A closed-form expression for the Sharma-Mittal entropy of exponential families

TL;DR

This paper derives a closed-form expression for Sharma-Mittal entropies applicable to exponential family distributions, including explicit formulas for multivariate Gaussian cases, facilitating easier entropy computation and estimation.

Contribution

It provides the first closed-form formula for Sharma-Mittal entropies of exponential families, expanding the tools for entropy analysis in statistical models.

Findings

Closed-form expression for Sharma-Mittal entropy of exponential families

Explicit formula for multivariate Gaussian distributions

Discussion on entropy estimation methods

Abstract

The Sharma-Mittal entropies generalize the celebrated Shannon, R\'enyi and Tsallis entropies. We report a closed-form formula for the Sharma-Mittal entropies and relative entropies for arbitrary exponential family distributions. We instantiate explicitly the formula for the case of the multivariate Gaussian distributions and discuss on its estimation.