# Matrix-valued random variates

December 1, 2021 — January 6, 2022

Distributions who support is a random matrix. There are many of these, surely? There are some particularly useful ones that I have encountered.

The most common matrix RV distributions I see are over positive-definite matrices in particular, which can be valid covariance functions. We also look at rotation matrices and matrices with i.i.d. elements.

## 1 “Random matrices”

Despite the general-sounding name, this is frequently used for a specific degenerate case, where the elements are i.i.d. random. See random matrices.

## 2 LKJ

Probability distribution for positive definite *correlation* matrices, or in practice, for their Cholesky factors.

## 3 Matrix Gaussian

Should look them up in Gupta and Nagar (1999).

## 4 Matrix Gamma

Currently handled under gamma processes.

## 5 Wishart

## 6 Inverse Wishart

## 7 Random rotations

See random rotations.

## 8 Matrix-*F*

Also introduced in Stephen R. Martin, Is the LKJ(1) prior uniform? “Yes”.

## 9 Matrix Beta/Dirichlet

The two wikipedia summaries are sparse:

Should look them up in Gupta and Nagar (1999).

## 10 References

*Foundations and Trends® in Machine Learning*.

*Symmetric Multivariate and Related Distributions*.

*Matrix Variate Distributions*. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 104.

*SIAM Journal on Matrix Analysis and Applications*.

*Journal of Multivariate Analysis*.

*Journal of Multivariate Analysis*.

*Linear Algebra and Its Applications*, Tenth Special Issue (Part 2) on Linear Algebra and Statistics,.

*Annals of the Institute of Statistical Mathematics*.

*Journal of Multivariate Analysis*.

*Journal of Multivariate Analysis*.

*arXiv:1201.3256 [Math]*.

*Scandinavian Journal of Statistics*.

*Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics*.

*Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*. UAI’11.