Rényi Divergence and Kullback-Leibler Divergence

Tim van Erven (1, 2) , Peter Harremoës (3)

1 SELECT - Model selection in statistical learning
2 Département de Mathématiques [ORSAY]
3 Niels Brock Copenhagen Business College

Rényi divergence is related to Rényi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by Rényi 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 Rényi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of Rényi divergence and Kullback-Leibler divergence, including convexity, continuity, limits of {\sigma}-algebras and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders), and present several other minimax results.

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]; Théorie [stat.TH]; Machine Learning [stat.ML]

Tim van Erven, Peter Harremoës. Rényi Divergence and Kullback-Leibler Divergence. 2013. ⟨hal-00758191⟩