Matrix-valued random variates

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.


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

Matrix Gaussian

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

Matrix Gamma

Currently handled under gamma processes.


Inverse Wishart

Random rotations

See random rotations.


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).


Bishop, Adrian N., Pierre Del Moral, and Angèle Niclas. 2018. An Introduction to Wishart Matrix Moments.” Foundations and Trends® in Machine Learning 11 (2): 97–218.
Edelman, Alan. 1989. Eigenvalues and Condition Numbers of Random Matrices.”
Fang, Kai-Tai, Samuel Kotz, and Kai Wang Ng. 2017. Symmetric Multivariate and Related Distributions. Boca Raton: Chapman and Hall/CRC.
Gupta, A. K., and D. K. Nagar. 1999. Matrix Variate Distributions. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 104. Boca Raton: Chapman and Hall/CRC.
Holmes, R. 1991. On Random Correlation Matrices.” SIAM Journal on Matrix Analysis and Applications 12 (2): 239–72.
Lewandowski, Daniel, Dorota Kurowicka, and Harry Joe. 2009. Generating Random Correlation Matrices Based on Vines and Extended Onion Method.” Journal of Multivariate Analysis 100 (9): 1989–2001.
Mathai, A. M., and P. G. Moschopoulos. 1991. On a Multivariate Gamma.” Journal of Multivariate Analysis 39 (1): 135–53.
Mathai, A. M., and Serge B. Provost. 2005. Some Complex Matrix-Variate Statistical Distributions on Rectangular Matrices.” Linear Algebra and Its Applications, Tenth Special Issue (Part 2) on Linear Algebra and Statistics, 410 (November): 198–216.
Mathal, A. M., and P. G. Moschopoulos. 1992. A Form of Multivariate Gamma Distribution.” Annals of the Institute of Statistical Mathematics 44 (1): 97–106.
Meier, Alexander. 2018. A matrix Gamma process and applications to Bayesian analysis of multivariate time series.”
Meier, Alexander, Claudia Kirch, and Renate Meyer. 2020. Bayesian Nonparametric Analysis of Multivariate Time Series: A Matrix Gamma Process Approach.” Journal of Multivariate Analysis 175 (January): 104560.
Pérez-Abreu, Victor, and Robert Stelzer. 2014. Infinitely Divisible Multivariate and Matrix Gamma Distributions.” Journal of Multivariate Analysis 130 (September): 155–75.
Pfaffel, Oliver. 2012. Wishart Processes.” arXiv:1201.3256 [Math], January.
Singpurwalla, Nozer D., and Mark A. Youngren. 1993. Multivariate Distributions Induced by Dynamic Environments.” Scandinavian Journal of Statistics 20 (3): 251–61.
Thibaux, Romain, and Michael I. Jordan. 2007. Hierarchical Beta Processes and the Indian Buffet Process.” In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, 564–71. PMLR.
Wilson, Andrew Gordon, and Zoubin Ghahramani. 2011. Generalised Wishart Processes.” In 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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