polish text

Author: penelopeysm

index.qmd

@@ -13,7 +13,7 @@ description: |
 
 ::: {.panel}
 ##### Expressive {.panel-title .pb-1}
-Turing models are easy to write and communicate — their syntax closely resembles mathematical notation.
+Turing models are easy to write and communicate.
 :::
 
 ::: {.panel}
@@ -123,16 +123,16 @@ sample(model, externalsampler(SliceSteppingOut(2.0)), 1000)
 <div class="fs-4 fw-bold pb-1">Composability with Julia</div>
 
 As Turing.jl models are simply Julia functions under the hood, they can contain arbitrary Julia code.
+This allows users to draw on the rich numerical and scientific computing ecosystem of Julia.
 
-For example, [differential equations](https://turinglang.org/docs/tutorials/bayesian-differential-equations/) can be added to a model using the `DifferentialEquations.jl` Julia package.
+In the example here, we define an [ordinary differential equations](https://turinglang.org/docs/tutorials/bayesian-differential-equations/) using the `DifferentialEquations.jl` Julia package and use it in a Turing.jl model.
 
 :::
 
 ::: {.example-code style="overflow-x: scroll;"}
 ```{.julia .code-overflow-scroll}
-using DifferentialEquations
-
 # Define the system of equations
+using DifferentialEquations
 function lotka_volterra(du, u, params, t)
     α, β, δ, γ = params
     x, y = u
@@ -143,14 +143,11 @@ prob = ODEProblem(lotka_volterra, ...)
 
 # Use it in a model
 @model function fit_lotka_volterra()
-    # Priors
-    α ~ Normal(0, 1)
-    # ...
-
+    # Priors on ODE parameters
+    α ~ Normal(0, 1) # and others...
     # Solve the ODE
-    predictions = solve(prob, Tsit5(); p=p)
-
-    # Likelihood
+    predictions = solve(prob, Tsit5(); params=(α, ...))
+    # Calculate likelihood
     data ~ Poisson.(predictions, ...)
 end
 ```