Fano's inequality for random variables
Résumé
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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Fano-HAL-final.pdf (516.26 Ko)
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Im-diff.pdf (15.68 Ko)
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Im-varphi.pdf (16.85 Ko)
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