Extropy’s Limitations as a Complementary Dual of Entropy and the Role of Informity

In their 2015 paper, “Extropy: Complementary Dual of Entropy,” Lad et al. introduced the concept of extropy as a proposed complementary dual of entropy, with distinct definitions for discrete and continuous distributions. This commentary argues that extropy suffers from two primary issues: an ad hoc transition from discrete to continuous extropy, lacking theoretical rigor, and a contradictory interpretation as a measure of certainty, undermined by its equivalence to entropy in the binary case and negative continuous form. Consequently, extropy is not a fully coherent complementary dual of entropy. We demonstrate that informity, introduced by Huang[1], quantifies certainty consistently across discrete and continuous distributions, serving as the true complementary dual to entropy’s uncertainty measure. We compare informity, entropy, and extropy with two examples.