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→Properties 属性
== Properties 属性 ==
== Properties 属性 ==
===Conditional entropy equals zero===
=== 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>\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>.
===Conditional entropy of independent random variables===
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 ===
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>\Eta(Y|X) = \Eta(Y)</math> if and only if <math>Y</math> and <math>X</math> are [[independent random variables]].
===Chain rule===
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>。
Assume that the combined system determined by two random variables <math>X</math> and <math>Y</math> has [[joint entropy]] <math>\Eta(X,Y)</math>, that is, we need <math>\Eta(X,Y)</math> bits of information on average to describe its exact state. Now if we first learn the value of <math>X</math>, we have gained <math>\Eta(X)</math> bits of information. Once <math>X</math> is known, we only need <math>\Eta(X,Y)-\Eta(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>\Eta(Y|X)</math>, which gives the ''chain rule'' of conditional entropy:
=== Chain rule 链式法则 ===
Assume that the combined system determined by two random variables <math>X</math> and <math>Y</math> has [[joint entropy]] <math>H(X,Y)</math>, that is, we need <math>H(X,Y)</math> bits of information on average to describe its exact state. Now if we first learn the value of <math>X</math>, we have gained <math>H(X)</math> bits of information. Once <math>X</math> is known, we only need <math>H(X,Y)-H(X)</math> bits to describe the state of the whole system. This quantity is exactly <math>H(Y|X)</math>, which gives the ''chain rule'' of conditional entropy:
假设由两个随机变量<math>X</math>和<math>Y</math>确定的组合系统具有联合熵<math>H(X,Y)</math>，也就是说，我们通常需要<math>H(X,Y)</math>位信息来描述其确切状态。现在，如果我们首先获得<math>X</math>的值，我们将知晓<math>H(X)</math>位信息。一旦知道了<math>X</math>的值，我们就可以通过<math>H(X,Y)</math>-<math>H(X)</math>位来描述整个系统的状态。这个数量恰好是<math>H(Y|X)</math>，它给出了条件熵的链式法则：
:<math>\Eta(Y|X)\, = \, \Eta(X,Y)- \Eta(X).</math><ref name=cover1991 />{{rp|17}}
:<math>\Eta(Y|X)\, = \, \Eta(X,Y)- \Eta(X).</math><ref name=cover1991 />{{rp|17}}
The chain rule follows from the above definition of conditional entropy:
The chain rule follows from the above definition of conditional entropy:
链式法则遵循以上条件熵的定义：
:<math>\begin{align}
:<math>\begin{align}
& = \Eta(X,Y) - \Eta(X).
& = \Eta(X,Y) - \Eta(X).
\end{align}</math>
\end{align}</math>
In general, a chain rule for multiple random variables holds:
In general, a chain rule for multiple random variables holds:
通常情况下，多个随机变量的链式法则表示为：
:<math> \Eta(X_1,X_2,\ldots,X_n) =
:<math> \Eta(X_1,X_2,\ldots,X_n) =
\sum_{i=1}^n \Eta(X_i | X_1, \ldots, X_{i-1}) </math><ref name=cover1991 />{{rp|22}}
\sum_{i=1}^n \Eta(X_i | X_1, \ldots, X_{i-1}) </math><ref name=cover1991 />{{rp|22}}
It has a similar form to [[Chain rule (probability)|chain rule]] in probability theory, except that addition instead of multiplication is used.
It has a similar form to [[Chain rule (probability)|chain rule]] in probability theory, except that addition instead of multiplication is used.
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===