lin_std.stan

// Gaussian linear model with standardized data
data {
  int<lower=0> N; // number of data points
  vector[N] x; //
  vector[N] y; //
  real xpred; // input location for prediction
}
transformed data {
  vector[N] x_std;
  vector[N] y_std;
  real xpred_std;
  x_std = (x - mean(x)) / sd(x);
  y_std = (y - mean(y)) / sd(y);
  xpred_std = (xpred - mean(x)) / sd(x);
}
parameters {
  real alpha;
  real beta;
  real<lower=0> sigma_std;
}
transformed parameters {
  vector[N] mu_std;
  mu_std = alpha + beta*x_std;
}
model {
  alpha ~ normal(0, 1);
  beta ~ normal(0, 1);
  y_std ~ normal(mu_std, sigma_std);
}
generated quantities {
  vector[N] mu;
  real<lower=0> sigma;
  real ypred;
  vector[N] log_lik;
  mu = mu_std*sd(y) + mean(y);
  sigma = sigma_std*sd(y);
  ypred = normal_rng((alpha + beta*xpred_std)*sd(y)+mean(y), sigma*sd(y));
  for (i in 1:N)
    log_lik[i] = normal_lpdf(y[i] | mu[i], sigma);
}