Extropy: a complementary dual of entropy

This article resolves a longstanding question in the axiomatisation of entropy as proposed by Shannon and highlighted in renewed concerns expressed by Jaynes. We introduce a companion measure of a probability distribution that we suggest be called the extropy of the distribution. The entropy and the extropy of an event distribution are identical. However, this identical measure bifurcates into distinct measures for any quantity that is not merely an event indicator. As for entropy, the maximum extropy distribution is also the uniform distribution. We display several theoretical and geometrical properties of the proposed extropy measure, discussing in detail the difference between its assessment of a refined probability distribution and the axiom that characterises the Shannon entropy in this regard. This is what resolves the concerns of Shannon and Jaynes. In a discrete context, the extropy measure is approximated by a variant of Gini’s index of heterogeneity when the maximum probability mass is small. This is related to the “repeat rate” of a mass function as studied by Turing and Good. The continuous analogue of extropy turns out to equal the negative integral of the square of the density function. We conclude with a consideration of a rescaled measure of extropy which identifies it as the dual of entropy. The structure of the duality suggests a general theory of complementary distributions.