Fano's inequality for random variables

Sebastien Gerchinovitz (1) , Pierre Ménard (1) , Gilles Stoltz (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC
3 LMO - Laboratoire de Mathématiques d'Orsay

We extend Fano's inequality, which controls the average probability of events in terms of the average of some f--divergences, to work with arbitrary events (not necessarily forming a partition) and even with arbitrary [0,1]--valued random variables, possibly in continuously infinite number. We provide two applications of these extensions, in which the consideration of random variables is particularly handy: we offer new and elegant proofs for existing lower bounds, on Bayesian posterior concentration (minimax or distribution-dependent) rates and on the regret in non-stochastic sequential learning.

Mots clés: Multiple-hypotheses testing; Lower bounds; Information theory; Bayesian posterior concentration

Domaines: Théorie [stat.TH]; Théorie de l'information [cs.IT]

Sebastien Gerchinovitz, Pierre Ménard, Gilles Stoltz. Fano's inequality for random variables. 2018. ⟨hal-01470862v2⟩