Variational Characterizations of Sibson's $\alpha$-Mutual Information

Sibson's $\alpha$ -mutual information has received renewed attention recently in several contexts: concentration of measure under dependence, statistical learning, hypothesis testing, and estimation theory. In this work, we introduce several variational representations of Sibson's $\alpha$ -mutual information: 1) as a supremum over joint distributions of (a combination of) KL divergences; and 2) as a supremum over functions of opportune expected values. Leveraging them, we produce a variety of novel and known results, including a generalization of transportation-cost inequalities and Fano's inequality.