Multivariate priors for hierarchical models confusion

Modeling
1

Hello, I have a confusion about this chapter.

Specifically, it states:

…the correlation matrix Omega and scale vector tau are more natural to inspect in the output;

Then later it also says:

The prior scale vector tau is unchanged, and furthermore, pre-multiplying the Cholesky factor by the scale produces the Cholesky factor of the final covariance matrix,
Sigma_beta = …

My understanding is that the off-diagonal elements of Omega are the correlations between the random effects (please confirm?).

But I see that the diagonal elements of Sigma_beta are quite different from tau. Are they supposed to be different? Which is supposed to represent the population variances of the random effects?

Thank you in advance.

2

I am answering my own quesiton.

The diagonal elements of Sigma_beta is the square of tau.

3

Thanks for answering your own question! I’m just piling on so this doesn’t look like an orphan and I’m getting scolded by Discourse because the question is already answered.

That’s correct. The off-diagonal elements of Sigma are the covariances.