Deformed statistics Kullback–Leibler divergence minimization within a scaled Bregman framework

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.