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== Properties 属性 ==

== Properties 属性 ==

=== Conditional entropy equals zero 条件熵等于零 ===

=== Conditional entropy equals zero 条件熵等于零 ===

<math>\Eta(Y|X)=0</math> if and only if the value of <math>Y</math> is completely determined by the value of <math>X</math>.

<math>H(Y|X)=0</math> if and only if the value of <math>Y</math> is completely determined by the value of <math>X</math>.

当且仅当<math>Y</math>的值完全由<math>X</math>的值确定时，才为<math>H(Y|X)=0</math>。

当且仅当<math>Y</math>的值完全由<math>X</math>的值确定时，才为<math>H(Y|X)=0</math>。

=== Conditional entropy of independent random variables 独立随机变量的条件熵 ===

=== Conditional entropy of independent random variables 独立随机变量的条件熵 ===

Conversely, <math>\Eta(Y|X) = \Eta(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].

Conversely, <math>H(Y|X) = \Eta(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].

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>。

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===

=== Bayes' rule 贝叶斯法则 ===

[[Bayes' rule]] for conditional entropy states

[[Bayes' rule]] for conditional entropy states

:<math>\Eta(Y|X) \,=\, \Eta(X|Y) - \Eta(X) + \Eta(Y).</math>

:<math>\Eta(Y|X) \,=\, \Eta(X|Y) - \Eta(X) + \Eta(Y).</math>

:<math>\Eta(Y|X,Z) \,=\, \Eta(Y|X).</math>

:<math>\Eta(Y|X,Z) \,=\, \Eta(Y|X).</math>

===Other properties===

=== Other properties 其他性质 ===

For any <math>X</math> and <math>Y</math>:

For any <math>X</math> and <math>Y</math>:

:<math display="block">\begin{align}

:<math display="block">\begin{align}

   \Eta(Y|X) &\le \Eta(Y) \, \\

   H(Y|X) &\le H(Y) \, \\

   \Eta(X,Y) &= \Eta(X|Y) + \Eta(Y|X) + \operatorname{I}(X;Y),\qquad \\

   H(X,Y) &= H(X|Y) + H(Y|X) + \operatorname{I}(X;Y),\qquad \\

   \Eta(X,Y) &= \Eta(X) + \Eta(Y) - \operatorname{I}(X;Y),\, \\

   H(X,Y) &= H(X) + H(Y) - \operatorname{I}(X;Y),\, \\

   \operatorname{I}(X;Y) &\le \Eta(X),\,

   \operatorname{I}(X;Y) &\le H(X),\,

\end{align}</math>

\end{align}</math>

For independent <math>X</math> and <math>Y</math>:

For independent <math>X</math> and <math>Y</math>:

:<math>\Eta(Y|X) = \Eta(Y) </math> and <math>\Eta(X|Y) = \Eta(X) \, </math>

:<math>H(Y|X) = H(Y) </math> and <math>H(X|Y) = H(X) \, </math>

Although the specific-conditional entropy <math>\Eta(X|Y=y)</math> can be either less or greater than <math>\Eta(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>\Eta(X|Y)</math> can never exceed <math>\Eta(X)</math>.

Although the specific-conditional entropy <math>H(X|Y=y)</math> can be either less or greater than <math>H(X)</math> for a given [[random variate]] <math>y</math> of <math>Y</math>, <math>H(X|Y)</math> can never exceed <math>H(X)</math>.

== Conditional differential entropy ==

== Conditional differential entropy ==