Scaled Bregman divergences in a Tsallis scenario

TL;DR

This paper explores the relationship between two versions of Kullback-Leibler divergence in Tsallis statistics, showing that one can be viewed as a scaled Bregman divergence, which unifies them.

Contribution

It establishes a condition for the consistency of generalized K-L divergences in Tsallis statistics and reveals that the dual form is a scaled Bregman divergence.

Findings

Dual generalized K-L divergence is a scaled Bregman divergence.

Dual generalized mutual information is a scaled Bregman information.

Conditions for consistency between generalized K-L divergences are derived.

Abstract

There exist two different versions of the Kullback-Leibler divergence (K-Ld) in Tsallis statistics, namely the usual generalized K-Ld and the generalized Bregman K-Ld. Problems have been encountered in trying to reconcile them. A condition for consistency between these two generalized K-Ld-forms by recourse to the additive duality of Tsallis statistics is derived. It is also shown that the usual generalized K-Ld subjected to this additive duality, known as the dual generalized K-Ld, is a scaled Bregman divergence. This leads to an interesting conclusion: the dual generalized mutual information is a scaled Bregman information. The utility and implications of these results are discussed.