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第10行： − 这里的维恩图显示了各种信息间的交并补运算关系关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的<font color="#ff8000"> '''联合熵 Joint Entropy'''</font><math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的<font color="#ff8000"> '''独立熵 Individual Entropy'''</font><math>H(X)</math>，红色（差集）部分表示X的<font color="#ff8000"> '''条件熵 Conditional Entropy'''</font><math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的<font color="#ff8000">'''互信息 Mutual Information，MI'''</font><math>\operatorname{I}(X;Y)</math>）。+ 第18行：
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第28行： − 不仅限于实值随机变量和线性依赖之类的的相关系数，互信息表示的关系其实更加普遍，它决定了一对变量<math>(X,Y)</math>的联合分布与<math>X</math>和<math>Y</math>的<font color="#ff8000">'''边缘分布 Marginal Distributions'''</font>之积的不同程度。互信息是'''点互信息 Pointwise Mutual Information，PMI'''的期望值。+ 第50行：
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第65行： − 需要注意的是，根据KL散度的性质，当两个随机变量的联合分布与其分别的边缘分布的乘积相等时，即当<math>X</math>和<math>Y</math>是相互独立的时，<math>I(X;Y)</math>等于零（因此已知<math>Y</math>的信息并不能得到任何关于<math>X</math>的信息）。'''<font color="#32CD32">一般来说，<math>I(X;Y)</math>是非负的，因为它是将<math>(X,Y)</math>作为一对独立随机变量来编码进而来进行价格（价值）度量的，但实际上它们并不一定是非负的。</font>'''+ 第73行：
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第127行： − 互信息是在独立假设下，<math>X</math和<math>Y</math>的联合分布相对于其内在相关性的度量。因此互信息是在以下条件下定义相关性的：<math>\operatorname{I}(X;Y)=0</math>当且仅当<math>X</math和<math>Y</math>是独立随机变量时。这很容易从一个方向看出：如果<math>X</math和<math>Y</math>是独立的，那么<math>p_{(X,Y)}(x,y)=p_X(x) \cdot p_Y(y)</math>，因此：+ + + + + 第152行：
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第195行： − 就输出<math>Y</math>是输入<math>X</math>的噪声版本的通信信道而言，这些关系如图中总结所示：+ 第204行：
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第218行： − 同理，上述其他恒等式的证明方法都是是相似的。一般情况(不仅仅是离散情况)的证明是类似的，用积分代替求和。+ 第224行：
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第248行： − 互信息是边缘分布乘积的KL散度<math>D_{KL}</math>，也就是联合分布<math>p_{(X,Y)}</math>的乘积，即：+ 第284行：
第288行： − 因此，互信息也可以理解为X的单变量分布<math>p_X</math>与给定<math>Y</math>的<math>X</math>的条件分布<math>p_{X|Y}</math>的KL散度的期望：平均分布<math>p_{X|Y}</math>和<math>p_X</math>的分布差异越大，信息增益越大。+ 第298行：
第302行： − 关于这方面的第一项工作是文献[2]。后来的研究人员重新推导了文献[3]中的内容，并扩展了关于[4]的分析。+ 第306行：
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第334行： − 另一方面，人们可能有兴趣知道在因子分解过程中，有<math>p(x,y)</math>携带了多少信息。在这种情况下，全分布<math>p(x,y)</math>通过矩阵因子分解所携带的多余信息由KL散度给出+ 第344行：
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第356行： − 为了适应不同的需要，已经提出了几种互信息的变形。其中包括对两个以上变量的规范化变量和泛化。+ 第388行：
第392行： − 满足度量的性质（三角形不等式、非负性、不可除性和对称性）。这种距离度量也称为信息的变化。+ 第410行：
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第471行： − 有时度量两个随机变量在以第三个随机变量为条件的情况下的相互信息也是有意义的。+ 第500行：
第504行： − 对于联合离散随机变量，这采用以下形式：+ 第532行：
第536行： − 对于联合连续的随机变量，其形式为：+ 第584行：
第588行： − 目前提出了许多将互信息推广到两个以上随机变量的方法，如'''<font color="#ff8000">全相关 Total Correlation</font>'''（或'''<font color="#ff8000">多信息 Multi-Information</font>'''）以及'''<font color="#ff8000">交互信息 Interaction Information</font>'''。多元高阶互信息的表达和研究是在两部看似独立的著作中实现的：McGill（1954年）在文献[8]中将这些函数统称为“互信息”，胡国亭（1962年）也在文献[9]中首次证明了大于2度的互信息可能是负的，并在文献[10]中用代数的方法证明了互信息和维恩图的直观对应关系。+ 第634行：
第638行： − （这个多元互信息的定义与互信息的定义相同，对于随机变量的数目为奇数时符号的变化除外。）+ 第647行：
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第662行： − 对于3个变量，Brenner等人将多变量互信息应用到神经编码中，并将其称为'''<font color="#ff8000">负面“协同作用” Negativity "Synergy"</font>'''，接着Watkinson 等人将其应用到基因表达上。对于任意k个变量，Tapia 等人将多元互信息应用于基因表达——'''<font color="#32CD32">正性对应于一般化成对相关性的关系，无效性对应于一个精确的独立性概念，负性检测高维“涌现”关系和聚合数据点</font>'''。+ 第666行：
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第708行： − 互信息的规范化变量由约束系数、不确定系数或熟练程度组成: + 第765行：
第772行： − 它表示两个不确定系数的'''<font color="#ff8000">调和平均数 Harmonic Mean</font>'''。+ 第785行：
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第810行： − 有一个标准化的名字——它起源于最初把互信息看作是'''<font color="#ff8000">协方差 Covariance</font>'''的类比（因此香农熵类似于方差）。然后计算归一化互信息类似于'''<font color="#ff8000">皮尔森相关系数 Pearson Product-moment</font>'''：+ 第813行：
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第879行： − 概率分布可以被看作是集合划分。然后有人可能会问: 如果一个集合被随机分割，概率的分布会是什么？相互信息的期望值是什么？我们用'''<font color="#ff8000">调整后的互信息 Adjusted Mutual Information</font>'''或 AMI 减去 MI 的期望值，这样当两个不同的分布是随机的时候 AMI 为零，当两个分布是相同的时候 AMI 为零。AMI的定义类似于一个集合的两个不同分区的'''<font color="#ff8000">调整后的Rand指数 Adjusted Rand Index</font>'''。+ 第972行：
