Information measures can be constructed from R\'enyi divergences much like mutual information from Kullback-Leibler divergence. One such information measure is known as Sibson's $\alpha$-mutual information and has received renewed attention recently in several contexts: concentration of measure under dependence, statistical learning, hypothesis testing, and estimation theory. In this paper, we survey and extend the state of the art. In particular, we introduce variational representations for Sibson's $\alpha$-mutual information and employ them in each of the contexts just described to derive novel results. Namely, we produce generalized Transportation-Cost inequalities and Fano-type inequalities. We also present an overview of known applications, spanning from learning theory and Bayesian risk to universal prediction.

翻译：信息测度可以从Rényi散度构建，类似于互信息从Kullback-Leibler散度构建。这类信息测度之一称为Sibson的$\alpha$-互信息，近年来在多个领域重新受到关注：依赖下的测度集中、统计学习、假设检验和估计理论。本文综述并扩展了现有研究进展。特别地，我们引入了Sibson的$\alpha$-互信息的变分表示，并将其应用于上述各领域以推导新颖结果。即，我们生成了广义运输成本不等式和Fano型不等式。此外，我们概述了已知应用，涵盖学习理论、贝叶斯风险及通用预测等领域。