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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Sanov's theorem ： ウィキペディア英語版 Sanov's theorem In information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution. Let ''A'' be a set of probability distributions over an alphabet ''X'', and let ''q'' be an arbitrary distribution over ''X'' (where ''q'' may or may not be in ''A''). Suppose we draw ''n'' i.i.d. samples from ''q'', represented by the vector $x^n\; =\; x\_1,\; x\_2,\; \backslash ldots,\; x\_n$. Further, let us ask that the empirical distribution, $\backslash hat\_$, of the samples falls within the set ''A''—formally, we write $\backslash \; \backslash in\; A\backslash \}$. Then, :$q^n(x^n)\; \backslash le\; (n+1)^\; 2^$, where * $q^n(x^n)$ is shorthand for $q(x\_1)q(x\_2)\; \backslash cdots\; q(x\_n)$, and * $p^$ * is the information projection of ''q'' onto ''A''. In words, the probability of drawing an atypical distribution is proportional to the KL distance from the true distribution to the atypical one; in the case that we consider a set of possible atypical distributions, there is a dominant atypical distribution, given by the information projection. Furthermore, if ''A'' is a closed set, :$\backslash lim\_\backslash frac\backslash log\; q^n(x^n)\; =\; -D\_\{\backslash mathrm\{KL\}\}(p^$ *||q). ==References== * *Sanov, I. N. (1957) "On the probability of large deviations of random variables". ''Mat. Sbornik'' 42, 11–44. Category:Probabilistic inequalities 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sanov's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.