Extropy: Complementary Dual of Entropy

TL;DR

This paper introduces 'extropy' as a complementary dual to Shannon's entropy, providing a new perspective on information measures, their properties, and their dual relationships, including applications to statistical forecasting.

Contribution

It formally defines extropy as the dual measure to entropy, explores its properties, and establishes its relationship with divergence measures like Kullback-Leibler, unifying these concepts within Bregman divergences.

Findings

Entropy and extropy are identical for binary distributions.

Maximum extropy distribution is the uniform distribution.

Relative extropy is dual to Kullback-Leibler divergence.

Abstract

This article provides a completion to theories of information based on entropy, resolving a longstanding question in its axiomatization as proposed by Shannon and pursued by Jaynes. We show that Shannon's entropy function has a complementary dual function which we call "extropy." The entropy and the extropy of a binary distribution are identical. However, the measure bifurcates into a pair of distinct measures for any quantity that is not merely an event indicator. As with entropy, the maximum extropy distribution is also the uniform distribution, and both measures are invariant with respect to permutations of their mass functions. However, they behave quite differently in their assessments of the refinement of a distribution, the axiom which concerned Shannon and Jaynes. Their duality is specified via the relationship among the entropies and extropies of course and fine partitions. We also analyze the extropy function for densities, showing that relative extropy constitutes a dual to the Kullback-Leibler divergence, widely recognized as the continuous entropy measure. These results are unified within the general structure of Bregman divergences. In this context they identify half the $L_2$ metric as the extropic dual to the entropic directed distance. We describe a statistical application to the scoring of sequential forecast distributions which provoked the discovery.