Hi everyone 
Following some chats with stan devs and others at StanCon, I wanted to start some discussion of kroneckerstructured covariances or array/matrix normal distributions. This is something I’ve wanted for work I’m doing, and I’ve seen some interest in the users’ list as well, e.g. https://groups.google.com/d/msg/stanusers/t9FksfRBdBo/Cp_fxLpyAAAJ, https://groups.google.com/d/msg/stanusers/iddShgJVwTQ/YmsPkJu52WQJ.
What is the current status? Here is what I have dug up:

It seems like there is code for the precisionparameterized matrix normal in https://github.com/standev/math/blob/develop/stan/math/prim/mat/prob/matrix_normal_prec_log.hpp. There is an issue https://github.com/standev/math/issues/86 that suggests the current botteleneck to exposing this into stan is having reversemode tests, but as far as I can tell some additional tests have been added since the last update on that issue that I think are reversemode tests (test/unit/math/rev/mat/prob/matrix_normal2_text.cpp). But I don’t understand stan’s autodiff or test structure well enough to be sure.

There is an issue for multiplying a kroneckerseparable matrix by a vector
(K1 %x% K2) %*% y
(https://github.com/standev/math/issues/393). From what I can tell based on https://github.com/standev/math/issues/60, the intent of this is to be able to construct kronecker quadratic forms for matrixnormal densitiest(Y) (K1 %x% K2) %*% Y
. In general for only a 2factor matrix normal I don’t think this is needed: I think you can rewrite the trace term in the likelihood asK1 %*% t(X) %*% K2 %*% X
, and the determinant terms can be split out as well. This is needed for more than two covariances, though. 
For higherorder separable covariance, there’s the additional question / design decision (mentioned in issue #60) of the data structure to hold what is basically a list of covariance matrices.
I don’t understand autodiff but can try to make headway on #1. I have code for doing #2 that I will try to port to C++ and contribute, unless someone’s working on it already. #3 I think requires broader consensus. Please let me know what I’m missing. Cheers,
Mike.