Properties of a Generalized Divergence Related to Tsallis Relative Entropy

TL;DR

This paper explores a generalized divergence related to Tsallis entropy, establishing conditions for its convexity, partition inequality, and Pinsker's inequality, and identifying when it reduces to the Tsallis relative entropy.

Contribution

It introduces a new generalized divergence based on a deformed exponential, providing necessary and sufficient conditions for key inequalities and characterizing when it matches Tsallis entropy.

Findings

The generalized divergence satisfies the partition inequality.

Joint convexity holds only when it coincides with Tsallis relative entropy.

A criterion for Pinsker's inequality based on the partition inequality was established.

Abstract

In this paper, we investigate the partition inequality, joint convexity, and Pinsker's inequality, for a divergence that generalizes the Tsallis Relative Entropy and Kullback-Leibler divergence. The generalized divergence is defined in terms of a deformed exponential function, which replaces the Tsallis $q$-exponential. We also constructed a family of probability distributions related to the generalized divergence. We found necessary and sufficient conditions for the partition inequality to be satisfied. A sufficient condition for the joint convexity was established. We proved that the generalized divergence satisfies the partition inequality, and is jointly convex, if, and only if, it coincides with the Tsallis relative entropy. As an application of partition inequality, a criterion for the Pinsker's inequality was found.