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第1,026行： − 在所有的通信信道中，信息的最大化分配是在所有的通信信道上进行的。+ − 第1,046行：
第1,058行： − 互信息用于确定数据集中两个不同'''<font color="#ff8000">聚类 Clusterings</font>'''的相似性。因此，它与传统的Rand指数相比具有一定的优势。+ 第1,052行：
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无编辑摘要
已由[[用户:Yillia Jing]]进行初步翻译。{{Information theory}}
已由[[用户:Yillia Jing]]进行初步翻译，已由[[用户:Flipped]]进行审校。{{Information theory}}
[[File:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|[[Venn diagram]] showing additive and subtractive relationships various information measures associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the [[joint entropy 这里的维恩图显示了各种信息间的交并补运算关系关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的<font color="#ff8000"> '''联合熵 Joint entropy'''</font><math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的<font color="#ff8000"> '''独立熵 Individual entropy'''</font><math>H(X)</math>，红色（差集）部分表示X的<font color="#ff8000"> '''条件熵 Conditional entropy'''</font><math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的<font color="#ff8000">'''互信息 Mutual information，MI'''</font><math>\operatorname{I}(X;Y)</math>）。]]
[[File:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|[[Venn diagram]] showing additive and subtractive relationships various information measures associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the [[joint entropy 这里的维恩图显示了各种信息间的交并补运算关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的<font color="#ff8000"> '''联合熵 Joint entropy'''</font><math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的<font color="#ff8000"> '''独立熵 Individual entropy'''</font><math>H(X)</math>，红色（差集）部分表示X的<font color="#ff8000"> '''条件熵 Conditional entropy'''</font><math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的<font color="#ff8000">'''互信息 Mutual information，（MI）'''</font><math>\operatorname{I}(X;Y)</math>）。]]
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）图片应该按照[图1：英文+中文]
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）图片应该按照[图1：英文+中文]
<math>H(X,Y)</math>. The circle on the left (red and violet) is the [[Entropy (information theory)|individual entropy]] <math>H(X)</math>, with the red being the [[conditional entropy]] <math>H(X|Y)</math>. The circle on the right (blue and violet) is <math>H(Y)</math>, with the blue being <math>H(Y|X)</math>. The violet is the [[mutual information]] <math>\operatorname{I}(X;Y)</math>. 这里的维恩图显示了各种信息间的交并补运算关系关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的<font color="#ff8000"> '''联合熵 Joint entropy'''</font><math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的<font color="#ff8000"> '''独立熵 Individual entropy'''</font><math>H(X)</math>，红色（差集）部分表示X的<font color="#ff8000"> '''条件熵 Conditional entropy'''</font><math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的<font color="#ff8000">'''互信息 Mutual information，MI'''</font><math>\operatorname{I}(X;Y)</math>）。]]
<math>H(X,Y)</math>. The circle on the left (red and violet) is the [[Entropy (information theory)|individual entropy]] <math>H(X)</math>, with the red being the [[conditional entropy]] <math>H(X|Y)</math>. The circle on the right (blue and violet) is <math>H(Y)</math>, with the blue being <math>H(Y|X)</math>. The violet is the [[mutual information]] <math>\operatorname{I}(X;Y)</math>. 这里的维恩图显示了各种信息间的交并补运算关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的<font color="#ff8000"> '''联合熵 Joint entropy'''</font><math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的<font color="#ff8000"> '''独立熵 Individual entropy'''</font><math>H(X)</math>，红色（差集）部分表示X的<font color="#ff8000"> '''条件熵 Conditional entropy'''</font><math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的<font color="#ff8000">'''互信息 Mutual information（MI） '''</font><math>\operatorname{I}(X;Y)</math>）。]]
Venn diagram showing additive and subtractive relationships various information measures associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the joint entropy <math>H(X,Y)</math>. The circle on the left (red and violet) is the individual entropy <math>H(X)</math>, with the red being the conditional entropy <math>H(X|Y)</math>. The circle on the right (blue and violet) is <math>H(Y)</math>, with the blue being <math>H(Y|X)</math>. The violet is the mutual information <math>\operatorname{I}(X;Y)</math>.
