条件互信息

此词条Jie翻译。

以上是三个变量[math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], 和 [math]\displaystyle{ z }[/math]信息理论测度的维恩图，分别由左下，右下和上部的圆圈表示。条件交互信息[math]\displaystyle{ I(x;z|y) }[/math], [math]\displaystyle{ I(y;z|x) }[/math] 和 [math]\displaystyle{ I(x;y|z) }[/math]分别由黄色，青色和品红色（注意：该图颜色标注错误，需要修改）区域表示。

In probability theory, particularly information theory, the conditional mutual information is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third.

在 概率论Probability theory中，特别是与 信息论Information theory相关的情况下，最基本形式的 条件交互信息Conditional mutual information ，是在给定第三个值的两个随机变量间交互信息的期望值。

Definition 定义

For random variables [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], and [math]\displaystyle{ Z }[/math] with support sets [math]\displaystyle{ \mathcal{X} }[/math], [math]\displaystyle{ \mathcal{Y} }[/math] and [math]\displaystyle{ \mathcal{Z} }[/math], we define the conditional mutual information as

对于具有支持集[math]\displaystyle{ \mathcal{X} }[/math], [math]\displaystyle{ \mathcal{Y} }[/math] 和 [math]\displaystyle{ \mathcal{Z} }[/math]的随机变量[math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], 和 [math]\displaystyle{ Z }[/math]，我们将条件交互信息定义为：

[math]\displaystyle{ I(X;Y|Z) = \int_\mathcal{Z} D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} ) dP_{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]\displaystyle{ I(X;Y|Z) = \mathbb{E}_Z [D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} )] }[/math].

Thus [math]\displaystyle{ I(X;Y|Z) }[/math] is the expected (with respect to [math]\displaystyle{ Z }[/math]) Kullback–Leibler divergence from the conditional joint distribution [math]\displaystyle{ P_{(X,Y)|Z} }[/math] to the product of the conditional marginals [math]\displaystyle{ P_{X|Z} }[/math] and [math]\displaystyle{ P_{Y|Z} }[/math]. Compare with the definition of mutual information.

因此，相较于交互信息的定义，[math]\displaystyle{ I(X;Y|Z) }[/math]可以表达为期望的 Kullback-Leibler散度（相对于[math]\displaystyle{ Z }[/math]），即从条件联合分布[math]\displaystyle{ P_{(X,Y)|Z} }[/math]到条件边际[math]\displaystyle{ P_{X|Z} }[/math] 和 [math]\displaystyle{ P_{Y|Z} }[/math]的乘积。

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]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], and [math]\displaystyle{ Z }[/math] with support sets [math]\displaystyle{ \mathcal{X} }[/math], [math]\displaystyle{ \mathcal{Y} }[/math] and [math]\displaystyle{ \mathcal{Z} }[/math], the conditional mutual information [math]\displaystyle{ I(X;Y|Z) }[/math] is as follows

对于具有支持集[math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], 和 [math]\displaystyle{ Z }[/math]的离散随机变量[math]\displaystyle{ \mathcal{X} }[/math], [math]\displaystyle{ \mathcal{Y} }[/math] 和 [math]\displaystyle{ \mathcal{Z} }[/math]，条件交互信息[math]\displaystyle{ I(X;Y|Z) }[/math]如下:

[math]\displaystyle{ 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)} }[/math]

where the marginal, joint, and/or conditional probability mass functions are denoted by [math]\displaystyle{ p }[/math] with the appropriate subscript. This can be simplified as

其中边缘概率密度函数，联合概率密度函数，和（或）条件概率质量函数可以由[math]\displaystyle{ p }[/math]加上适当的下标表示。这可以简化为:

[math]\displaystyle{ I(X;Y|Z) = \sum_{z\in \mathcal{Z}} \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}} p_{X,Y,Z}(x,y,z) \log \frac{p_Z(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}. }[/math]

In terms of pdf's for continuous distributions 关于连续分布的概率密度函数

For (absolutely) continuous random variables [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], and [math]\displaystyle{ Z }[/math] with support sets [math]\displaystyle{ \mathcal{X} }[/math], [math]\displaystyle{ \mathcal{Y} }[/math] and [math]\displaystyle{ \mathcal{Z} }[/math], the conditional mutual information [math]\displaystyle{ I(X;Y|Z) }[/math] is as follows

对于具有支持集[math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], 和 [math]\displaystyle{ Z }[/math]的（绝对）连续随机变量[math]\displaystyle{ \mathcal{X} }[/math], [math]\displaystyle{ \mathcal{Y} }[/math] 和 [math]\displaystyle{ \mathcal{Z} }[/math]，条件交互信息[math]\displaystyle{ I(X;Y|Z) }[/math]如下:

[math]\displaystyle{ I(X;Y|Z) = \int_{\mathcal{Z}} \bigg( \int_{\mathcal{Y}} \int_{\mathcal{X}} \log \left(\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}\right) p_{X,Y|Z}(x,y|z) dx dy \bigg) p_Z(z) dz }[/math]

where the marginal, joint, and/or conditional probability density functions are denoted by [math]\displaystyle{ p }[/math] with the appropriate subscript. This can be simplified as

