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We derive a closed form solution for the Kullback-Leibler divergence between two Weibull distributions. These notes are meant as reference material and intended to provide a guided tour towards a result that is often mentioned but seldom…

Information Theory · Computer Science 2013-10-15 Christian Bauckhage

We derive the Kullback-Leibler divergence for the normal-gamma distribution and show that it is identical to the Bayesian complexity penalty for the univariate general linear model with conjugate priors. Based on this finding, we provide…

Statistics Theory · Mathematics 2016-11-07 Joram Soch , Carsten Allefeld

Discrete normal distributions are defined as the distributions with prescribed means and covariance matrices which maximize entropy on the integer lattice support. The set of discrete normal distributions form an exponential family with…

Information Theory · Computer Science 2022-01-25 Frank Nielsen

We derive a closed-form solution for the Kullback-Leibler divergence between two Fr\'echet extreme-value distributions. The resulting expression is rather simple and involves the Euler-Mascheroni constant.

Statistics Theory · Mathematics 2023-03-28 Jean-Christophe Pain

We report a closed-form expression for the Kullback-Leibler divergence between Cauchy distributions which involves the calculation of a novel definite integral. The formula shows that the Kullback-Leibler divergence between Cauchy densities…

Information Theory · Computer Science 2019-05-29 Frédéric Chyzak , Frank Nielsen

In this note, we characterize the Gompertz distribution in terms of extreme value distributions and point out that it implicitly models the interplay of two antagonistic growth processes. In addition, we derive a closed form expressions for…

Information Theory · Computer Science 2014-02-14 Christian Bauckhage

The Kullback-Leibler (KL) divergence is a fundamental equation of information theory that quantifies the proximity of two probability distributions. Although difficult to understand by examining the equation, an intuition and understanding…

Information Theory · Computer Science 2014-04-09 Jonathon Shlens

Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably…

Machine Learning · Computer Science 2017-02-01 Frank Nielsen , Ke Sun

This document shows how to obtain the Jacobian and Hessian matrices of the Kullback-Leibler divergence between two multivariate Gaussian distributions, using the first and second-order differentials. The presented derivations are based on…

Machine Learning · Computer Science 2025-07-01 Juan Maroñas

This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution $U$ which maximizes the KL…

Machine Learning · Computer Science 2020-12-11 Vincenzo Bonnici

This short note is on a property of the Kullback-Leibler (KL) divergence which indicates that independent Gaussian distributions minimize the KL divergence from given independent Gaussian distributions. The primary purpose of this note is…

Information Theory · Computer Science 2020-12-04 Song Fang , Quanyan Zhu

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be…

Probability · Mathematics 2016-05-20 Daniel Sanz-Alonso , Andrew M. Stuart

In many applications in biology, engineering and economics, identifying similarities and differences between distributions of data from complex processes requires comparing finite categorical samples of discrete counts. Statistical…

Methodology · Statistics 2023-07-11 Francesco Camaglia , Ilya Nemenman , Thierry Mora , Aleksandra M. Walczak

Kullback-Leibler (KL) divergence is a fundamental concept in information theory that quantifies the discrepancy between two probability distributions. In the context of Variational Autoencoders (VAEs), it serves as a central regularization…

Machine Learning · Computer Science 2026-04-14 Andrés Muñoz , Rodrigo Ramele

The Kullback-Leibler divergence or relative entropy is an information-theoretic measure between statistical models that play an important role in measuring a distance between random variables. In the study of complex systems, random fields…

Information Theory · Computer Science 2022-03-25 Alexandre L. M. Levada

The generalized gamma distribution shows up in many problems related to engineering, hydrology as well as survival analysis. Earlier work has been done that estimated the deviation of the exponential and the Weibull distribution from…

Probability · Mathematics 2022-11-10 Zelong Bi , Irfan Durmić , Steven J. Miller

Kullback-Leibler (KL) divergence is one of the most important divergence measures between probability distributions. In this paper, we prove several properties of KL divergence between multivariate Gaussian distributions. First, for any two…

Information Theory · Computer Science 2023-01-24 Yufeng Zhang , Wanwei Liu , Zhenbang Chen , Ji Wang , Kenli Li

We generalise the classical Pinsker inequality which relates variational divergence to Kullback-Liebler divergence in two ways: we consider arbitrary f-divergences in place of KL divergence, and we assume knowledge of a sequence of values…

Information Theory · Computer Science 2009-06-09 Mark D. Reid , Robert C. Williamson

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by Bach [2022]. Unlike the classical Kullback-Leibler (KL) divergence that involves…

Machine Learning · Statistics 2025-03-12 Clémentine Chazal , Anna Korba , Francis Bach

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a` la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler…

Optimization and Control · Mathematics 2009-11-04 Augusto Ferrante , Federico Ramponi , Francesco Ticozzi
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