Numerical Stability of Generalized Entropies

TL;DR

This paper investigates how generalized entropies, including Shannon, Tsallis, and Rényi, depend continuously on experimental errors measured by generalized L^p distances, providing stability conditions for their robustness.

Contribution

It introduces a framework linking generalized entropies to L^p distances and establishes stabilizing conditions ensuring entropy stability under experimental errors.

Findings

Generalized entropies depend continuously on experimental errors measured in L^p spaces.

Stabilizing conditions are derived for classical and generalized entropies.

Shannon entropy requires more restrictive conditions for stability.

Abstract

In many applications, the probability density function is subject to experimental errors. In this work the continuos dependence of a class of generalized entropies on the experimental errors is studied. This class includes the C. Shannon, C. Tsallis, A. R\'{e}nyi and generalized R\'{e}nyi entropies. By using the connection between R\'{e}nyi or Tsallis entropies, and the \textit{distance} in a the Lebesgue functional spaces, we introduce a further extensive generalizations of the R\'{e}nyi entropy. In this work we suppose that the experimental error is measured by some generalized $L^{p}$ distance. In line with the methodology normally used for treating the so called \textit{ill-posed problems}, auxiliary stabilizing conditions are determined, such that small - in the sense of $L^{p}$ metric - experimental errors provoke small variations of the classical and generalized entropies. These stabilizing conditions are formulated in terms of $L^{p}$ metric in a class of generalized $L^{p}$ spaces of functions. Shannon's entropy requires, however, more restrictive stabilizing conditions.