On Sibson's $\alpha$-Mutual Information

Abstract

We explore a family of information measures that stems from Rényi'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.

Submitted to ISIT 2022