Venn diagram showing additive and subtractive relationships various information measures associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the joint entropy <math>H(X,Y)</math>. The circle on the left (red and violet) is the individual entropy <math>H(X)</math>, with the red being the conditional entropy <math>H(X|Y)</math>. The circle on the right (blue and violet) is <math>H(Y)</math>, with the blue being <math>H(Y|X)</math>. The violet is the mutual information <math>\operatorname{I}(X;Y)</math>.
Venn diagram showing additive and subtractive relationships various information measures associated with correlated variables 𝑋 and 𝑌. The area contained by both circles is the joint entropy H(𝑋,𝑌). The circle on the left (red and violet) is the individual entropy H(𝑋), with the red being the conditional entropy H(𝑋|𝑌). The circle on the right (blue and violet) is H(𝑌), with the blue being H(𝑌|𝑋). The violet is the mutual information I(𝑋;𝑌).
Venn diagram showing additive and subtractive relationships various information measures associated with correlated variables 𝑋 and 𝑌. The area contained by both circles is the joint entropy H(𝑋,𝑌). The circle on the left (red and violet) is the individual entropy H(𝑋), with the red being the conditional entropy H(𝑋|𝑌). The circle on the right (blue and violet) is H(𝑌), with the blue being H(𝑌|𝑋). The violet is the mutual information I(𝑋;𝑌).
这里的维恩图显示了各种信息间的交并补运算关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的<font color="#ff8000"> '''联合熵 Joint Entropy'''</font><math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的<font color="#ff8000"> '''独立熵 Individual Entropy'''</font><math>H(X)</math>，红色（差集）部分表示X的<font color="#ff8000"> '''条件熵 Conditional Entropy'''</font><math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的<font color="#ff8000">'''互信息 Mutual Information，（MI）'''</font><math>\operatorname{I}(X;Y)</math>）。
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons, commonly called bits) obtained about one random variable through observing the other random variable. The concept of mutual information is intricately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons, commonly called bits) obtained about one random variable through observing the other random variable. The concept of mutual information is intricately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.
在<font color="#ff8000"> '''概率论 Probability Theory'''</font>和<font color="#ff8000"> '''信息论 Information Theory'''</font>理论中，两个随机变量的<font color="#ff8000"> '''互信息 Mutual Information，MI'''</font>是两个变量之间相互依赖程度的度量。更具体地说，它量化了通过观察一个随机变量而可以获得的关于另一个随机变量的“信息量”（单位如''香农 Shannons''，通常称为比特）。互信息的概念与随机变量的熵之间有着错综复杂的联系，熵是信息论中的一个基本概念，它量化了随机变量中所包含的预期“信息量”。
在<font color="#ff8000"> '''概率论 Probability Theory'''</font>和<font color="#ff8000"> '''信息论 Information Theory'''</font>理论中，两个随机变量的互信息是两个变量之间相互依赖程度的度量。更具体地说，通过观察一个随机变量而可以获得的关于另一个随机变量的“信息量”，互信息将其量化（单位如''香农 Shannons''，通常称为比特）。互信息的概念与随机变量的熵之间有着错综复杂的联系，熵是信息论中的一个基本概念，它量化了随机变量中所包含的预期“信息量”。
Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair <math>(X,Y)</math> is to the product of the marginal distributions of <math>X</math> and <math>Y</math>. MI is the expected value of the pointwise mutual information (PMI).
Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair <math>(X,Y)</math> is to the product of the marginal distributions of <math>X</math> and <math>Y</math>. MI is the expected value of the pointwise mutual information (PMI).
不仅限于实值随机变量和线性相关性（如相关系数），互信息表示的关系其实更加普遍，它决定了一对变量<math>(X,Y)</math>的联合分布与<math>X</math>和<math>Y</math>的<font color="#ff8000">'''边缘分布 Marginal Distributions'''</font>之积的不同程度。互信息是'''点互信息 Pointwise Mutual Information，PMI'''的期望值。
Let <math>(X,Y)</math> be a pair of random variables with values over the space <math>\mathcal{X}\times\mathcal{Y}</math>. If their joint distribution is <math>P_{(X,Y)}</math> and the marginal distributions are <math>P_X</math> and <math>P_Y</math>, the mutual information is defined as
Let <math>(X,Y)</math> be a pair of random variables with values over the space <math>\mathcal{X}\times\mathcal{Y}</math>. If their joint distribution is <math>P_{(X,Y)}</math> and the marginal distributions are <math>P_X</math> and <math>P_Y</math>, the mutual information is defined as
设一对随机变量<math>(X,Y)</math>的参数空间为<math>\mathcal{X}\times\mathcal{Y}</math>。若它们之间的的联合概率分布为<math>P_{(X,Y)}</math>，边缘分布分别为<math>P_X</math>和<math>P_Y</math>，则它们之间的互信息定义为：
设一对随机变量<math>(X,Y)</math>的参数空间为<math>\mathcal{X}\times\mathcal{Y}</math>。若它们之间的联合分布为<math>P_{(X,Y)}</math>，边缘分布分别为<math>P_X</math>和<math>P_Y</math>，则它们之间的互信息定义为：
Notice, as per property of the [[Kullback–Leibler divergence]], that <math>I(X;Y)</math> is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when <math>X</math> and <math>Y</math> are independent (and hence observing <math>Y</math> tells you nothing about <math>X</math>). '''<font color="#32CD32">In general <math>I(X;Y)</math> is non-negative, it is a measure of the price for encoding <math>(X,Y)</math> as a pair of independent random variables, when in reality they are not.</font>'''
