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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 Chain rule (probability) ： ウィキペディア英語版 Chain rule (probability) In probability theory, the chain rule (also called the general product rule) permits the calculation of any member of the joint distribution of a set of random variables using only conditional probabilities. The rule is useful in the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities. Consider an indexed set of sets $A\_1,\; \backslash ldots\; ,\; A\_n$. To find the value of this member of the joint distribution, we can apply the definition of conditional probability to obtain: ::$\backslash mathrm\; P(A\_n,\; \backslash ldots\; ,\; A\_1)\; =\; \backslash mathrm\; P(A\_n\; |\; A\_,\; \backslash ldots\; ,\; A\_1)\; \backslash cdot\backslash mathrm\; P(\; A\_,\; \backslash ldots\; ,\; A\_1)$ Repeating this process with each final term creates the product: ::$\backslash mathrm\; P\backslash left(\backslash bigcap\_^n\; A\_k\backslash right)\; =\; \backslash prod\_^n\; \backslash mathrm\; P\backslash left(A\_k\; \backslash ,\backslash Bigg|\backslash ,\; \backslash bigcap\_^\; A\_j\backslash right)$ With four variables, the chain rule produces this product of conditional probabilities: ::$\backslash mathrm\; P(A\_4,\; A\_3,\; A\_2,\; A\_1)\; =\; \backslash mathrm\; P(A\_4\; \backslash mid\; A\_3,\; A\_2,\; A\_1)\backslash cdot\; \backslash mathrm\; P(A\_3\; \backslash mid\; A\_2,\; A\_1)\backslash cdot\; \backslash mathrm\; P(A\_2\; \backslash mid\; A\_1)\backslash cdot\; \backslash mathrm\; P(A\_1)$ This rule is illustrated in the following example. Urn 1 has 1 black ball and 2 white balls and Urn 2 has 1 black ball and 3 white balls. Suppose we pick an urn at random and then select a ball from that urn. Let event A be choosing the first urn: P(A) = P(~A) = 1/2. Let event B be the chance we choose a white ball. The chance of choosing a white ball, given that we've chosen the first urn, is P(B|A) = 2/3. Event A, B would be their intersection: choosing the first urn and a white ball from it. The probability can be found by the chain rule for probability: ::$\backslash mathrm\; P(A,\; B)=\backslash mathrm\; P(B\; \backslash mid\; A)\; \backslash mathrm\; P(A)\; =\; 2/3\; \backslash times\; 1/2\; =\; 1/3$. == Footnotes == 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「Chain rule (probability)」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.