Fano’s Inequality for Random Variables

• Gerchinovitz, Sébastien ^[1] ; Ménard, Pierre ^[1] ; Stoltz, Gilles ^[2]
  1. [1] Paul Sabatier University

     Paul Sabatier University

     Arrondissement de Toulouse, Francia

     
  2. [2] Univerité Paris-Sud
• Localización: Statistical science, ISSN 0883-4237, Vol. 35, Nº. 2, 2020, págs. 178-201
• Idioma: inglés
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• Resumen

  • 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.