A new class of entropic information measures, formal group theory and information geometry

TL;DR

This paper develops a group-theoretical framework for generalized entropies and their information geometric properties, proposing methods to create new entropies and divergences with specific structural features.

Contribution

It introduces a novel group-theoretical approach to generalized entropies and divergences, linking algebraic structures with information geometry.

Findings

Established conditions for natural properties of entropies within a group framework

Proposed methods to generate new entropies and divergences from known ones

Analyzed the Riemannian structure associated with the divergences

Abstract

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated.