Conditional Rényi Divergence Saddlepoint and the Maximization of α-Mutual Information

Rényi-type generalizations of entropy, relative entropy and mutual information have found numerous applications throughout information theory and beyond. While there is consensus that the ways A. Rényi generalized entropy and relative entropy in 1961 are the “right” ones, several candidates have been put forth as possible mutual informations of order α. In this paper we lend further evidence to the notion that a Bayesian measure of statistical distinctness introduced by R. Sibson in 1969 (closely related to Gallager’s E0 function) is the most natural generalization, lending itself to explicit computation and maximization, as well as closed-form formulas. This paper considers general (not necessarily discrete) alphabets and extends the major analytical results on the saddle-point and saddle-level of the conditional relative entropy to the conditional Rényi divergence. Several examples illustrate the main application of these results, namely, the maximization of α-mutual information with and without constraints.