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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 Possibility theory ： ウィキペディア英語版 Possibility theory Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier in the 50s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise. ==Formalization of possibility== For simplicity, assume that the universe of discourse Ω is a finite set, and assume that all subsets are measurable. A distribution of possibility is a function $\backslash operatorname$ from $2^\backslash Omega$ to () such that: :Axiom 1: $\backslash operatorname(\backslash varnothing)\; =\; 0$ :Axiom 2: $\backslash operatorname(\backslash Omega)\; =\; 1$ :Axiom 3: $\backslash operatorname(U\; \backslash cup\; V)\; =\; \backslash max\; \backslash left(\; \backslash operatorname(U),\; \backslash operatorname(V)\; \backslash right)$ for any disjoint subsets $U$ and $V$. It follows that, like probability, the possibility measure is determined by its behavior on singletons: :$\backslash operatorname(U)\; =\; \backslash max\_\; \backslash operatorname(\backslash )$ provided that ''U'' is finite or countably infinite. Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω. Axiom 2 could be interpreted as the assumption that the evidence from which $\backslash operatorname$ was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1. Axiom 3 corresponds to the additivity axiom in probabilities. However there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that: :$\backslash operatorname(U\; \backslash cup\; V)\; =\; \backslash max\; \backslash left(\; \backslash operatorname(U),\; \backslash operatorname(V)\; \backslash right)$ for ''any'' subsets $U$ and $V$. Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is ''compositional'' with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally: :$\backslash operatorname(U\; \backslash cap\; V)\; \backslash leq\; \backslash min\; \backslash left(\; \backslash operatorname(U),\; \backslash operatorname(V)\; \backslash right)$ When Ω is not finite, Axiom 3 can be replaced by: :For all index sets $I$, if the subsets $U\_$ are pairwise disjoint, $\backslash operatorname\backslash left(\backslash cup\_\; U\_i\backslash right)\; =\; \backslash sup\_\backslash operatorname(U\_i).$ 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「Possibility theory」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.