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Intuitive explanation of the definition :  

Intuitive explanation of the definition : According to the definition, <math>\displaystyle H( Y|X) =\mathbb{E}( \ f( X,Y) \ )</math> where <math>\displaystyle f:( x,y) \ \rightarrow -\log( \ p( y|x) \ ) .</math> <math>\displaystyle f</math> associates to  <math>\displaystyle ( x,y)</math> the information content of <math>\displaystyle ( Y=y)</math> given <math>\displaystyle (X=x)</math>, which is the amount of information needed to describe the event <math>\displaystyle (Y=y)</math> given <math>(X=x)</math>.  According to the law of large numbers, <math>\displaystyle H(Y|X)</math> is the arithmetic mean of a large number of independent realizations of <math>\displaystyle f(X,Y)</math>.

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，H（Y ǀ X）是<math>\displaystyle f(X,Y)</math>的大量独立实现的算术平均值。

 

 

 

 

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，H（Y ǀ X）是<math>\displaystyle f(X,Y)</math>的大量独立实现的算术平均值。

== Motivation ==

== Motivation ==