Issue/870 affine xform clarification

src/reference-manual/transforms.qmd

@@ -330,7 +330,7 @@ $$
 
 The default value for the offset $\mu$ is $0$ and for the multiplier $\sigma$ is
 $1$ in case not both are specified.
-
+For a container variable, the affine transform is applied to each element of that variable.
 
 ### Affine inverse transform {-}
 

src/reference-manual/types.qmd

@@ -472,6 +472,27 @@ model {
 }
 ```
 
+For a container variable, the affine transform is applied to each element of that variable.
+As an example, the non-centered parameterization of Neal's Funnel in the
+[Stan User's Guide reparameterization section](https://mc-stan.org/docs/stan-users-guide/reparameterization.html),
+$$
+p(y,x) = \textsf{normal}(y \mid 0,3) \times \prod_{n=1}^9
+\textsf{normal}(x_n \mid 0,\exp(y/2)).
+$$
+can be written as:
+
+```stan
+parameters {
+  real<multiplier=3> y;
+  vector<multiplier=exp(y/2)>[9] x;
+}
+model {
+  y ~ std_normal();
+  x ~ std_normal();
+}
+```
+where the affine transform is applied to every element of vector `x`.
+
 ### Expressions as bounds and offset/multiplier {-}
 
 Bounds (and offset and multiplier)

src/stan-users-guide/efficiency-tuning.qmd

@@ -266,8 +266,6 @@ funnel's neck is particularly sharp because of the exponential
 function applied to $y$.  A plot of the log marginal density of $y$
 and the first dimension $x_1$ is shown in the following plot.
 
-
-
 The funnel can be implemented directly in Stan as follows.
 
 ```stan
@@ -327,14 +325,26 @@ model {
 ```
 
 In this second model, the parameters `x_raw` and `y_raw` are
-sampled as independent standard normals, which is easy for Stan.  These
-are then transformed into samples from the funnel.  In this case, the
-same transform may be used to define Monte Carlo samples directly
-based on independent standard normal samples; Markov chain Monte Carlo
-methods are not necessary. If such a reparameterization were used in
-Stan code, it is useful to provide a comment indicating what the
-distribution for the parameter implies for the distribution of the
-transformed parameter.
+sampled as independent standard normals, which is easy for Stan,
+and then transformed into samples from the funnel.
+When this reparameterization is used in Stan code, a comment indicating what the
+distribution for the parameter implies for the distribution of the transformed parameter
+will improve readibility and maintainability.
+
+As of Stan release v2.19.0, this program can be written using Stan's
+[affinely transformed real type](https://mc-stan.org/docs/reference-manual/types.html#affine-transform.section).
+The affine transform on the vector `x` is applied to each element of `x`.
+
+```stan
+parameters {
+  real<multiplier=3> y;
+  vector<multiplier=exp(y/2)>[9] x;
+}
+model {
+  y ~ std_normal(); // implies y ~ normal(0, 3)
+  x ~ std_normal(); // implies x ~ normal(0, exp(y/2))
+}
+```
 
 ### Reparameterizing the Cauchy {-}