Notice, as per property of the [[Kullback–Leibler divergence]], that <math>I(X;Y)</math> is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when <math>X</math> and <math>Y</math> are independent (and hence observing <math>Y</math> tells you nothing about <math>X</math>). '''<font color="#32CD32">In general <math>I(X;Y)</math> is non-negative, it is a measure of the price for encoding <math>(X,Y)</math> as a pair of independent random variables, when in reality they are not.</font>'''
需要注意的是，根据KL散度的性质，当两个随机变量的联合分布与其分别的边缘分布的乘积相等时，如当<math>X</math>和<math>Y</math>是相互独立时（因此观察y不会得到x的信息），<math>I(X;Y)</math>等于零（因此已知<math>Y</math>的信息并不能得到任何关于<math>X</math>的信息）。'''<font color="#32CD32">一般来说，<math>I(X;Y)</math>是非负的，因为它是将<math>(X,Y)</math>作为一对独立随机变量来编码进而进行价值度量的，但实际上它们并不一定是非负的。</font>'''
== 关于离散分布的PMF In terms of PMFs for discrete distributions ==
== 关于离散分布的PMF In terms of PMFs for discrete distributions ==
The mutual information of two jointly discrete random variables <math>X</math> and <math>Y</math> is calculated as a double sum:
The mutual information of two jointly discrete random variables <math>X</math> and <math>Y</math> is calculated as a double sum:
两个联合分布的离散型随机变量X和Y的互信息计算表现为双和的形式：
两个联合分布的离散型随机变量X和Y的互信息计算表现为双和的形式：<ref name=cover1991>{{cite book|last1=Cover|first1=T.M.|last2=Thomas|first2=J.A.|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration|date=1991|isbn=978-0-471-24195-9|edition=Wiley}}</ref>{{rp|20}}
[[文件:MI pic2.png|居中|缩略图]]
[[文件:MI pic2.png|居中|缩略图]]
where <math>p_{(X,Y)}</math> is the joint probability mass function of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the marginal probability mass functions of <math>X</math> and <math>Y</math> respectively.
where <math>p_{(X,Y)}</math> is the joint probability mass function of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the marginal probability mass functions of <math>X</math> and <math>Y</math> respectively.
<math>p_{(X,Y)}</math>是<math>X</math>和<math>Y</math>的'''<font color="#ff8000">概率质量函数 Probability Mass Functions</font>'''，而<math>p_X</math>和<math>p_Y</math>分别是数学<math>X</math>和<math>Y</math>的'''<font color="#ff8000">边缘概率质量函数 Marginal Probability Mass Functions</font>'''。
<math>p_{(X,Y)}</math>是<math>X</math>和<math>Y</math>的'''<font color="#ff8000">联合概率质量函数 Probability Mass Functions</font>'''，而<math>p_X</math>和<math>p_Y</math>分别是数学<math>X</math>和<math>Y</math>的'''<font color="#ff8000">边缘概率质量函数 Marginal Probability Mass Functions</font>'''。
== 连续分布的PDF In terms of PDFs for continuous distributions ==
== 连续分布的PDF In terms of PDFs for continuous distributions ==
In the case of jointly continuous random variables, the double sum is replaced by a double integral:
In the case of jointly continuous random variables, the double sum is replaced by a double integral:
在联合分布的随机变量为连续型的情况下，公式中的二重求和用二重积分代替:
在联合分布的随机变量为连续型的情况下，公式中的二重求和用二重积分代替: <ref name=cover1991 />{{rp|251}}
[[文件:MI pic3.png|居中|缩略图]]
[[文件:MI pic3.png|居中|缩略图]]
where <math>p_{(X,Y)}</math> is now the joint probability density function of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the marginal probability density functions of <math>X</math> and <math>Y</math> respectively.
where <math>p_{(X,Y)}</math> is now the joint probability density function of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the marginal probability density functions of <math>X</math> and <math>Y</math> respectively.
式中，<math>p_{(X,Y)}</math>是<math>X</math>和<math>Y</math>的联合概率密度函数，而<math>p_X</math>和<math>p_Y</math>分别是<math>X</math>和<math>Y</math>的'''<font color="#ff8000">边缘概率密度函数 Probability Density Function</font>'''。
式中，<math>p_{(X,Y)}</math>是<math>X</math>和<math>Y</math>的联合概率密度函数，而<math>p_X</math>和<math>p_Y</math>分别是<math>X</math>和<math>Y</math>的边缘概率密度函数。
Intuitively, mutual information measures the information that <math>X</math> and <math>Y</math> share: It measures how much knowing one of these variables reduces uncertainty about the other. For example, if <math>X</math> and <math>Y</math> are independent, then knowing <math>X</math> does not give any information about <math>Y</math> and vice versa, so their mutual information is zero. At the other extreme, if <math>X</math> is a deterministic function of <math>Y</math> and <math>Y</math> is a deterministic function of <math>X</math> then all information conveyed by <math>X</math> is shared with <math>Y</math>: knowing <math>X</math> determines the value of <math>Y</math> and vice versa. As a result, in this case the mutual information is the same as the uncertainty contained in <math>Y</math> (or <math>X</math>) alone, namely the entropy of <math>Y</math> (or <math>X</math>). Moreover, this mutual information is the same as the entropy of <math>X</math> and as the entropy of <math>Y</math>. (A very special case of this is when <math>X</math> and <math>Y</math> are the same random variable.)
