Properties of a Generalized Divergence Related to Tsallis Relative Entropy

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 th...

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Bibliographic Details
Published in:arXiv.org 2020-04
Main Authors: Vigelis, Rui F, Luiza H F de Andrade, Cavalcante, Charles C
Format: Article
Language:English
Subjects:
Convexity
Deformation
Divergence
Entropy
Normalizing (statistics)
Statistical analysis
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1810.09503