SUG 1.13 ("Multivariate priors for hierarchical models"): missing prior & serious identification issue?

Modeling
9

@mike-lawrence

I also had trouble understanding 1.13, even though I coded the model and put it into an R package. Here are some things I did to understand the model:

  1. I recoded the model using longer names rather than letters. This helped me avoid confusion around variables such as Ns and Ks.
  2. I have simulated fake data, but do not trust my simulation process, so I used known data. Specifically, the radon data from Gelman and Hill that is in the chapter 13. The data is also in Datasets for rstanarm examples — rstanarm-datasets • rstanarm (mc-stan.org)
  3. I ended up working through the model line-by-line with with lme4::lmer()and posted the solution in my answer to my own question: Understanding output from 1.13 Multivariate priors for hierarchical models example - Modeling - The Stan Forums (mc-stan.org)
  4. Realize that this is a regression on regression coefficients. This fact took me weeks to understand the first time and wrap my head around.
  5. Think about how matrices such as gamma and beta map from data to parameter or parameters to other parameters. I’ve had to think about linear algebra a lot to figure this out.

My sticking point was understanding the Cholesky factor and how to back transform, hence my question.

@Bob_Carpenter and the Stan Core team: Would the Stan team please consider show how to get Omega back in in the 1.13 tutorial. You include it for the Gaussian Process tutorial, but I found this but only through a Google search of the manual.

@andrewgelman Please consider using a side-by-side comparison between lme4::lmear() and the 1.16 example for Applied Regression and Multilevel Models See my linked, side by side example here: Understanding output from 1.13 Multivariate priors for hierarchical models example - Modeling - The Stan Forums (mc-stan.org)

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