A Probabilistic Upper Bound on Differential Entropy

A novel, non-trivial, probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution, is presented. No knowledge beyond the support of the unknown distribution is required, nor is the distribution required to have a density. Previous distribution-free bounds on the cumulative distribution function of a random variable given a sample of that variable are used to construct the bound. A simple, fast, and intuitive algorithm for computing the entropy bound from a sample is provided

Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities

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 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 and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures.Comment: 20 pages, 3 figure

Determining the Number of Samples Required to Estimate Entropy in Natural Sequences

Calculating the Shannon entropy for symbolic sequences has been widely considered in many fields. For descriptive statistical problems such as estimating the N-gram entropy of English language text, a common approach is to use as much data as possible to obtain progressively more accurate estimates. However in some instances, only short sequences may be available. This gives rise to the question of how many samples are needed to compute entropy. In this paper, we examine this problem and propose a method for estimating the number of samples required to compute Shannon entropy for a set of ranked symbolic natural events. The result is developed using a modified Zipf-Mandelbrot law and the Dvoretzky-Kiefer-Wolfowitz inequality, and we propose an algorithm which yields an estimate for the minimum number of samples required to obtain an estimate of entropy with a given confidence level and degree of accuracy

A Probabilistic Upper Bound on Differential Entropy

Abstract — The differential entropy is a quantity employed ubiquitously in communications, statistical learning, physics, and many other fields. We present, to our knowledge, the first non-trivial probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution. The bound is completely general in that it does not depend in any way on the form of the unknown distribution (among other things, it does not require that the distribution have a density). Our bound uses previous distribution-free bounds on the cumulative distribution function of a random variable given a sample of that variable. We provide a simple, fast, and intuitive algorithm for computing the entropy bound from a sample. I

A Probabilistic Upper Bound on Differential Entropy

Abstract — A novel, non-trivial, probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution, is presented. No knowledge beyond the support of the unknown distribution is required. Previous distribution-free bounds on the cumulative distribution function of a random variable given a sample of that variable are used to construct the bound. A simple, fast, and intuitive algorithm for computing the entropy bound from a sample is provided. I