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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 Inequalities in information theory ： ウィキペディア英語版 Inequalities in information theory Inequalities are very important in the study of information theory. There are a number of different contexts in which these inequalities appear. ==Shannon-type inequalities== Consider a finite collection of finitely (or at most countably) supported random variables on the same probability space. For a collection of ''n'' random variables, there are 2''n'' − 1 such non-empty subsets for which entropies can be defined. For example, when ''n'' = 2, we may consider the entropies $H(X\_1),$ $H(X\_2),$ and $H(X\_1,\; X\_2),$ and express the following inequalities (which together characterize the range of the marginal and joint entropies of two random variables): * $H(X\_1)\; \backslash ge\; 0$ * $H(X\_2)\; \backslash ge\; 0$ * $H(X\_1)\; \backslash le\; H(X\_1,\; X\_2)$ * $H(X\_2)\; \backslash le\; H(X\_1,\; X\_2)$ * $H(X\_1,\; X\_2)\; \backslash le\; H(X\_1)\; +\; H(X\_2).$ In fact, these can all be expressed as special cases of a single inequality involving the conditional mutual information, namely :$I(A;B|C)\; \backslash ge\; 0,$ where $A$, $B$, and $C$ each denote the joint distribution of some arbitrary (possibly empty) subset of our collection of random variables. Inequalities that can be derived from this are known as Shannon-type inequalities. More formally (following the notation of Yeung 〔)〕), define $\backslash Gamma^$ *_n to be the set of all ''constructible'' points in $\backslash mathbb\; R^,$ where a point is said to be constructible if and only if there is a joint, discrete distribution of ''n'' random variables such that each coordinate of that point, indexed by a non-empty subset of , is equal to the joint entropy of the corresponding subset of the ''n'' random variables. The closure of $\backslash Gamma^$ *_n is denoted $\backslash overline.$ In general :$\backslash Gamma^$ *_n \subseteq \overline \subseteq \Gamma_n. The cone in $\backslash mathbb\; R^$ characterized by all Shannon-type inequalities among ''n'' random variables is denoted $\backslash Gamma\_n.$ Software has been developed to automate the task of proving such inequalities . Given an inequality, such software is able to determine whether the given inequality contains the cone $\backslash Gamma\_n,$ in which case the inequality can be verified, since $\backslash Gamma^$ *_n \subseteq \Gamma_n. 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「Inequalities in information 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.