Adding WishartCholesky and InverseWishartCholesky

[sethaxen]

The logpdf function of Wishart computes the Cholesky factorization of the input, while its rand function constructs a Cholesky factor before computing the full matrix, so, like LKJCholesky, it would be convenient to have an implementation of WishartCholesky. This would e.g. allow Turing users to perform inference on parameters with Wishart priors without computing the Cholesky factorization.

Similarly, we could implement an InverseWishartCholesky to avoid Cholesky factorizing the input in logpdf. However, rand does not benefit in this case, since it would be implemented in terms of Wishart, and the Cholesky factorization of the inverse of a matrix is not related to the Cholesky factorization of the matrix.

Side benefits are that random sampling in InverseWishart and MatrixBeta could be sped up using WishartCholesky, though for the latter, this could be a breaking change (currently the struct stores Wishart distributions).

[devmotion]

Completely agree. I have been thinking about it for a few months but never found the time to put together a PR.

[sethaxen]

I have some local implementations I'll polish for a PR. I also looked at the same for MatrixBeta, but there's no way to use the factorization for its logpdf, just rand I think.

[sethaxen]

Similarly, we could implement an InverseWishartCholesky to avoid Cholesky factorizing the input in logpdf. However, rand does not benefit in this case, since it would be implemented in terms of Wishart, and the Cholesky factorization of the inverse of a matrix is not related to the Cholesky factorization of the matrix.

I worked out an algorithm for directly sampling the Cholesky factor of Inverse-Wishart without going through Wishart: https://arxiv.org/abs/2310.15884