Expand ForwardDiff thing

Author: penelopeysm

core-functionality/index.qmd

@@ -344,13 +344,9 @@ For example, let `c` be a `Chain`:
 
 #### Variable Types and Type Parameters
 
-The element type of a vector (or matrix) of random variables should match the `eltype` of its prior distribution, `<: Integer` for discrete distributions and `<: AbstractFloat` for continuous distributions. Moreover, if the continuous random variable is to be sampled using a Hamiltonian sampler, the vector's element type needs to either be:
+The element type of a vector (or matrix) of random variables should match the `eltype` of its prior distribution, `<: Integer` for discrete distributions and `<: AbstractFloat` for continuous distributions.
 
-1. `Real` to enable auto-differentiation through the model which uses special number types that are sub-types of `Real`, or
-
-2. Some type parameter `T` defined in the model header using the type parameter syntax, e.g. `function gdemo(x, ::Type{T} = Float64) where {T}`.
-
-Similarly, when using a particle sampler, the Julia variable used should either be:
+Moreover, when using a particle sampler, the Julia variable used should either be:
 
 1. An `Array`, or
 

usage/automatic-differentiation/index.qmd

@@ -175,7 +175,7 @@ A common error with ForwardDiff looks like this:
     x[1] = a
     b ~ MvNormal(x, I)
 end
-sample(forwarddiff_fail(), NUTS(; adtype=AutoForwardDiff()), 100)
+sample(forwarddiff_fail(), NUTS(; adtype=AutoForwardDiff()), 10)
 ```
 
 The problem here is the line `x[1] = a`.
@@ -187,16 +187,30 @@ In more depth: the basic premise of ForwardDiff is that functions have to accept
 Here, the line `x[1] = a` is equivalent to `setindex!(x, a, 1)`, and although the method `setindex!(::Vector{Float64}, ::Real, ...)` does exist, it attempts to convert the `Real` into a `Float64`, which is where it fails.
 :::
 
-The way around this is to pass a type as a parameter to the model:
+There are two ways around this.
+
+Firstly, you could broaden the type of the container:
+
+```{julia}
+@model function forwarddiff_working1()
+    x = Real[0.0, 1.0]
+    a ~ Normal()
+    x[1] = a
+    b ~ MvNormal(x, I)
+end
+sample(forwarddiff_working1(), NUTS(; adtype=AutoForwardDiff()), 10)
+```
+
+Or, you can pass a type as a parameter to the model:
 
 ```{julia}
-@model function forwarddiff_working(::Type{T}=Float64) where T
+@model function forwarddiff_working2(::Type{T}=Float64) where T
     x = T[0.0, 1.0]
     a ~ Normal()
     x[1] = a
     b ~ MvNormal(x, I)
 end
-sample(forwarddiff_working(), NUTS(), 100) # runs fine
+sample(forwarddiff_working2(), NUTS(; adtype=AutoForwardDiff()), 10)
 ```
 
 ## For AD Backend Developers