Open Access
May 2020 Fano’s Inequality for Random Variables
Sébastien Gerchinovitz, Pierre Ménard, Gilles Stoltz
Statist. Sci. 35(2): 178-201 (May 2020). DOI: 10.1214/19-STS716


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.


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Sébastien Gerchinovitz. Pierre Ménard. Gilles Stoltz. "Fano’s Inequality for Random Variables." Statist. Sci. 35 (2) 178 - 201, May 2020.


Published: May 2020
First available in Project Euclid: 3 June 2020

MathSciNet: MR4106600
Digital Object Identifier: 10.1214/19-STS716

Keywords: Bayesian posterior concentration , information theory , lower bounds , Multiple-hypotheses testing

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.35 • No. 2 • May 2020
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