On some properties of Tsallis hypoentropies and hypodivergences

TL;DR

This paper extends Ferreri's hypoentropy to Tsallis statistics, introducing Tsallis hypoentropy and hypodivergence, and explores their fundamental mathematical properties to address limitations of traditional divergence measures.

Contribution

It introduces the Tsallis hypoentropy and hypodivergence, expanding Ferreri's concept within Tsallis statistics and analyzing their key properties.

Findings

Established nonnegativity of Tsallis hypoentropy and hypodivergence.

Proved monotonicity and chain rule properties.

Demonstrated subadditivity of the new measures.

Abstract

Both the Kullback-Leibler and the Tsallis divergence have a strong limitation: if the value $0$ appears in probability distributions $\left( p_{1},\cdots ,p_{n}\right)$ and $\left( q_{1},\cdots ,q_{n}\right)$, it must appear in the same positions for the sake of significance. In order to avoid that limitation in the framework of Shannon statistics, Ferreri introduced in 1980 the hypoentropy: "such conditions rarely occur in practice". The aim of the present paper is to extend Ferreri's hypoentropy to the Tsallis statistics. We introduce the Tsallis hypoentropy and the Tsallis hypodivergence and describe their mathematical behavior. Fundamental properties like nonnegativity, monotonicity, the chain rule and subadditivity are established.