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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 Goodman–Nguyen–van Fraassen algebra ： ウィキペディア英語版 Goodman–Nguyen–van Fraassen algebra A Goodman–Nguyen–van Fraassen algebra is a type of conditional event algebra (CEA) that embeds the standard Boolean algebra of unconditional events in a larger algebra which is itself Boolean. The goal (as with all CEAs) is to equate the conditional probability ''P''(''A'' ∩ ''B'') / ''P''(''A'') with the probability of a conditional event, ''P''(''A'' → ''B'') for more than just trivial choices of ''A'', ''B'', and ''P''. ==Construction of the algebra== Given set Ω, which is the set of possible outcomes, and set ''F'' of subsets of Ω—so that ''F'' is the set of possible events—consider an infinite Cartesian product of the form ''E''1 × ''E''2 × … × ''E''''n'' × Ω × Ω × Ω × …, where ''E''1, ''E''2, … ''E''''n'' are members of ''F''. Such a product specifies the set of all infinite sequences whose first element is in ''E''1, whose second element is in ''E''2, …, and whose ''n''th element is in ''E''''n'', and all of whose elements are in Ω. Note that one such product is the one where ''E''1 = ''E''2 = … = ''E''''n'' = Ω, i.e., the set Ω × Ω × Ω × Ω × …. Designate this set as $\backslash hat$; it is the set of all infinite sequences whose elements are in Ω. A new Boolean algebra is now formed, whose elements are subsets of $\backslash hat$. To begin with, any event which was formerly represented by subset ''A'' of Ω is now represented by $\backslash hat$ = ''A'' × Ω × Ω × Ω × …. Additionally, however, for events ''A'' and ''B'', let the conditional event ''A'' → ''B'' be represented as the following infinite union of disjoint sets: :(∩ ''B'') × Ω × Ω × Ω × … ) ∪ :(× (''A'' ∩ ''B'') × Ω × Ω × Ω × … ) ∪ :(× ''A'' ′ × (''A'' ∩ ''B'') × Ω × Ω × Ω × … ) ∪ …. The motivation for this representation of conditional events will be explained shortly. Note that the construction can be iterated; ''A'' and ''B'' can themselves be conditional events. Intuitively, unconditional event ''A'' ought to be representable as conditional event Ω → ''A''. And indeed: because Ω ∩ ''A'' = ''A'' and Ω′ = ∅, the infinite union representing Ω → ''A'' reduces to ''A'' × Ω × Ω × Ω × …. Let $\backslash hat$ now be a set of subsets of $\backslash hat$, which contains representations of all events in ''F'' and is otherwise just large enough to be closed under construction of conditional events and under the familiar Boolean operations. $\backslash hat$ is a Boolean algebra of conditional events which contains a Boolean algebra corresponding to the algebra of ordinary events. 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「Goodman–Nguyen–van Fraassen algebra」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.