Information Measures: the Curious Case of the Binary Alphabet

TL;DR

This paper investigates unique properties of information divergence measures on binary and larger alphabets, revealing surprising differences and clarifying the special roles of KL divergence and f-divergences.

Contribution

It demonstrates that certain divergence characterizations do not extend from larger alphabets to binary cases, and establishes the unique invariance properties of KL divergence.

Findings

f-divergences are not the only decomposable divergences satisfying data processing on binary alphabets

KL divergence is the only Bregman divergence that is also an f-divergence for any alphabet size

KL divergence is uniquely invariant under sufficient transformations when alphabet size is at least three

Abstract

Four problems related to information divergence measures defined on finite alphabets are considered. In three of the cases we consider, we illustrate a contrast which arises between the binary-alphabet and larger-alphabet settings. This is surprising in some instances, since characterizations for the larger-alphabet settings do not generalize their binary-alphabet counterparts. Specifically, we show that $f$-divergences are not the unique decomposable divergences on binary alphabets that satisfy the data processing inequality, thereby clarifying claims that have previously appeared in the literature. We also show that KL divergence is the unique Bregman divergence which is also an $f$-divergence for any alphabet size. We show that KL divergence is the unique Bregman divergence which is invariant to statistically sufficient transformations of the data, even when non-decomposable divergences are considered. Like some of the problems we consider, this result holds only when the alphabet size is at least three.