Fano’s inequality for random variables Article - Juin 2019

Sebastien Gerchinovitz, Pierre Ménard, Gilles Stoltz

Sebastien Gerchinovitz, Pierre Ménard, Gilles Stoltz, « Fano’s inequality for random variables  », Statistical Science, à paraître. ISSN 0883-4237

Abstract

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.

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