A Note on Mathai's Entropy Measure

TL;DR

This paper explores Mathai's entropy measure, demonstrating its invariance across variable types and its applicability in deriving various distributions without the need for escort distributions, unlike Tsallis' entropy.

Contribution

It introduces the properties of Mathai's entropy, showing its consistency across different variable types and its direct application in deriving diverse distributions.

Findings

Mathai's entropy is invariant for real, complex, vector, and matrix variables.

Optimization of Mathai's entropy yields various applicable distributions.

Mathai's entropy does not require escort distributions unlike Tsallis' entropy.

Abstract

In a paper [8] the authors classify entropy into three categories, as a thermodynamics quantity, as a measure of information production, and as a means of statistical inference. An entropy measure introduced by Mathai falls into the second and third categories. It is shown that this entropy measure is the same whether the variables involved are real or complex scalar, vector, or matrix variables. If the entropy measure is optimized under some moment-like conditions then one can obtain various types of densities which are applicable in different areas. Unlike Tsallis' entropy [9], it does not need an intermediary escort distribution to yield the desired results. Calculus of variation can be directly applied to obtain the desired results under Mathai's entropy. Tsallis' entropy, which is the basis of the area of non-extensive statistical mechanics, is a modified version of the $\alpha$-generalized entropy of Havrda-Charvat considered in [7]. Various types of distributions that can be obtained through optimization of Mathai's entropy, are illustrated in this paper.