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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 Importance sampling ： ウィキペディア英語版 Importance sampling In statistics, importance sampling is a general technique for estimating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. It is related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both. == Basic theory == Let $X:\backslash Omega\backslash to\; \backslash mathbb$ be a random variable in some probability space $(\backslash Omega,\backslash mathcal,P)$. We wish to estimate the expected value of ''X'' under ''P'', denoted E(). If we have random samples $x\_1,\; \backslash ldots,\; x\_n$, generated according to ''P'', then an empirical estimate of E() is : $$ \widehat() = \frac \sum_^n x_i. The basic idea of importance sampling is to change the probability measure ''P'' so that the estimation of E() is easier. Choose a random variable $L\backslash geq\; 0$ such that E''()=1'' and that ''P''-almost everywhere $L(\backslash omega)\backslash neq\; 0$. With the variate ''L'' we define another probability $P^:=L\backslash ,\; P$ that satisfies : $$ \mathbf() = \mathbf\left(). The variable ''X/L'' will thus be sampled under ''P(L)'' to estimate $\backslash mathbf()$ as above. This procedure will improve the estimation when . Another case of interest is when ''X/L'' is easier to sample under ''P(L)'' than ''X'' under ''P''. When ''X'' is of constant sign over Ω, the best variable ''L'' would clearly be $L^$ *=\frac\geq 0, so that ''X/L *'' is the searched constant E''()'' and a single sample under ''P(L *)'' suffices to give its value. Unfortunately we cannot take that choice, because E''()'' is precisely the value we are looking for! However this theoretical best case ''L *'' gives us an insight into what importance sampling does: : $$ \begin\forall a\in\mathbb, \; P^(X\in()) &= \int_dP(\omega) \\ &= \frac\; a\,P(X\in()) \end to the right, $a\backslash ,P(X\backslash in())$ is one of the infinitesimal elements that sum up to E''()'': : $E()\; =\; \backslash int\_^\; a\backslash ,P(X\backslash in())$ therefore, a good probability change ''P(L)'' in importance sampling will redistribute the law of ''X'' so that its samples' frequencies are sorted directly according to their weights in E''()''. Hence the name "importance sampling." Note that whenever $P$ is the uniform distribution and $\backslash Omega\; =\backslash mathbb$, we are just estimating the integral of the real function $X:\backslash mathbb\backslash to\backslash mathbb$, so the method can also be used for estimating simple integrals. 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「Importance sampling」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.