Fano’s Inequality for Random Variables

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 nonstochastic sequential learning.