'rpart', mode = 'regression'
@@ -1655,7 +1655,7 @@ explanation_list_str$sep_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_separate",
   regression.model = "parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = 'ranger', mode = 'regression'
@@ -1672,7 +1672,7 @@ explanation_list_str$sur_rf_cv <- explain(
   model = model,
   x_explain = x_explain,
   x_train = x_train,
-  prediction_zero = p0,
+  phi0 = p0,
   approach = "regression_surrogate",
   regression.model = "parsnip::rand_forest(
     mtry = hardhat::tune(), trees = hardhat::tune(), engine = 'ranger', mode = 'regression'
diff --git a/vignettes/understanding_shapr_vaeac.Rmd b/vignettes/understanding_shapr_vaeac.Rmd
index dcf30355e..91ca7f772 100644
--- a/vignettes/understanding_shapr_vaeac.Rmd
+++ b/vignettes/understanding_shapr_vaeac.Rmd
@@ -135,7 +135,7 @@ model <- xgboost(
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 ```
 
 
@@ -155,7 +155,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.epochs = vaeac.epochs,
   vaeac.n_vaeacs_initialize = vaeac.n_vaeacs_initialize
@@ -206,7 +206,7 @@ expl_pretrained_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = explanation$internal$parameters$vaeac
@@ -238,7 +238,7 @@ expl_pretrained_vaeac_path <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = explanation$internal$parameters$vaeac$models$best
@@ -276,7 +276,7 @@ expl_batches_combinations <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_coalitions = 10,
   n_MC_samples = n_MC_samples,
   vaeac.extra_parameters = list(
@@ -331,7 +331,7 @@ expl_batches_combinations_2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_coalitions = 10,
   n_MC_samples = n_MC_samples,
   vaeac.n_vaeacs_initialize = 1,
@@ -385,7 +385,7 @@ expl_paired_sampling_TRUE <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.epochs = 10,
   vaeac.n_vaeacs_initialize = 1,
@@ -402,7 +402,7 @@ expl_paired_sampling_FALSE <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.epochs = 10,
   vaeac.n_vaeacs_initialize = 1,
@@ -467,7 +467,7 @@ expl_with_messages <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   verbose = c("basic","vS_details"),
   vaeac.epochs = 5,
@@ -524,7 +524,7 @@ progressr::with_progress({
     x_explain = x_explain,
     x_train = x_train,
     approach = "vaeac",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_MC_samples = n_MC_samples,
     verbose = "vS_details",
     vaeac.epochs = 5,
@@ -583,7 +583,7 @@ expl_little_training <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.epochs = 3,
   vaeac.n_vaeacs_initialize = 2
@@ -622,7 +622,7 @@ expl_train_more_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = expl_train_more$internal$parameters$vaeac
@@ -654,7 +654,7 @@ expl_train_even_more_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = expl_train_even_more$internal$parameters$vaeac
@@ -725,7 +725,7 @@ expl_early_stopping <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   verbose = c("basic","vS_details"),
   vaeac.epochs = 1000, # Set it to a big number
@@ -827,7 +827,7 @@ expl_early_stopping_train_more <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = expl_early_stopping_train_more$internal$parameters$vaeac
@@ -876,7 +876,7 @@ expl_group <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   group = list(A = c("Temp", "Month"), B = c("Wind", "Solar.R")),
   n_MC_samples = n_MC_samples,
   verbose = "vS_details",
@@ -943,7 +943,7 @@ model <- ranger(as.formula(paste0(y_var, " ~ ", paste0(x_var_cat, collapse = " +
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(data_train_cat[, get(y_var)])
+phi0 <- mean(data_train_cat[, get(y_var)])
 ```
 Then we compute explanations using the `ctree` and `vaeac` approaches. For the `vaeac` approach, we
 consider two setups: the default architecture, and a simpler one without skip connections. We do this
@@ -957,7 +957,7 @@ expl_ctree <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250
 )
 #> Note: Feature classes extracted from the model contains NA.
@@ -972,7 +972,7 @@ expl_vaeac_with <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.epochs = 50,
   vaeac.n_vaeacs_initialize = 4
@@ -990,7 +990,7 @@ expl_vaeac_without <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.epochs = 50,
   vaeac.n_vaeacs_initialize = 4,
@@ -1091,7 +1091,7 @@ x_explain <- dt_explain[, -1]
 model <- lm(y ~ ., dt_train)
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 # Fit vaeac model using the CPU
 time_cpu <- system.time({
@@ -1100,7 +1100,7 @@ time_cpu <- system.time({
     x_explain = x_explain,
     x_train = x_train,
     approach = "vaeac",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_MC_samples = 100,
     vaeac.epochs = 50,
     vaeac.n_vaeacs_initialize = 2,
@@ -1115,7 +1115,7 @@ time_cuda <- system.time({
     x_explain = x_explain,
     x_train = x_train,
     approach = "vaeac",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_MC_samples = 100,
     vaeac.epochs = 50,
     vaeac.n_vaeacs_initialize = 2,
diff --git a/vignettes/understanding_shapr_vaeac.Rmd.orig b/vignettes/understanding_shapr_vaeac.Rmd.orig
index 20499c9b7..3d621ff48 100644
--- a/vignettes/understanding_shapr_vaeac.Rmd.orig
+++ b/vignettes/understanding_shapr_vaeac.Rmd.orig
@@ -147,7 +147,7 @@ model <- xgboost(
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 ```
 
 
@@ -166,7 +166,7 @@ explanation <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.epochs = vaeac.epochs,
   vaeac.n_vaeacs_initialize = vaeac.n_vaeacs_initialize
@@ -200,7 +200,7 @@ expl_pretrained_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = explanation$internal$parameters$vaeac
@@ -225,7 +225,7 @@ expl_pretrained_vaeac_path <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = explanation$internal$parameters$vaeac$models$best
@@ -256,7 +256,7 @@ expl_batches_combinations <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_coalitions = 10,
   n_MC_samples = n_MC_samples,
   vaeac.extra_parameters = list(
@@ -282,7 +282,7 @@ expl_batches_combinations_2 <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_coalitions = 10,
   n_MC_samples = n_MC_samples,
   vaeac.n_vaeacs_initialize = 1,
@@ -312,7 +312,7 @@ expl_paired_sampling_TRUE <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.epochs = 10,
   vaeac.n_vaeacs_initialize = 1,
@@ -324,7 +324,7 @@ expl_paired_sampling_FALSE <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   vaeac.epochs = 10,
   vaeac.n_vaeacs_initialize = 1,
@@ -370,7 +370,7 @@ expl_with_messages <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = n_MC_samples,
   verbose = c("basic","vS_details"),
   vaeac.epochs = 5,
@@ -390,7 +390,7 @@ progressr::with_progress({
     x_explain = x_explain,
     x_train = x_train,
     approach = "vaeac",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_MC_samples = n_MC_samples,
     verbose = "vS_details",
     vaeac.epochs = 5,
@@ -423,7 +423,7 @@ expl_little_training <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.epochs = 3,
   vaeac.n_vaeacs_initialize = 2
@@ -452,7 +452,7 @@ expl_train_more_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = expl_train_more$internal$parameters$vaeac
@@ -479,7 +479,7 @@ expl_train_even_more_vaeac <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = expl_train_even_more$internal$parameters$vaeac
@@ -534,7 +534,7 @@ expl_early_stopping <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   verbose = c("basic","vS_details"),
   vaeac.epochs = 1000, # Set it to a big number
@@ -593,7 +593,7 @@ expl_early_stopping_train_more <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.extra_parameters = list(
     vaeac.pretrained_vaeac_model = expl_early_stopping_train_more$internal$parameters$vaeac
@@ -629,7 +629,7 @@ expl_group <- explain(
   x_explain = x_explain,
   x_train = x_train,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   group = list(A = c("Temp", "Month"), B = c("Wind", "Solar.R")),
   n_MC_samples = n_MC_samples,
   verbose = "vS_details",
@@ -669,7 +669,7 @@ model <- ranger(as.formula(paste0(y_var, " ~ ", paste0(x_var_cat, collapse = " +
 )
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(data_train_cat[, get(y_var)])
+phi0 <- mean(data_train_cat[, get(y_var)])
 ```
 Then we compute explanations using the `ctree` and `vaeac` approaches. For the `vaeac` approach, we
 consider two setups: the default architecture, and a simpler one without skip connections. We do this
@@ -682,7 +682,7 @@ expl_ctree <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "ctree",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250
 )
 