Intuitively, mutual information measures the information that <math>X</math> and <math>Y</math> share: It measures how much knowing one of these variables reduces uncertainty about the other. For example, if <math>X</math> and <math>Y</math> are independent, then knowing <math>X</math> does not give any information about <math>Y</math> and vice versa, so their mutual information is zero. At the other extreme, if <math>X</math> is a deterministic function of <math>Y</math> and <math>Y</math> is a deterministic function of <math>X</math> then all information conveyed by <math>X</math> is shared with <math>Y</math>: knowing <math>X</math> determines the value of <math>Y</math> and vice versa. As a result, in this case the mutual information is the same as the uncertainty contained in <math>Y</math> (or <math>X</math>) alone, namely the entropy of <math>Y</math> (or <math>X</math>). Moreover, this mutual information is the same as the entropy of <math>X</math> and as the entropy of <math>Y</math>. (A very special case of this is when <math>X</math> and <math>Y</math> are the same random variable.)
直观地说，互信息衡量了<math>X</math> and <math>Y</math>的信息共享程度：它衡量了当已知其中一个变量后可以减少另一个变量多少的不确定性。例如，若<math>X</math>和<math>Y</math>是相互独立的，那么已知<math>X</math>不会得到关于<math>Y</math>的任何信息，反之亦然，因此它们之间的互信息为零。而另一种极端情况就是，若<math>X</math>是<math>Y</math>的'''<font color="#32CD32">确定函数</font>'''，而<math>X</math>也是<math>X</math>自身的确定函数，则<math>X</math>传递的所有信息都与<math>Y</math>共享：即已知<math>X</math>就可以知道<math>Y</math>的值，反之亦然。因此，在这种情况下，互信息与仅包含在<math>Y</math>（或<math>X</math>）中的不确定性相同，即<math>Y</math>（或<math>X</math>）的熵相同。此外，这种情况下互信息与<math>X</math>的熵和<math>Y</math>的熵相同。（一个非常特殊的情况是当<math>X</math>和<math>Y</math>是相同的随机变量。）
直观地说，互信息衡量了<math>X</math> 和 <math>Y</math>的信息共享程度：当已知其中一个变量后，它可以衡量了另一个变量减少的不确定性。例如，若<math>X</math>和<math>Y</math>是相互独立的，那么已知<math>X</math>不会得到关于<math>Y</math>的任何信息，反之亦然，因此它们之间的互信息为零。而另一种极端情况就是，若<math>X</math>是<math>Y</math>的确定函数，而<math>Y</math>也是<math>X</math>的确定函数，则<math>X</math>传递的所有信息都与<math>Y</math>共享：即已知<math>X</math>就可以知道<math>Y</math>的值，反之亦然。因此，在这种情况下，互信息与仅包含在<math>Y</math>（或<math>X</math>）中的不确定性相同，即<math>Y</math>（或<math>X</math>）的熵相同。此外，这种情况下互信息与<math>X</math>的熵，<math>Y</math>的熵相同。（一个非常特殊的情况是当<math>X</math>和<math>Y</math>是相同的随机变量。）
Mutual information is a measure of the inherent dependence expressed in the joint distribution of 𝑋 and 𝑌 relative to the joint distribution of 𝑋 and 𝑌 under the assumption of independence. Mutual information therefore measures dependence in the following sense: I(𝑋;𝑌)=0 if and only if 𝑋 and 𝑌 are independent random variables. This is easy to see in one direction: if 𝑋 and 𝑌 are independent, then 𝑝(𝑋,𝑌)(𝑥,𝑦)=𝑝𝑋(𝑥)⋅𝑝𝑌(𝑦), and therefore:
Mutual information is a measure of the inherent dependence expressed in the joint distribution of 𝑋 and 𝑌 relative to the joint distribution of 𝑋 and 𝑌 under the assumption of independence. Mutual information therefore measures dependence in the following sense: I(𝑋;𝑌)=0 if and only if 𝑋 and 𝑌 are independent random variables. This is easy to see in one direction: if 𝑋 and 𝑌 are independent, then 𝑝(𝑋,𝑌)(𝑥,𝑦)=𝑝𝑋(𝑥)⋅𝑝𝑌(𝑦), and therefore:
--[[用户:flipped| flipped]]（[[用户讨论: flipped |第一句话有一点点不理解
in the [[joint distribution]] of <math>X</math> and <math>Y</math> relative to the joint distribution of <math>X</math> and <math>Y</math>]]）
<font color="#32cd32"> </font>
互信息是在独立假设下，<math>X</math> 和<math>Y</math>的联合分布相对于其内在相关性的度量。因此互信息是在以下条件下定义相关性的：当且仅当<math>X</math和<math>Y</math>是独立随机变量时，<math>\operatorname{I}(X;Y)=0</math>。这很容易得出：如果<math>X</math和<math>Y</math>是独立的，那么<math>p_{(X,Y)}(x,y)=p_X(x) \cdot p_Y(y)</math>，因此：
Using Jensen's inequality on the definition of mutual information we can show that <math>\operatorname{I}(X;Y)</math> is non-negative, i.e.
