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→Motivation
= -\sum_{y\in\mathcal Y} {\Pr(Y = y|X=x) \log_2{\Pr(Y = y|X=x)}}.</math>
= -\sum_{y\in\mathcal Y} {\Pr(Y = y|X=x) \log_2{\Pr(Y = y|X=x)}}.</math>
Note that <math>\Eta(Y|X)</math> is the result of averaging <math>\Eta(Y|X=x)</math> over all possible values <math>x</math> that <math>X</math> may take. Also, if the above sum is taken over a sample <math>y_1, \dots, y_n</math>, the expected value <math>E_X[ \Eta(y_1, \dots, y_n \mid X = x)]</math> is known in some domains as '''equivocation'''.<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref>
Note that <math>\Eta(Y|X)</math> is the result of averaging <math>\Eta(Y|X=x)</math> over all possible values <math>x</math> that <math>X</math> may take. Also, if the above sum is taken over a sample <math>y_1, \dots, y_n</math>, the expected value <math>E_X[ \Eta(y_1, \dots, y_n \mid X = x)]</math> is known in some domains as '''equivocation'''.<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref>
注意，H（Y ǀ X）是在<math>X</math>可能取的所有可能值<math>x</math>上对H（Y ǀ X = x）求平均值的结果。同样，如果将上述总和接管到样本<math>y_1, \dots, y_n</math>上，则预期值<math>E_X[ \Eta(y_1, \dots, y_n \mid X = x)]</math>在某些领域中会变得不那么明确。<ref>{{cite journal|author1=Hellman, M.|author2=Raviv, J.|year=1970|title=Probability of error, equivocation, and the Chernoff bound|journal=IEEE Transactions on Information Theory|volume=16|issue=4|pp=368-372}}</ref>
Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the weighted sum of <math>\Eta(Y|X=x)</math> for each possible value of <math>x</math>, using <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}
Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the weighted sum of <math>\Eta(Y|X=x)</math> for each possible value of <math>x</math>, using <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}