其中边缘概率密度函数，联合概率密度函数，和（或）条件概率密度函数可以由p加上适当的下标表示。这可以简化为

[math]\displaystyle{ I(X;Y|Z) = \int_{\mathcal{Z}} \int_{\mathcal{Y}} \int_{\mathcal{X}} \log \left(\frac{p_Z(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}\right) p_{X,Y,Z}(x,y,z) dx dy dz. }[/math]

Some identities 部分特性

Alternatively, we may write in terms of joint and conditional entropies as

同时我们也可以将联合和条件熵写为：

[math]\displaystyle{ I(X;Y|Z) = H(X,Z) + H(Y,Z) - H(X,Y,Z) - H(Z) = H(X|Z) - H(X|Y,Z) = H(X|Z)+H(Y|Z)-H(X,Y|Z). }[/math]

This can be rewritten to show its relationship to mutual information

这么表达以显示其与交互信息的关系

[math]\displaystyle{ I(X;Y|Z) = I(X;Y,Z) - I(X;Z) }[/math]

usually rearranged as the chain rule for mutual information

通常情况下，表达式被重新整理为“交互信息的链式法则”

[math]\displaystyle{ I(X;Y,Z) = I(X;Z) + I(X;Y|Z) }[/math]

Another equivalent form of the above is

上述的另一种等效形式是：

[math]\displaystyle{ I(X;Y|Z) = H(Z|X) + H(X) + H(Z|Y) + H(Y) - H(Z|X,Y) - H(X,Y) - H(Z) = I(X;Y) + H(Z|X) + H(Z|Y) - H(Z|X,Y) - H(Z) }[/math]

Like mutual information, conditional mutual information can be expressed as a Kullback–Leibler divergence:

类似交互信息一样，条件交互信息可以表示为Kullback-Leibler散度：

[math]\displaystyle{ I(X;Y|Z) = D_{\mathrm{KL}}[ p(X,Y,Z) \| p(X|Z)p(Y|Z)p(Z) ]. }[/math]

Or as an expected value of simpler Kullback–Leibler divergences:

或作为更简单的Kullback-Leibler差异的期望值：

[math]\displaystyle{ I(X;Y|Z) = \sum_{z \in \mathcal{Z}} p( Z=z ) D_{\mathrm{KL}}[ p(X,Y|z) \| p(X|z)p(Y|z) ] }[/math],

[math]\displaystyle{ 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 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. )

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]\displaystyle{ (\Omega, \mathcal F, \mathfrak P) }[/math] be a probability space, and let the random variables [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], and [math]\displaystyle{ Z }[/math] each be defined as a Borel-measurable function from [math]\displaystyle{ \Omega }[/math] to some state space endowed with a topological structure.

令[math]\displaystyle{ (\Omega, \mathcal F, \mathfrak P) }[/math]为一个概率空间，并将随机变量[math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], 和 [math]\displaystyle{ Z }[/math]分别定义为一个从[math]\displaystyle{ \Omega }[/math]到具有拓扑结构的状态空间的 波莱尔可测函数Borel-measurable function 。

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]\displaystyle{ \mathfrak P }[/math]-measure of its preimage in [math]\displaystyle{ \mathcal F }[/math]. This is called the pushforward measure [math]\displaystyle{ X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big). }[/math] The support of a random variable is defined to be the topological support of this measure, i.e. [math]\displaystyle{ \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 波莱尔测度Borel measure（关于开放集生成的σ代数），是由每个波莱尔集分配到的[math]\displaystyle{ \mathcal F }[/math]中的原像[math]\displaystyle{ \mathfrak P }[/math]测度来确定的。这被称为 前推测度Pushforward measure [math]\displaystyle{ 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]\displaystyle{ \mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P. }[/math]。

Now we can formally define the conditional probability measure given the value of one (or, via the product topology, more) of the random variables. Let [math]\displaystyle{ M }[/math] be a measurable subset of [math]\displaystyle{ \Omega, }[/math] (i.e. [math]\displaystyle{ M \in \mathcal F, }[/math]) and let [math]\displaystyle{ 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]\displaystyle{ M }[/math]为[math]\displaystyle{ \Omega, }[/math]的可测子集（即[math]\displaystyle{ M \in \mathcal F, }[/math]，），令[math]\displaystyle{ x \in \mathrm{supp}\,X. }[/math]。然后，使用分解定理：

[math]\displaystyle{ \mathfrak P(M | X=x) = \lim_{U \ni x} \frac {\mathfrak P(M \cap \{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]

where the limit is taken over the open neighborhoods [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x }[/math], as they are allowed to become arbitrarily smaller with respect to set inclusion.

