Fano's inequality for random variables
Fano's inequality for random variables
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随机变量的 Fano 不等式
作者：
Sébastien Gerchinovitz;Pierre Ménard;Gilles Stoltz
我们延长法诺不等式，控制事件的平均概率的平均值的一些$f$-发散，工作与任意事件（不一定形成一个分区），甚至与任意的$[0，1]$-值随机变量，可能在连续无穷多。我们提供了这些扩展的两个应用程序，其中考虑随机变量是特别方便：我们提供了新的和优雅的证明现有的下界，贝叶斯后验浓度（极大极小或分布依赖）率和非随机顺序学习的遗憾。
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.