Deformed Statistics Kullback-Leibler Divergence Minimization within a Scaled Bregman Framework

TL;DR

This paper unifies a generalized form of Kullback-Leibler divergence from Tsallis statistics with Bregman divergence theory, revealing new geometric and minimization properties within a measure-theoretic framework.

Contribution

It demonstrates that the dual generalized K-L divergence is a scaled Bregman divergence and derives a Pythagorean theorem for this measure, highlighting unique minimization features.

Findings

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

Derived Pythagorean theorem for the divergence.

Identified unique features in the minimization process.

Abstract

The generalized Kullback-Leibler divergence (K-Ld) in Tsallis statistics [constrained by the additive duality of generalized statistics (dual generalized K-Ld)] is here reconciled with the theory of Bregman divergences for expectations defined by normal averages, within a measure-theoretic framework. Specifically, it is demonstrated that the dual generalized K-Ld is a scaled Bregman divergence. The Pythagorean theorem is derived from the minimum discrimination information-principle using the dual generalized K-Ld as the measure of uncertainty, with constraints defined by normal averages. The minimization of the dual generalized K-Ld, with normal averages constraints, is shown to exhibit distinctly unique features.