Guaranteed bounds on information-theoretic measures of univariate mixtures using piecewise log-sum-exp inequalities

Abstract

Information-theoreticmeasures, such as the entropy, the cross-entropy and the Kullback-Leibler divergence between two mixture models, are core primitives in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte Carlo stochastic integration, approximated or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy, the Kullback-Leibler and the α-divergences of mixtures. We illustrate the versatile method by reporting our experiments for approximating the Kullback-Leibler and the α-divergences between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures and Gamma mixtures.