On Sibson's $\alpha$-Mutual Information

TL;DR

This paper extends Sibson's $ ext{α}$-Mutual Information to negative $ ext{α}$ values, exploring its properties and connections to functional inequalities relevant for Bayesian estimation risk bounds.

Contribution

It introduces a novel extension of Sibson's $ ext{α}$-Mutual Information to negative $ ext{α}$, analyzing its properties and applications in functional inequalities.

Findings

Extended the definition of Sibson's $ ext{α}$-Mutual Information to negative $ ext{α}$ values.

Established properties of the extended information measure.

Linked the measure to functional inequalities used in Bayesian risk bounds.

Abstract

We explore a family of information measures that stems from R\'enyi's $\alpha$-Divergences with $\alpha<0$. In particular, we extend the definition of Sibson's $\alpha$-Mutual Information to negative values of $\alpha$ and show several properties of these objects. Moreover, we highlight how this family of information measures is related to functional inequalities that can be employed in a variety of fields, including lower-bounds on the Risk in Bayesian Estimation Procedures.