更改

因此，相较于交互信息的定义，<math>I(X;Y|Z)</math>可以表达为期望的'''<font color="#ff8000"> Kullback-Leibler散度</font>'''（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。

因此，相较于交互信息的定义，<math>I(X;Y|Z)</math>可以表达为期望的'''<font color="#ff8000"> Kullback-Leibler散度</font>'''（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。

==In terms of pmf's for discrete distributions==

== In terms of pmf's for discrete distributions 关于离散分布的概率质量函数 ==

For discrete 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>, the conditional mutual information <math>I(X;Y|Z)</math> is as follows

For discrete 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>, the conditional mutual information <math>I(X;Y|Z)</math> is as follows

:<math>

:<math>

I(X;Y|Z) = \sum_{z\in \mathcal{Z}} p_Z(z) \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}}

I(X;Y|Z) = \sum_{z\in \mathcal{Z}} p_Z(z) \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}}

       p_{X,Y|Z}(x,y|z) \log \frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}

       p_{X,Y|Z}(x,y|z) \log \frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}

</math>

</math>

where the marginal, joint, and/or conditional [[probability mass function]]s are denoted by <math>p</math> with the appropriate subscript. This can be simplified as

where the marginal, joint, and/or conditional [[probability mass function]]s are denoted by <math>p</math> with the appropriate subscript. This can be simplified as

{{Equation box 1

{{Equation box 1