Using Jensen's inequality on the definition of mutual information we can show that <math>\operatorname{I}(X;Y)</math> is non-negative, i.e.
利用'''<font color="#ff8000">琴生不等式 Jensen's Inequality</font>'''对互信息的定义进行推导，我们可以证明<math>\operatorname{I}(X;Y)</math>是非负的，即:
利用'''<font color="#ff8000">琴生不等式 Jensen's Inequality</font>'''对互信息的定义进行推导，我们可以证明<math>\operatorname{I}(X;Y)</math>是非负的，即: <ref name=cover1991 />{{rp|28}}
<math>\operatorname{I}(X;Y) \ge 0</math>
<math>\operatorname{I}(X;Y) \ge 0</math>
对于输出<math>Y</math>是输入<math>X</math>的噪声版本的通信通道而言，这些关系如图中总结所示：
因为<math>\operatorname{I}(X;Y)</math>是非负的，因此<math>H(X) \ge H(X|Y)</math>。这里我们给出了联合离散随机变量情形下结论<math>\operatorname{I}(X;Y)=H(Y)-H(Y|X)</math>的详细推导过程：
因为<math>\operatorname{I}(X;Y)</math>是非负的，因此<math>H(X) \ge H(X|Y)</math>。这里我们给出了联合离散随机变量情形下，结论<math>\operatorname{I}(X;Y)=H(Y)-H(Y|X)</math>的详细推导过程：
The proofs of the other identities above are similar. The proof of the general case (not just discrete) is similar, with integrals replacing sums.
The proofs of the other identities above are similar. The proof of the general case (not just discrete) is similar, with integrals replacing sums.
同理，上述其他恒等式的证明方法都是相似的。一般情况(不仅仅是离散情况)的证明是类似的，用积分代替求和。
Intuitively, if entropy 𝐻(𝑌) is regarded as a measure of uncertainty about a random variable, then 𝐻(𝑌|𝑋) is a measure of what 𝑋 does not say about 𝑌. This is "the amount of uncertainty remaining about 𝑌 after 𝑋 is known", and thus the right side of the second of these equalities can be read as "the amount of uncertainty in 𝑌, minus the amount of uncertainty in 𝑌 which remains after 𝑋 is known", which is equivalent to "the amount of uncertainty in 𝑌 which is removed by knowing 𝑋". This corroborates the intuitive meaning of mutual information as the amount of information (that is, reduction in uncertainty) that knowing either variable provides about the other.
Intuitively, if entropy 𝐻(𝑌) is regarded as a measure of uncertainty about a random variable, then 𝐻(𝑌|𝑋) is a measure of what 𝑋 does not say about 𝑌. This is "the amount of uncertainty remaining about 𝑌 after 𝑋 is known", and thus the right side of the second of these equalities can be read as "the amount of uncertainty in 𝑌, minus the amount of uncertainty in 𝑌 which remains after 𝑋 is known", which is equivalent to "the amount of uncertainty in 𝑌 which is removed by knowing 𝑋". This corroborates the intuitive meaning of mutual information as the amount of information (that is, reduction in uncertainty) that knowing either variable provides about the other.
理论上来说，如果熵<math>H(Y)</math>被视为随机变量不确定性的度量，那么<math>H(Y|X)</math>则是对<math>X</math>没有说明<math>Y</math>的程度的度量。也就是“已知<math>X</math>后，关于<math>Y</math>剩余的不确定性”的度量，因此这些等式中第二个等式的右侧可以解读为“<math>Y</math>中的不确定性量，减去已知<math>X</math>后仍然存在的不确定性的量”，相当于“已知后消除的<math>Y</math>中的不确定性量”<math>X</math>".这证实了相互信息的直观含义就是了解其中一个变量提供的关于另一个变量的信息量（即不确定性的减少程度）。
理论上来说，如果熵<math>H(Y)</math>被视为随机变量不确定性的度量，那么<math>H(Y|X)</math>则是对<math>X</math>没有说明<math>Y</math>的程度的度量。也就是“已知<math>X</math>后，关于<math>Y</math>剩余的不确定性”的度量，因此这些等式中第二个等式的右侧可以解读为“<math>Y</math>的不确定性的量，减去已知<math>X</math>后的<math>Y</math>中仍然存在不确定性的量”，相当于“已知<math>X</math>后消除的<math>Y</math>中的不确定性量” .这证实了互信息的直观含义就是了解其中一个变量提供的关于另一个变量的信息量（即不确定性的减少量）。
注意，在离散情况下，<math>H(X|X) = 0</math>，因此<math>H(X) = \operatorname{I}(X;X)</math>。所以，<math>\operatorname{I}(X; X) \ge \operatorname{I}(X; Y)</math>，据此我们可以得到一个基本结论，那就是一个变量至少包含与任何其他变量所能提供的关于自身的信息量的这么多信息。
注意，在离散情况下，<math>H(X|X) = 0</math>，因此<math>H(X) = \operatorname{I}(X;X)</math>。所以，<math>\operatorname{I}(X; X) \ge \operatorname{I}(X; Y)</math>，据此我们可以得到一个基本结论，那就是一个变量包含的信息量至少与任何其他变量所能提供的关于自身的信息量的一样多。
=== 与相对熵的关系 Relation to Kullback–Leibler divergence ===
=== 与相对熵的关系 Relation to Kullback–Leibler divergence ===
mutual information is the Kullback–Leibler divergence of the product of the marginal distributions, 𝑝𝑋⋅𝑝𝑌, from the joint distribution 𝑝(𝑋,𝑌), that is,
mutual information is the Kullback–Leibler divergence of the product of the marginal distributions, 𝑝𝑋⋅𝑝𝑌, from the joint distribution 𝑝(𝑋,𝑌), that is,
互信息是边缘分布乘积<math>p_X \cdot p_Y</math>的KL散度<math>D_{KL}</math>，也就是联合分布<math>p_{(X,Y)}</math>的乘积，即：
Note that here the Kullback–Leibler divergence involves integration over the values of the random variable <math>X</math> only, and the expression <math>D_\text{KL}(p_{X|Y} \parallel p_X)</math> still denotes a random variable because <math>Y</math> is random. Thus mutual information can also be understood as the expectation of the Kullback–Leibler divergence of the univariate distribution <math>p_X</math> of <math>X</math> from the conditional distribution <math>p_{X|Y}</math> of <math>X</math> given <math>Y</math>: the more different the distributions <math>p_{X|Y}</math> and <math>p_X</math> are on average, the greater the information gain.
