An entropy functional bounded from above by one

TL;DR

This paper introduces a new entropy functional based on Jensen-Shannon divergence that is bounded by one and effectively reflects alphabet size, improving the characterization of discrete random variables.

Contribution

It proposes a novel entropy measure bounded by one that better captures alphabet size compared to normalized Shannon entropy.

Findings

The new entropy functional is strictly increasing with alphabet size under uniformity.

It provides a more accurate measure of uncertainty for discrete variables.

The functional is bounded from above by one, unlike traditional normalized Shannon entropy.

Abstract

Shannon entropy is widely used to quantify the uncertainty of discrete random variables. But when normalized to the unit interval, as is often done in practice, it no longer conveys the alphabet sizes of the random variables being studied. This work introduces an entropy functional based on Jensen-Shannon divergence that is naturally bounded from above by one. Unlike normalized Shannon entropy, this new functional is strictly increasing in alphabet size under uniformity and is thus well suited to the characterization of discrete random variables.