Entropy Bounds for Discrete Random Variables via Maximal Coupling

TL;DR

This paper introduces new entropy difference bounds for discrete random variables using maximal coupling, applicable to finite or countably infinite alphabets, with applications demonstrated in Poisson approximation.

Contribution

It provides novel bounds on entropy differences based on local and total variation distances, extending previous results through the use of maximal coupling.

Findings

New bounds relate entropy differences to variation distances.

Bounds are applicable to finite and countably infinite alphabets.

Application to Poisson approximation using Stein's method.

Abstract

This paper derives new bounds on the difference of the entropies of two discrete random variables in terms of the local and total variation distances between their probability mass functions. The derivation of the bounds relies on maximal coupling, and they apply to discrete random variables which are defined over finite or countably infinite alphabets. Loosened versions of these bounds are demonstrated to reproduce some previously reported results. The use of the new bounds is exemplified for the Poisson approximation, where bounds on the local and total variation distances follow from Stein's method.