在[math]\displaystyle{ x }[/math]的开放邻域[math]\displaystyle{ U }[/math]处采用极限，因为相对于 集包含Set inclusion，它们可以任意变小。

Finally we can define the conditional mutual information via Lebesgue integration:

最后，我们可以通过 勒贝格积分Lebesgue integration来定义条件交互信息：

[math]\displaystyle{ I(X;Y|Z) = \int_\Omega \log \Bigl( \frac {d \mathfrak P(\omega|X,Z)\, d\mathfrak P(\omega|Y,Z)} {d \mathfrak P(\omega|Z)\, d\mathfrak P(\omega|X,Y,Z)} \Bigr) d \mathfrak P(\omega), }[/math]

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 拉东-尼科迪姆导数Radon–Nikodym derivative的对数，涉及我们刚刚定义的一些条件概率测度。

Note on notation 注释符号

In an expression such as [math]\displaystyle{ I(A;B|C), }[/math] [math]\displaystyle{ A, }[/math] [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ C }[/math] need not necessarily be restricted to representing individual random variables, but could also represent the joint distribution of any collection of random variables defined on the same probability space. As is common in probability theory, we may use the comma to denote such a joint distribution, e.g. [math]\displaystyle{ I(A_0,A_1;B_1,B_2,B_3|C_0,C_1). }[/math] Hence the use of the semicolon (or occasionally a colon or even a wedge [math]\displaystyle{ \wedge }[/math]) to separate the principal arguments of the mutual information symbol. (No such distinction is necessary in the symbol for joint entropy, since the joint entropy of any number of random variables is the same as the entropy of their joint distribution.)

在诸如[math]\displaystyle{ I(A;B|C) }[/math]的表达式中，[math]\displaystyle{ A }[/math] [math]\displaystyle{ B }[/math] 和 [math]\displaystyle{ C }[/math]不限于表示单个随机变量，它们同时可以表示在同一概率空间上定义的任意随机变量集合的联合分布。类似概率论中的表达方式，我们可以使用逗号来表示这种联合分布，例如[math]\displaystyle{ I(A_0,A_1;B_1,B_2,B_3|C_0,C_1). }[/math]。因此，使用分号（或有时用冒号或楔形[math]\displaystyle{ \wedge }[/math]）来分隔交互信息符号的主要参数。（在联合熵的符号中，不需要作这样的区分，因为任意数量随机变量的联合熵与它们联合分布的熵相同。）

Properties

Nonnegativity

It is always true that

[math]\displaystyle{ I(X;Y|Z) \ge 0 }[/math],

for discrete, jointly distributed random variables [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ Z }[/math]. This result has been used as a basic building block for proving other inequalities in information theory, in particular, those known as Shannon-type inequalities. Conditional mutual information is also non-negative for continuous random variables under certain regularity conditions.

Interaction information

Conditioning on a third random variable may either increase or decrease the mutual information: that is, the difference [math]\displaystyle{ I(X;Y) - I(X;Y|Z) }[/math], called the interaction information, may be positive, negative, or zero. This is the case even when random variables are pairwise independent. Such is the case when: [math]\displaystyle{ X \sim \mathrm{Bernoulli}(0.5), Z \sim \mathrm{Bernoulli}(0.5), \quad Y=\left\{\begin{array}{ll} X & \text{if }Z=0\\ 1-X & \text{if }Z=1 \end{array}\right. }[/math]in which case [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ Z }[/math] are pairwise independent and in particular [math]\displaystyle{ I(X;Y)=0 }[/math], but [math]\displaystyle{ I(X;Y|Z)=1. }[/math]

Chain rule for mutual information

[math]\displaystyle{ I(X;Y,Z) = I(X;Z) + I(X;Y|Z) }[/math]

Multivariate mutual information

Main article: Multivariate mutual information

The conditional mutual information can be used to inductively define a multivariate mutual information in a set- or measure-theoretic sense in the context of information diagrams. In this sense we define the multivariate mutual information as follows:

[math]\displaystyle{ I(X_1;\ldots;X_{n+1}) = I(X_1;\ldots;X_n) - I(X_1;\ldots;X_n|X_{n+1}), }[/math]

where

[math]\displaystyle{ I(X_1;\ldots;X_n|X_{n+1}) = \mathbb{E}_{X_{n+1}} [D_{\mathrm{KL}}( P_{(X_1,\ldots,X_n)|X_{n+1}} \| P_{X_1|X_{n+1}} \otimes\cdots\otimes P_{X_n|X_{n+1}} )]. }[/math]

This definition is identical to that of interaction information except for a change in sign in the case of an odd number of random variables. A complication is that this multivariate mutual information (as well as the interaction information) can be positive, negative, or zero, which makes this quantity difficult to interpret intuitively. In fact, for [math]\displaystyle{ n }[/math] random variables, there are [math]\displaystyle{ 2^n-1 }[/math] degrees of freedom for how they might be correlated in an information-theoretic sense, corresponding to each non-empty subset of these variables. These degrees of freedom are bounded by various Shannon- and non-Shannon-type inequalities in information theory.