Weighted Relative Group Entropies and Associated Fisher Metrics

Abstract

A large family of new α-weighted group entropy functionals is defined and associated Fisher-like metrics are considered. All these notions are well-suited semi-Riemannian tools for the geometrization of entropy-related statistical models, where they may act as sensitive controlling invariants. The main result of the paper establishes a link between such a metric and a canonical one. A sufficient condition is found, in order that the two metrics be conformal (or homothetic). In particular, we recover a recent result, established for α=1 and for non-weighted relative group entropies. Our conformality condition is “universal”, in the sense that it does not depend on the group exponential.

Some Information Measures Properties of the GOS-Concomitants from the FGM Family

“…When , Tsallis entropy approaches to Shannon entropy. Tsallis entropy has, in turn, various generalizations, see, for example, [ 32 ].…”

Permutation group entropy: A new route to complexity for real-valued processes

“…In this section, we shall construct new complexity measures based on the notion of group entropy. As we mentioned in the Introduction, they represent a large class of generalized entropies, first introduced and discussed in [42], and further extended in a series of papers (see, e.g., [9,10,22,26,28,29,40,[43][44][45]49]). In principle, group entropy represents a versatile tool, that can be used to classify many known entropies and to design new ones, useful for specific purposes.…”

“…This generalized entropy includes very many of the trace-form entropies defined in the literature in the last decades. It represents a weak formulation of the notion of group entropy, which can still be used for many information-theoretical purposes (see for instance the recent works [22], [26]).…”

"Lie symmetries of the nonlinear Fokker-Planck equation based on weighted Tsallis entropy"

“…Entropy is too general a subject to be treated superficially here, even if only historically. We recall only the papers related to the NFPE associated to the Tsallis, Kaniadakis, Sharma-Taneja-Mittal entropies [56,62] with references therein and the surveys [27,59].…”

“…Subsequently, the weighted entropy was characterized axiomatically ( [22,23]). Some recent contributions about the weighted entropies include [2,8,24,27,31,35,47,48,57,58], to quote but a few.…”