Change the coefficients of conditional entropies in extensivity

TL;DR

This paper investigates how modifying the coefficients in the extensivity property of conditional entropies affects their form, showing that only power functions preserve extensivity and analyzing the differences for general functionals.

Contribution

It proves the impossibility of replacing coefficients with non-power functions in extensivity and provides estimates for general functionals.

Findings

Only power functions preserve extensivity in conditional entropies.

Non-power coefficient functions cannot maintain extensivity.

Provides bounds on the difference for general entropy functionals.

Abstract

The Boltzmann--Gibbs entropy is a functional on the space of probability measures. When a state space is countable, one characterization of the Boltzmann--Gibbs entropy is given by the Shannon--Khinchin axioms, which consist of continuity, maximality, expandability and extensivity. Among these four properties, the extensivity is generalized in various ways. The extensivity of a functional is interpreted as the property that, for any random variables $(X,Y)$ taking finitely many values in $\mathbb{N}$, the difference between the value of the functional at the joint law of $(X,Y)$ and that at the law of $X$ coincides with the linear combinations of the values at the conditional laws of $Y$ given $X=n$ with coefficients given by the probabilities of each event $X=n$. A generalization of the extensivity obtained by replacing the coefficients with a power of the probabilities of the events $X=n$ provides a characterization of the Tsallis entropy. In this paper, we first prove the impossibility to replace the coefficients with a non-power function of the probabilities of the events $X=n$. Then we estimate the difference between the value at the joint law of $(X,Y)$ and that at the law of $X$ for a general functional.