Note that here the Kullback–Leibler divergence involves integration over the values of the random variable <math>X</math> only, and the expression <math>D_\text{KL}(p_{X|Y} \parallel p_X)</math> still denotes a random variable because <math>Y</math> is random. Thus mutual information can also be understood as the expectation of the Kullback–Leibler divergence of the univariate distribution <math>p_X</math> of <math>X</math> from the conditional distribution <math>p_{X|Y}</math> of <math>X</math> given <math>Y</math>: the more different the distributions <math>p_{X|Y}</math> and <math>p_X</math> are on average, the greater the information gain.
请注意，此处的KL散度仅涉及对随机变量<math>X</math>的值进行积分，并且表达式<math>D_\text{KL}(p_{X|Y} \parallel p_X)</math> 仍然表示随机变量，因为y是随机的。因此，互信息也可以理解为X的单变量分布<math>p_X</math>与给定<math>Y</math>的<math>X</math>的条件分布<math>p_{X|Y}</math>的KL散度的期望：平均分布<math>p_{X|Y}</math>和<math>p_X</math>的分布差异越大，信息增益越大。
=== 互信息的贝叶斯估计 Bayesian estimation of mutual information ===
=== 互信息的贝叶斯估计 Bayesian estimation of mutual information ===
关于这方面的第一项工作<ref>{{cite journal | last1 = Wolpert | first1 = D.H. | last2 = Wolf | first2 = D.R. | year = 1995 | title = Estimating functions of probability distributions from a finite set of samples | journal = Physical Review E | volume = 52 | issue = 6 | pages = 6841–6854 | doi = 10.1103/PhysRevE.52.6841 | pmid = 9964199 | citeseerx = 10.1.1.55.7122 | bibcode = 1995PhRvE..52.6841W }}</ref>,它还展示了如何对贝叶斯估计进行除互信息之外的许多其他信息理论性质。后来的研究人员重新推导了<ref>{{cite journal | last1 = Hutter | first1 = M. | year = 2001 | title = Distribution of Mutual Information | journal = Advances in Neural Information Processing Systems 2001 }}</ref>内容，并扩展了<ref>{{cite journal | last1 = Archer | first1 = E. | last2 = Park | first2 = I.M. | last3 = Pillow | first3 = J. | year = 2013 | title = Bayesian and Quasi-Bayesian Estimators for Mutual Information from Discrete Data | journal = Entropy| volume = 15 | issue = 12 | pages = 1738–1755 | doi = 10.3390/e15051738 | citeseerx = 10.1.1.294.4690 | bibcode = 2013Entrp..15.1738A }}</ref>分析。
最近的一篇论文[5]，该论文基于一个专门针对相互信息本身估计的先验知识。
最近的一篇论文<ref>{{cite journal | last1 = Wolpert | first1 = D.H | last2 = DeDeo | first2 = S. | year = 2013 | title = Estimating Functions of Distributions Defined over Spaces of Unknown Size | journal = Entropy | volume = 15 | issue = 12 | pages = 4668–4699 | doi = 10.3390/e15114668 | arxiv = 1311.4548 | bibcode = 2013Entrp..15.4668W }}</ref>中基于一个专门针对互信息本身估计的先验知识。
Besides, recently an estimation method accounting for continuous and multivariate outputs, <math>Y</math>, was proposed in <ref>{{citation| journal = [[PLOS Computational Biology]]|volume = 15|issue = 7|pages = e1007132|doi = 10.1371/journal.pcbi.1007132|pmid = 31299056|pmc = 6655862|title=Information-theoretic analysis of multivariate single-cell signaling responses|author1= Tomasz Jetka|author2= Karol Nienaltowski|author3= Tomasz Winarski| author4=Slawomir Blonski| author5= Michal Komorowski|year=2019|bibcode = 2019PLSCB..15E7132J|arxiv = 1808.05581}}</ref>.
