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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 Fano inequality ： ウィキペディア英語版 Fano's inequality In information theory, Fano's inequality (also known as the Fano converse and the Fano lemma) relates the average information lost in a noisy channel to the probability of the categorization error. It was derived by Robert Fano in the early 1950s while teaching a Ph.D. seminar in information theory at MIT, and later recorded in his 1961 textbook. It is used to find a lower bound on the error probability of any decoder as well as the lower bounds for minimax risks in density estimation. Let the random variables ''X'' and ''Y'' represent input and output messages with a joint probability $P(x,y)$. Let ''e'' represent an occurrence of error; i.e., that $X\backslash neq\; \backslash tilde$, being $\backslash tilde=f(Y)$ a noise approximate version of $X$. Fano's inequality is :$H(X|Y)\backslash leq\; H(e)+P(e)\backslash log(|\backslash mathcal|-1),$ where $\backslash mathcal$ denotes the support of ''X'', :$H\backslash left(X|Y\backslash right)=-\backslash sum\_\; P(x\_i,y\_j)\backslash log\; P\backslash left(x\_i|y\_j\backslash right)$ is the conditional entropy, :$P(e)=P(X\backslash neq\; \backslash tilde)$ is the probability of the communication error, and :$H(e)=-P(e)\backslash log\; P(e)-(1-P(e))\backslash log(1-P(e))$ is the corresponding binary entropy. ==Alternative formulation== Let ''X'' be a random variable with density equal to one of $r+1$ possible densities $f\_1,\backslash ldots,f\_$. Furthermore, the Kullback–Leibler divergence between any pair of densities cannot be too large, :$D\_(f\_i\backslash |f\_j)\backslash leq\; \backslash beta$ for all $i\backslash not\; =\; j.$ Let $\backslash psi(X)\backslash in\backslash$ be an estimate of the index. Then :$\backslash sup\_i\; P\_i(\backslash psi(X)\backslash not\; =\; i)\; \backslash geq\; 1-\backslash frac$ where $P\_i$ is the probability induced by $f\_i$ 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「Fano's inequality」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.