Recommended prior for covariance matrix

Is there a recommended default prior for a covariance matrix? I’m asking because the Prior Choice Recommendations page has contradictory information which is virtually certain to confuse a reader.

I would say there is a consensus now to decompose a covariance matrix into a correlation matrix and something else. But there is less consensus on whether the something else should be standard deviations or variances and less consensus on what the prior should be.

I think people should do what rstanarm does, but that is somewhat complicated.

I’ll add that to the wiki, unless someone beats me to it.

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Would you have some explanation/reference on why decomposing the covariance into a correlation matrix and something else is better? I noticed in my simulations that I get way more precise results, and faster computations using LKJ priors (following the manual’s reparametrization advices) instead of any sort of invert Wishart or Wishart prior (though I have not tried any reparametrization using them). I would like to understand why!
Thank you!

The inverse wishart shows correlation between correlation and variance of the cov, which make it hard to sample from. There is a wide discussion about in literature.

I like Ben’s paper on the topic (with a bunch of co-authors):

This is far better than the hack found in the Gelman and Hill regression book, which rescales a Wishart prior!