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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 Rényi's divergence ： ウィキペディア英語版 Rényi entropy In information theory, the Rényi entropy generalizes the Hartley entropy, the Shannon entropy, the collision entropy and the min-entropy. Entropies quantify the diversity, uncertainty, or randomness of a system. The Rényi entropy is named after Alfréd Rényi. The Rényi entropy is important in ecology and statistics as indices of diversity. The Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement. In the Heisenberg XY spin chain model, the Rényi entropy as a function of ''α'' can be calculated explicitly by virtue of the fact that it is an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in the context of randomness extractors. == Definition == The Rényi entropy of order $\backslash alpha$, where $\backslash alpha\; \backslash geq\; 0$ and $\backslash alpha\; \backslash neq\; 1$, is defined as :$H\_\backslash alpha(X)\; =\; \backslash frac\backslash log\backslash Bigg(\backslash sum\_^n\; p\_i^\backslash alpha\backslash Bigg)$ .〔 Here, $X$ is a discrete random variable with possible outcomes $1,2,...,n$ and corresponding probabilities $p\_i\; \backslash doteq\; \backslash Pr(X=i)$ for $i=1,\backslash dots,n$, and the logarithm is base 2. If the probabilities are $p\_i=1/n$ for all $i=1,\backslash dots,n$, then all the Rényi entropies of the distribution are equal: $H\_\backslash alpha(X)=\backslash log\; n$. In general, for all discrete random variables $X$, $H\_\backslash alpha(X)$ is a non-increasing function in $\backslash alpha$. Applications often exploit the following relation between the Rényi entropy and the ''p''-norm of the vector of probabilities: :$H\_\backslash alpha(X)=\backslash frac\; \backslash log\; \backslash left(\backslash |P\backslash |\_\backslash alpha\backslash right)$ . Here, the discrete probability distribution $P=(p\_1,\backslash dots,p\_n)$ is interpreted as a vector in $\backslash R^n$ with $p\_i\backslash geq\; 0$ and $\backslash sum\_^\; p\_i\; =1$. The Rényi entropy for any $\backslash alpha\; \backslash geq\; 0$ is Schur concave. 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「Rényi entropy」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.