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Entropy
Volume 26
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10.3390/e26080663
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Article

Exact Expressions for Kullback–Leibler Divergence for Multivariate and Matrix-Variate Distributions

by
Victor Nawa
1 and
Saralees Nadarajah
2,*
1
Department of Mathematics and Statistics, University of Zambia, Lusaka 10101, Zambia
2
Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
*
Author to whom correspondence should be addressed.
Entropy 2024, 26(8), 663; https://doi.org/10.3390/e26080663
Submission received: 9 July 2024 / Revised: 2 August 2024 / Accepted: 3 August 2024 / Published: 4 August 2024
(This article belongs to the Section Information Theory, Probability and Statistics)
Versions Notes

Abstract

The Kullback–Leibler divergence is a measure of the divergence between two probability distributions, often used in statistics and information theory. However, exact expressions for it are not known for multivariate or matrix-variate distributions apart from a few cases. In this paper, exact expressions for the Kullback–Leibler divergence are derived for over twenty multivariate and matrix-variate distributions. The expressions involve various special functions.
Keywords: beta function; gamma function; matrix-variate beta function; matrix-variate confluent hypergeometric function; matrix-variate gamma function; matrix-variate Gauss hypergeometric function; matrix-variate type I Dirichlet density; type I Dirichlet density beta function; gamma function; matrix-variate beta function; matrix-variate confluent hypergeometric function; matrix-variate gamma function; matrix-variate Gauss hypergeometric function; matrix-variate type I Dirichlet density; type I Dirichlet density

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MDPI and ACS Style

Nawa, V.; Nadarajah, S. Exact Expressions for Kullback–Leibler Divergence for Multivariate and Matrix-Variate Distributions. Entropy 2024, 26, 663. https://doi.org/10.3390/e26080663

AMA Style

Nawa V, Nadarajah S. Exact Expressions for Kullback–Leibler Divergence for Multivariate and Matrix-Variate Distributions. Entropy. 2024; 26(8):663. https://doi.org/10.3390/e26080663

Chicago/Turabian Style

Nawa, Victor, and Saralees Nadarajah. 2024. "Exact Expressions for Kullback–Leibler Divergence for Multivariate and Matrix-Variate Distributions" Entropy 26, no. 8: 663. https://doi.org/10.3390/e26080663

APA Style

Nawa, V., & Nadarajah, S. (2024). Exact Expressions for Kullback–Leibler Divergence for Multivariate and Matrix-Variate Distributions. Entropy, 26(8), 663. https://doi.org/10.3390/e26080663

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