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第151行： − + − \Eta(Y|X) &\le \Eta(Y) \, \\ + − \Eta(X,Y) &= \Eta(X|Y) + \Eta(Y|X) + \operatorname{I}(X;Y),\qquad \\ + − \Eta(X,Y) &= \Eta(X) + \Eta(Y) - \operatorname{I}(X;Y),\, \\ + − + 第164行：
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→Properties 属性
== 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}
H(Y|X) &\le H(Y) \, \\
H(X,Y) &= H(X|Y) + H(Y|X) + \operatorname{I}(X;Y),\qquad \\
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 ==