A Characterization of Entropy in Terms of Information Loss

TL;DR

This paper characterizes Shannon and Tsallis entropy through the concept of information loss, demonstrating that Shannon entropy uniquely satisfies functoriality, convex-linearity, and continuity in this context.

Contribution

It provides a simple, property-based characterization of Shannon entropy using information loss, extending naturally to Tsallis entropy.

Findings

Shannon entropy is uniquely characterized by functoriality, convex-linearity, and continuity.

The concept of information loss is a unifying framework for entropy.

The characterization generalizes to Tsallis entropy.

Abstract

There are numerous characterizations of Shannon entropy and Tsallis entropy as measures of information obeying certain properties. Using work by Faddeev and Furuichi, we derive a very simple characterization. Instead of focusing on the entropy of a probability measure on a finite set, this characterization focuses on the `information loss', or change in entropy, associated with a measure-preserving function. Information loss is a special case of conditional entropy: namely, it is the entropy of a random variable conditioned on some function of that variable. We show that Shannon entropy gives the only concept of information loss that is functorial, convex-linear and continuous. This characterization naturally generalizes to Tsallis entropy as well.