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.

## “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.

## LKJ

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

## Matrix Gamma

Currently handled under gamma processes.

## Wishart

## Inverse Wishart

## Random rotations

See random rotations.

## Matrix-*F*

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

## Matrix Beta/Dirichlet

The two wikipedia summaries are sparse:

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

## References

*Foundations and Trends® in Machine Learning*11 (2): 97–218.

*Symmetric Multivariate and Related Distributions*. Boca Raton: Chapman and Hall/CRC.

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

*SIAM Journal on Matrix Analysis and Applications*12 (2): 239–72.

*Journal of Multivariate Analysis*100 (9): 1989–2001.

*Journal of Multivariate Analysis*39 (1): 135–53.

*Linear Algebra and Its Applications*, Tenth Special Issue (Part 2) on Linear Algebra and Statistics, 410 (November): 198–216.

*Annals of the Institute of Statistical Mathematics*44 (1): 97–106.

*Journal of Multivariate Analysis*175 (January): 104560.

*Journal of Multivariate Analysis*130 (September): 155–75.

*arXiv:1201.3256 [Math]*, January.

*Scandinavian Journal of Statistics*20 (3): 251–61.

*Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics*, 564–71. PMLR.

*Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*, 736–44. UAI’11. Arlington, Virginia, United States: AUAI Press.

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