On $f$-Divergences: Integral Representations, Local Behavior, and Inequalities

TL;DR

This paper explores $f$-divergences by providing integral representations, new inequality derivations, and analyzing their local behavior, enhancing understanding and application in statistical hypothesis testing.

Contribution

It introduces integral representations of $f$-divergences, a novel approach for deriving inequalities, and studies their local behavior, advancing theoretical understanding.

Findings

Integral representations via the relative information spectrum

New inequalities for $f$-divergences derived

Insights into local behavior of $f$-divergences

Abstract

This paper is focused on $f$-divergences, consisting of three main contributions. The first one introduces integral representations of a general $f$-divergence by means of the relative information spectrum. The second part provides a new approach for the derivation of $f$-divergence inequalities, and it exemplifies their utility in the setup of Bayesian binary hypothesis testing. The last part of this paper further studies the local behavior of $f$-divergences.