  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.
