# Matrix-valued random variates

December 1, 2021 — January 6, 2022

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Distributions who support is a random matrix. There are many of these, surely? We generally care about a small subset of possible random matrices.

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.

## 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

Bishop, Del Moral, and Niclas. 2018. Foundations and Trends® in Machine Learning.
Edelman. 1989.
Fang, Kotz, and Ng. 2017. Symmetric Multivariate and Related Distributions.
Gupta, and Nagar. 1999. Matrix Variate Distributions. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 104.
Holmes. 1991. SIAM Journal on Matrix Analysis and Applications.
Lewandowski, Kurowicka, and Joe. 2009. Journal of Multivariate Analysis.
Mathai, and Moschopoulos. 1991. Journal of Multivariate Analysis.
Mathai, and Provost. 2005. Linear Algebra and Its Applications, Tenth Special Issue (Part 2) on Linear Algebra and Statistics,.
Mathal, and Moschopoulos. 1992. Annals of the Institute of Statistical Mathematics.
Meier, Kirch, and Meyer. 2020. Journal of Multivariate Analysis.
Pérez-Abreu, and Stelzer. 2014. Journal of Multivariate Analysis.
Pfaffel. 2012. arXiv:1201.3256 [Math].
Singpurwalla, and Youngren. 1993. Scandinavian Journal of Statistics.
Thibaux, and Jordan. 2007. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics.
Wilson, and Ghahramani. 2011. In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence. UAI’11.