Numerical Stability of Generalized Entropies

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.