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→Definition 定义
== Definition 定义 ==
== Definition 定义 ==
For random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, we define the conditional mutual information as
For random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, we define the conditional mutual information as
对于具有支持集<math>\mathcal{X}</math>, <math>\mathcal{Y}</math> 和 <math>\mathcal{Z}</math>的随机变量<math>X</math>, <math>Y</math>, 和 <math>Z</math>，我们将条件交互信息定义为：
{{Equation box 1
{{Equation box 1
|background colour=#F5FFFA}}
|background colour=#F5FFFA}}
This may be written in terms of the expectation operator: <math>I(X;Y|Z) = \mathbb{E}_Z [D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} )]</math>.
This may be written in terms of the expectation operator:
这可以用期望运算符来表示：
<math>I(X;Y|Z) = \mathbb{E}_Z [D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} )]</math>.
Thus <math>I(X;Y|Z)</math> is the expected (with respect to <math>Z</math>) [[Kullback–Leibler divergence]] from the conditional joint distribution <math>P_{(X,Y)|Z}</math> to the product of the conditional marginals <math>P_{X|Z}</math> and <math>P_{Y|Z}</math>. Compare with the definition of [[mutual information]].
Thus <math>I(X;Y|Z)</math> is the expected (with respect to <math>Z</math>) [[Kullback–Leibler divergence]] from the conditional joint distribution <math>P_{(X,Y)|Z}</math> to the product of the conditional marginals <math>P_{X|Z}</math> and <math>P_{Y|Z}</math>. Compare with the definition of [[mutual information]].
因此，相较于交互信息的定义，<math>I(X;Y|Z)</math>可以表达为预期的Kullback-Leibler散度（相对于<math>Z</math>），即从条件联合分布<math>P_{(X,Y)|Z}</math>到条件边际<math>P_{X|Z}</math> 和 <math>P_{Y|Z}</math>的乘积的。
==In terms of pmf's for discrete distributions==
==In terms of pmf's for discrete distributions==