Maximizable informational entropy as measure of probabilistic uncertainty

TL;DR

This paper investigates a variationally defined entropy measure, called varentropy, analyzing its maximization properties across various probability distributions to understand its role as a measure of uncertainty.

Contribution

It provides an analytical study of the maximizable varentropy for different distributions, extending the understanding of entropy measures based on variational principles.

Findings

Varentropy underpins a generalized virtual work principle.

Maximum entropy condition derived from variational relationship.

Analytical results for stretched exponential, kappa-exponential, and Cauchy distributions.

Abstract

In this work, we consider a recently proposed entropy S (called varentropy) defined by a variational relationship dI=beta*(d<x>-<dx>) as a measure of uncertainty of random variable x. By definition, varentropy underlies a generalized virtual work principle <dx>=0 leading to maximum entropy d(I-beta*<x>)=0. This paper presents an analytical investigation of this maximizable entropy for several distributions such as stretched exponential distribution, kappa-exponential distribution and Cauchy distribution.