@@ -692,7 +692,7 @@ expl_vaeac_with <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.epochs = 50,
   vaeac.n_vaeacs_initialize = 4
@@ -704,7 +704,7 @@ expl_vaeac_without <- explain(
   x_explain = x_explain_cat,
   x_train = x_train_cat,
   approach = "vaeac",
-  prediction_zero = prediction_zero,
+  phi0 = phi0,
   n_MC_samples = 250,
   vaeac.epochs = 50,
   vaeac.n_vaeacs_initialize = 4,
@@ -786,7 +786,7 @@ x_explain <- dt_explain[, -1]
 model <- lm(y ~ ., dt_train)
 
 # Specifying the phi_0, i.e. the expected prediction without any features
-prediction_zero <- mean(y_train)
+phi0 <- mean(y_train)
 
 # Fit vaeac model using the CPU
 time_cpu <- system.time({
@@ -795,7 +795,7 @@ time_cpu <- system.time({
     x_explain = x_explain,
     x_train = x_train,
     approach = "vaeac",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_MC_samples = 100,
     vaeac.epochs = 50,
     vaeac.n_vaeacs_initialize = 2,
@@ -810,7 +810,7 @@ time_cuda <- system.time({
     x_explain = x_explain,
     x_train = x_train,
     approach = "vaeac",
-    prediction_zero = prediction_zero,
+    phi0 = phi0,
     n_MC_samples = 100,
     vaeac.epochs = 50,
     vaeac.n_vaeacs_initialize = 2,