discuss pinning parameters in workflow in User's Guide

This is a request from "Somebody" on @andrewgelman's blog:

https://statmodeling.stat.columbia.edu/2026/05/04/expanding-the-stan-users-guide/#comment-2414300.

I'll summarize the first request here.

We should discuss the process of pinning parameters when doing development. This can make the models much faster to fit and also give us some hints at relevant hyper priors. For example, suppose we have a hierarchical model

data {
  int<lower=0> N;
}
parameters {
  vector[N] alpha;
  real<lower=0> sigma;
}
model {
  alpha ~ normal(0, sigma);
  sigma ~ ???
}

This is going to try to fit alpha and sigma and create a pure funnel posterior which has varying geometry and is not log concave.

We can replace this with a model with a fixed value, say sigma = 1.7. But this requires a new Stan model:

data {
  int<lower=0> N;
}
parameters {
  vector[N] alpha;
}
model {
  alpha ~ normal(0, 1.7);
}

If we fit this second model, we will get a posterior distribution over alpha. I have added a generated quantities block that shows how we can recover the actual scale of the parameter variation per draw, and hence get a posterior over it which can help inform our hyper prior on sigma.

generated quantities {
  real<lower=0> sigma_alpha = sd(alpha);
}

Start by locating the relevant section of the Stan User's Guide and review the linked blog request and the example hierarchical model. Add a discussion of pinning parameters during development, including the fixed-scale model and generated-quantities approach shown here. Done means the workflow and its use for informing hyperpriors are clearly explained in the guide.