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对该定义的直观解释是：根据定义<math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math>，其中<math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) </math>. <math>\displaystyle f</math>将给定<math>\displaystyle (X=x)</math>的<math>\displaystyle ( Y=y)</math>的信息内容与<math>\displaystyle ( x,y)</math>相关联，这是描述在给定<math>(X=x)</math>条件下的事件<math>\displaystyle (Y=y)</math>所需的信息量。根据大数定律，<math>H（Y ǀ X）</math>是<math>\displaystyle f(X,Y)</math>的大量独立实现的算术平均值。
对该定义的直观解释是：根据定义<math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math>，其中<math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) </math>. <math>\displaystyle f</math>将给定<math>\displaystyle (X=x)</math>的<math>\displaystyle ( Y=y)</math>的信息内容与<math>\displaystyle ( x,y)</math>相关联，这是描述在给定<math>(X=x)</math>条件下的事件<math>\displaystyle (Y=y)</math>所需的信息量。根据大数定律，<math>H（Y ǀ X）</math>是<math>\displaystyle f(X,Y)</math>的大量独立实现的算术平均值。
== Motivation 动机 ==
== Motivation 动机 ==
这里当取值为<math>y_i</math>时，<math>\operatorname{I}(y_i)</math>是其结果<math>Y</math>的信息内容。类似地以<math>X</math>为条件的<math>Y</math>的熵，当值为<math>x</math>时，也可以通过条件期望来定义：
这里当取值为<math>y_i</math>时，<math>\operatorname{I}(y_i)</math>是其结果<math>Y</math>的信息内容。类似地以<math>X</math>为条件的<math>Y</math>的熵，当值为<math>x</math>时，也可以通过条件期望来定义：
<!-- This paragraph is incorrect; the last line is not the KL divergence between any two distributions, since p(x) is [in general] not a valid distribution over the domains of X and Y. The last formula above is the [[Kullback-Leibler divergence]], also known as relative entropy. Relative entropy is always positive, and vanishes if and only if <math>p(x,y) = p(x)</math>. This is when knowing <math>x</math> tells us everything about <math>y</math>. ADDED: Could this comment be out of date since the KL divergence is not mentioned above? November 2014 -->
<!-- This paragraph is incorrect; the last line is not the KL divergence between any two distributions, since p(x) is [in general] not a valid distribution over the domains of X and Y. The last formula above is the [[Kullback-Leibler divergence]], also known as relative entropy. Relative entropy is always positive, and vanishes if and only if <math>p(x,y) = p(x)</math>. This is when knowing <math>x</math> tells us everything about <math>y</math>. ADDED: Could this comment be out of date since the KL divergence is not mentioned above? November 2014 -->
== Properties 属性 ==
== Properties 属性 ==
相反，当且仅当<math>Y</math>和<math>X</math>是独立随机变量时，则为<math>H(Y|X) =H(Y)</math>。
相反，当且仅当<math>Y</math>和<math>X</math>是独立随机变量时，则为<math>H(Y|X) =H(Y)</math>。
=== Chain rule 链式法则 ===
=== Chain rule 链式法则 ===
除了使用加法而不是乘法之外，它具有与概率论中的链式法则类似的形式。
除了使用加法而不是乘法之外，它具有与概率论中的链式法则类似的形式。
=== Bayes' rule 贝叶斯法则 ===
=== Bayes' rule 贝叶斯法则 ===
:<math>H(Y|X,Z) \,=\, H(Y|X).</math>
:<math>H(Y|X,Z) \,=\, H(Y|X).</math>
=== Other properties 其他性质 ===
=== Other properties 其他性质 ===