I am looking for an explanation of how Cholesky Covariance priors work in the context of mixed effects regression. In particular, when they are applied to the correlations among random effects. What exactly happens when the LKJ Cholesky prior is set to 1 vs. 1.5, for instance.

Edit: This is coming from the "brms" manual, which says:

"If there is more than one group-level effect per grouping factor, the correlations between those effects have to be estimated. The prior "lkj_corr_cholesky(eta)" or in short "lkj(eta)" with eta > 0 is essentially the only prior for (Cholesky factors) of correlation matrices. If eta = 1 (the default) all correlations matrices are equally likely a priori. If eta > 1, extreme correlations become less likely, whereas 0 < eta < 1 results in higher probabilities for extreme correlations. Correlation matrix parameters in brms models are named as cor_, (e.g., cor_g if g is the grouping factor). To set the same prior on every correlation matrix, use for instance set_prior("lkj(2)", class = "cor"). Internally, the priors are transformed to be put on the Cholesky factors of the correlation matrices to improve efficiency and numerical stability. The corresponding parameter class of the Cholesky factors is L, but it is not recommended to specify priors for this parameter class directly."

Edit 2: I also found a great post here. However, I'd still appreciate an intuitive explanation if someone has one.

  • $\begingroup$ Cholesky factorization is just an expedient way to represent a PD object like a covariance matrix, similar to how orthogonal decompositions can be helpful in other contexts. The underlying prior is still LKJ, the Cholesky part is just a choice of how to represent that information. $\endgroup$ – Sycorax Apr 17 '18 at 21:24
  • $\begingroup$ I'm not clear on what orthogonal decomposition is. $\endgroup$ – Dave Apr 17 '18 at 22:20
  • $\begingroup$ Examples include eigenvalue decomposition and singular value decomposition. $\endgroup$ – Sycorax Apr 17 '18 at 22:22
  • $\begingroup$ Does this (docs.pymc.io/notebooks/LKJ.html) help you? At least the second link. $\endgroup$ – David Nov 15 '18 at 3:31

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