A two-parameter entropy and its fundamental properties
Supriyo Dutta, Shigeru Furuichi, Partha Guha
TL;DR
This paper introduces a two-parameter generalization of entropy, denoted $S_{k,r}$, built from a two-parameter deformed logarithm $\ln_{k,r}$ and paired with a generalized divergence $D_{k,r}$. It shows that $S_{k,r}$ reduces to the Tsallis entropy for $k=r=(q-1)/2$ and that $D_{k,r}$ extends Sharma–Mittal and Tsallis divergences, establishing fundamental properties including chain rule, sub-additivity, strong sub-additivity, joint convexity, and information monotonicity. It then analyzes the information-geometric structure by deriving a Hessian metric on the probability simplex, showing the induced manifold is Hassian. These results provide a flexible two-parameter framework for information theory in complex systems and suggest future work on mutual information and two-parameter data-processing inequalities.
Abstract
This article proposes a new two-parameter generalized entropy, which can be reduced to the Tsallis and the Shannon entropy for specific values of its parameters. We develop a number of information-theoretic properties of this generalized entropy and divergence, for instance, the sub-additive property, strong sub-additive property, joint convexity, and information monotonicity. This article presents an exposit investigation on the information-theoretic and information-geometric characteristics of the new generalized entropy and compare them with the properties of the Tsallis and the Shannon entropy.