A counterexample against the Lesche stability of a generic entropy functional

TL;DR

This paper presents a counterexample demonstrating that a broad class of entropy functionals, defined as sums of a function g over probabilities, are not necessarily Lesche stable, challenging assumptions about their robustness.

Contribution

The paper provides the first counterexample showing that generic entropy functionals may lack Lesche stability without additional conditions on g.

Findings

Counterexample disproves Lesche stability for certain entropy forms

Stability depends on specific properties of the function g

Lesche stability cannot be assumed for all concave, analytic functions g

Abstract

We provide a counterexample to show that the generic form of entropy S(p)=sum_i g(p_i) is not always stable against small variation of probability distribution (Lesche stability) even if is concave function on [0,1] and analytic on ]0,1]. Our conclusion is that the stability of such a generic functional needs more hypotheses on the property of the function g, or in other words, the stability of entropy cannot be discussed at this formal stage.