Fixing elements of a covariance matrix vs. parameters in univariate priors

There seem to be a few different posts describing how to fix parameters to specific values that are (not) elements of a covariance matrix. They conflict at times and the especially confusing part is whether fixing elements of a covariance matrix is different from fixing single parameters from univariate priors.

Can someone summarize recommended approaches for fixing parameters in univariate vs. multivariate priors?

Example Posts

Below are some example posts that provide different (competing at times) solutions:

Here is a post advocating fixing a parameter this is not part of a covariance matrix (thus not PSD constrained) in the model block using assignment like param = 5:

Here’s a post discussing fixing an element of correlation matrix to 0 that first suggests using the transformed parameters but then meanders into discussion that isn’t very clear:

Meanwhile, the guide’s section on partially known parameters fixes elements of a covariance matrix in the transformed parameters and says no transform is needed because of some constraints.

It’s ‘different’ for covariance matrices because you can’t sample parameters for them directly. For most univariate scenarios I expect you can just declare the parameters you want directly and things will be fine, but I’m sure there are other complicated cases out there. So it’s not so much a univariate vs multivariate distinction but a ‘can parameterize directly’ vs ‘can’t paramaterize directly’ issue.

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You can parameterize a correlation matrix directly but it’s annoying and often brittle. Brittle, in that, the positive definite constraint is really strong and too strong priors will frustrate exploration by the sampler. Think funnels upon funnels. The issue is that as the dimension of a correlation matrix increases the determinant tends to 0. Here’s a pic of 1000 draws from an lkj with dimensions 5, 25, 50, 100, 500.

Anyway, with typical low dimension corr mats it is achievable and ive done it with hyperspherical parameterization of a cholesky factors of corr mats. It does still require some fiddling and you have to think in terms of correlative angles.

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I may misunderstand how to fix parameters in Stan. I’ll post it as a different question, however.