Mean Entropies

TL;DR

This paper explores the mathematical foundations of entropy measures, linking them to specific types of means and their properties, and clarifies their additivity, invariance, and relation to distribution measures.

Contribution

It establishes the connection between entropies and Kolmogorov-Nagumo means, clarifies the properties of pseudo-additive entropies, and discusses the limitations of escort averages.

Findings

Shannon entropy corresponds to the weighted arithmetic mean.

Renyi entropy corresponds to the exponential mean.

Exponential mean error functions measure the extent of a distribution.

Abstract

Entropies must correspond to mean values for them to be measurable. The Shannon entropy corresponds to the weighted arithmetic mean, whereas the Renyi entropy corresponds to the exponential mean. These means refer to code lengths, which are converted into entropies by replacing the length of a sequence by the negative logarithm of the probability of its occurrence. Only affine and exponential generating functions of means preserve the property of additivity and invariance under translations, and hence are Kolmogorov-Nagumo functions, resulting in the Shannon and Renyi entropies, respectively. Pseudo-additive entropies are generating functions of means of order 0\le\tau<1, which is the exponential Renyi entropy, or, in the \tau=0 limit, the Shannon entropy. Means of any order cannot be expressed as escort averages because such averages contradict the fact that the means are monotonically increasing functions of their order. Exponential mean error functions of Renyi, in general, and Shannon, in particular, are shown to be measures of the extent of a distribution.