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→More general definition
:<math> I(X;Y|Z) = \sum_{y \in \mathcal{Y}} p( Y=y ) D_{\mathrm{KL}}[ p(X,Z|y) \| p(X|Z)p(Z|y) ]</math>.
:<math> I(X;Y|Z) = \sum_{y \in \mathcal{Y}} p( Y=y ) D_{\mathrm{KL}}[ p(X,Z|y) \| p(X|Z)p(Z|y) ]</math>.
==More general definition==
== More general definition 其他定义==
A more general definition of conditional mutual information, applicable to random variables with continuous or other arbitrary distributions, will depend on the concept of '''[[regular conditional probability]]'''. (See also.<ref>[http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]</ref><ref>D. Leao, Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>)
A more general definition of conditional mutual information, applicable to random variables with continuous or other arbitrary distributions, will depend on the concept of '''[[regular conditional probability]]'''. (See also. <ref>[http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]</ref><ref>D. Leao, Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>)
条件交互信息的其他通用定义（适用于具有连续或其他任意分布的随机变量）将取决于正则条件概率的概念。
Let <math>(\Omega, \mathcal F, \mathfrak P)</math> be a [[probability space]], and let the random variables <math>X</math>, <math>Y</math>, and <math>Z</math> each be defined as a Borel-measurable function from <math>\Omega</math> to some state space endowed with a topological structure.
Let <math>(\Omega, \mathcal F, \mathfrak P)</math> be a [[probability space]], and let the random variables <math>X</math>, <math>Y</math>, and <math>Z</math> each be defined as a Borel-measurable function from <math>\Omega</math> to some state space endowed with a topological structure.
令<math>(\Omega, \mathcal F, \mathfrak P)</math>为一个概率空间，并将随机变量<math>X</math>, <math>Y</math>, 和 <math>Z</math>分别定义为一个从<math>\Omega</math>到具有拓扑结构的状态空间的'''<font color="#ff8000"> 波莱尔可测函数Borel-measurable function </font>'''。
Consider the Borel measure (on the σ-algebra generated by the open sets) in the state space of each random variable defined by assigning each Borel set the <math>\mathfrak P</math>-measure of its preimage in <math>\mathcal F</math>. This is called the [[pushforward measure]] <math>X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big).</math> The '''support of a random variable''' is defined to be the [[Support (measure theory)|topological support]] of this measure, i.e. <math>\mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P.</math>
Consider the Borel measure (on the σ-algebra generated by the open sets) in the state space of each random variable defined by assigning each Borel set the <math>\mathfrak P</math>-measure of its preimage in <math>\mathcal F</math>. This is called the [[pushforward measure]] <math>X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big).</math> The '''support of a random variable''' is defined to be the [[Support (measure theory)|topological support]] of this measure, i.e. <math>\mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P.</math>
考虑到在每个随机变量状态空间中的'''<font color="#ff8000"> 波莱尔测度Borel measure</font>'''（关于开放集生成的σ代数），是由每个波莱尔集分配到的<math>\mathcal F</math>中的原像<math>\mathfrak P</math>测度来确定的。这被称为'''<font color="#ff8000"> 前推测度Pushforward measure </font>''' <math>X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big).</math>。随机变量的支撑集定义为该测度的拓扑支撑集，即<math>\mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P.</math>。
Now we can formally define the [[conditional probability distribution|conditional probability measure]] given the value of one (or, via the [[product topology]], more) of the random variables. Let <math>M</math> be a measurable subset of <math>\Omega,</math> (i.e. <math>M \in \mathcal F,</math>) and let <math>x \in \mathrm{supp}\,X.</math> Then, using the [[disintegration theorem]]:
Now we can formally define the [[conditional probability distribution|conditional probability measure]] given the value of one (or, via the [[product topology]], more) of the random variables. Let <math>M</math> be a measurable subset of <math>\Omega,</math> (i.e. <math>M \in \mathcal F,</math>) and let <math>x \in \mathrm{supp}\,X.</math> Then, using the [[disintegration theorem]]:
现在，我们可以在给定其中一个随机变量值（或通过积拓扑获得更多）的情况下正式定义条件概率测度。令<math>M</math>为<math>\Omega,</math>的可测子集（即<math>M \in \mathcal F,</math>，），令<math>x \in \mathrm{supp}\,X.</math>。然后，使用分解定理：
:<math>\mathfrak P(M | X=x) = \lim_{U \ni x}
:<math>\mathfrak P(M | X=x) = \lim_{U \ni x}
\frac {\mathfrak P(M \cap \{X \in U\})}
\frac {\mathfrak P(M \cap \{X \in U\})}
{\mathfrak P(\{X \in U\})}
{\mathfrak P(\{X \in U\})}
\qquad \textrm{and} \qquad \mathfrak P(M|X) = \int_M d\mathfrak P\big(\omega|X=X(\omega)\big),</math>
\qquad \textrm{and} \qquad \mathfrak P(M|X) = \int_M d\mathfrak P\big(\omega|X=X(\omega)\big),</math>
where the limit is taken over the open neighborhoods <math>U</math> of <math>x</math>, as they are allowed to become arbitrarily smaller with respect to [[Subset|set inclusion]].
where the limit is taken over the open neighborhoods <math>U</math> of <math>x</math>, as they are allowed to become arbitrarily smaller with respect to [[Subset|set inclusion]].
在<math>x</math>的开放邻域<math>U</math>处采用极限，因为相对于'''<font color="#ff8000"> 集包含Set inclusion</font>'''，它们可以任意变小。
Finally we can define the conditional mutual information via [[Lebesgue integration]]:
Finally we can define the conditional mutual information via [[Lebesgue integration]]:
最后，我们可以通过'''<font color="#ff8000"> 勒贝格积分Lebesgue integration</font>'''来定义条件交互信息：
:<math>I(X;Y|Z) = \int_\Omega \log
:<math>I(X;Y|Z) = \int_\Omega \log
\Bigl(
\Bigl(
d \mathfrak P(\omega),
d \mathfrak P(\omega),
</math>
</math>
where the integrand is the logarithm of a [[Radon–Nikodym derivative]] involving some of the conditional probability measures we have just defined.
where the integrand is the logarithm of a [[Radon–Nikodym derivative]] involving some of the conditional probability measures we have just defined.
其中被积函数是'''<font color="#ff8000"> 拉东-尼科迪姆导数Radon–Nikodym derivative</font>'''的对数，涉及我们刚刚定义的一些条件概率测度。
==Note on notation==
==Note on notation==