Matrix-valued random variates

December 1, 2021 — January 6, 2022

high d
linear algebra
signal processing
stochastic processes
Figure 1

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.


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

Bishop, Del Moral, and Niclas. 2018. An Introduction to Wishart Matrix Moments.” Foundations and Trends® in Machine Learning.
Edelman. 1989. Eigenvalues and Condition Numbers of Random Matrices.”
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. On Random Correlation Matrices.” SIAM Journal on Matrix Analysis and Applications.
Lewandowski, Kurowicka, and Joe. 2009. Generating Random Correlation Matrices Based on Vines and Extended Onion Method.” Journal of Multivariate Analysis.
Mathai, and Moschopoulos. 1991. On a Multivariate Gamma.” Journal of Multivariate Analysis.
Mathai, and 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,.
Mathal, and Moschopoulos. 1992. A Form of Multivariate Gamma Distribution.” Annals of the Institute of Statistical Mathematics.
Meier. 2018. A matrix Gamma process and applications to Bayesian analysis of multivariate time series.”
Meier, Kirch, and Meyer. 2020. Bayesian Nonparametric Analysis of Multivariate Time Series: A Matrix Gamma Process Approach.” Journal of Multivariate Analysis.
Pérez-Abreu, and Stelzer. 2014. Infinitely Divisible Multivariate and Matrix Gamma Distributions.” Journal of Multivariate Analysis.
Pfaffel. 2012. Wishart Processes.” arXiv:1201.3256 [Math].
Singpurwalla, and Youngren. 1993. Multivariate Distributions Induced by Dynamic Environments.” Scandinavian Journal of Statistics.
Thibaux, and Jordan. 2007. Hierarchical Beta Processes and the Indian Buffet Process.” In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics.
Wilson, and Ghahramani. 2011. Generalised Wishart Processes.” In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence. UAI’11.