R\'enyi divergence is related to R\'enyi entropy much like Kullback-Leibler
divergence is related to Shannon's entropy, and comes up in many settings. It
was introduced by R\'enyi as a measure of information that satisfies almost the
same axioms as Kullback-Leibler divergence, and depends on a parameter that is
called its order. In particular, the R\'enyi divergence of order 1 equals the
Kullback-Leibler divergence.
We review and extend the most important properties of R\'enyi divergence and
Kullback-Leibler divergence, including convexity, continuity, limits of
σ-algebras and the relation of the special order 0 to the Gaussian
dichotomy and contiguity. We also show how to generalize the Pythagorean
inequality to orders different from 1, and we extend the known equivalence
between channel capacity and minimax redundancy to continuous channel inputs
(for all orders) and present several other minimax results.Comment: To appear in IEEE Transactions on Information Theor