Besides, recently an estimation method accounting for continuous and multivariate outputs, <math>Y</math>, was proposed in <ref>{{citation| journal = [[PLOS Computational Biology]]|volume = 15|issue = 7|pages = e1007132|doi = 10.1371/journal.pcbi.1007132|pmid = 31299056|pmc = 6655862|title=Information-theoretic analysis of multivariate single-cell signaling responses|author1= Tomasz Jetka|author2= Karol Nienaltowski|author3= Tomasz Winarski| author4=Slawomir Blonski| author5= Michal Komorowski|year=2019|bibcode = 2019PLSCB..15E7132J|arxiv = 1808.05581}}</ref>.
此外，最近文献[6]提出了一种考虑连续多种输出变量𝑌的估计方法。
此外，最近文献<ref>{{citation| journal = [[PLOS Computational Biology]]|volume = 15|issue = 7|pages = e1007132|doi = 10.1371/journal.pcbi.1007132|pmid = 31299056|pmc = 6655862|title=Information-theoretic analysis of multivariate single-cell signaling responses|author1= Tomasz Jetka|author2= Karol Nienaltowski|author3= Tomasz Winarski| author4=Slawomir Blonski| author5= Michal Komorowski|year=2019|bibcode = 2019PLSCB..15E7132J|arxiv = 1808.05581}}</ref>提出了一种考虑连续多种输出变量𝑌的估计方法。
=== 独立性假设 Independence assumptions ===
=== 独立性假设 Independence assumptions ===
The Kullback-Leibler divergence formulation of the mutual information is predicated on that one is interested in comparing 𝑝(𝑥,𝑦) to the fully factorized outer product 𝑝(𝑥)⋅𝑝(𝑦). In many problems, such as non-negative matrix factorization, one is interested in '''<font color="#32CD32">less extreme factorizations</font>'''; specifically, one wishes to compare 𝑝(𝑥,𝑦) to a low-rank matrix approximation in some unknown variable 𝑤; that is, to what degree one might have
The Kullback-Leibler divergence formulation of the mutual information is predicated on that one is interested in comparing 𝑝(𝑥,𝑦) to the fully factorized outer product 𝑝(𝑥)⋅𝑝(𝑦). In many problems, such as non-negative matrix factorization, one is interested in '''<font color="#32CD32">less extreme factorizations</font>'''; specifically, one wishes to compare 𝑝(𝑥,𝑦) to a low-rank matrix approximation in some unknown variable 𝑤; that is, to what degree one might have
互信息的KL散度公式是基于这样一个结论的：人们会更关注将<math>p(x,y)</math>与完全分解的'''<font color="#ff8000">外积 Outer Product</font>'''<math>p(x) \cdot p(y)</math>进行比较。在许多问题中，例如'''<font color="#ff8000">非负矩阵因式分解 Non-negative matrix factorization</font>'''中，人们对'''<font color="#32CD32">较不极端的</font>'''因式分解感兴趣；具体地说，人们希望将<math>p(x,y)</math>与某个未知变量<math>w</math>中的低秩矩阵近似进行比较；也就是说，在多大程度上可能会有这样的结果：
互信息的KL散度公式是基于这样一个结论的：人们会更关注将<math>p(x,y)</math>与完全分解的'''<font color="#ff8000">外积 Outer Product</font>'''<math>p(x) \cdot p(y)</math>进行比较。在许多问题中，例如'''<font color="#ff8000">非负矩阵因式分解 Non-negative matrix factorization</font>'''，人们对较不极端的因式分解感兴趣；具体地说，人们希望将<math>p(x,y)</math>与某个未知变量<math>w</math>中的低秩矩阵近似进行比较；也就是说，在多大程度上可能会有这样的结果：
:<math>p(x,y)\approx \sum_w p^\prime (x,w) p^{\prime\prime}(w,y)</math>
:<math>p(x,y)\approx \sum_w p^\prime (x,w) p^{\prime\prime}(w,y)</math>
Alternately, one might be interested in knowing how much more information 𝑝(𝑥,𝑦) carries over its factorization. In such a case, the excess information that the full distribution 𝑝(𝑥,𝑦) carries over the matrix factorization is given by the Kullback-Leibler divergence
Alternately, one might be interested in knowing how much more information 𝑝(𝑥,𝑦) carries over its factorization. In such a case, the excess information that the full distribution 𝑝(𝑥,𝑦) carries over the matrix factorization is given by the Kullback-Leibler divergence
另一方面，人们可能有兴趣了解在因式分解过程中， <math>p(x,y)</math>携带了多少信息。在这种情况下，全分布<math>p(x,y)</math>通过矩阵因式分解所携带的多余信息由KL散度给出
:<math>\operatorname{I}_{LRMA} = \sum_{y \in \mathcal{Y}} \sum_{x \in \mathcal{X}}
:<math>\operatorname{I}_{LRMA} = \sum_{y \in \mathcal{Y}} \sum_{x \in \mathcal{X}}
The conventional definition of the mutual information is recovered in the extreme case that the process <math>W</math> has only one value for <math>w</math>.
The conventional definition of the mutual information is recovered in the extreme case that the process <math>W</math> has only one value for <math>w</math>.
在过程<math>w</math>只有一个值的极端情况下，可以使用传统的互信息定义。
在过程<math> W </math>中，<math>w</math>只有一个值的极端情况下，可以使用传统的互信息定义。
== 变形 Variations ==
== 变形 Variations ==
Several variations on mutual information have been proposed to suit various needs. Among these are normalized variants and generalizations to more than two variables.
Several variations on mutual information have been proposed to suit various needs. Among these are normalized variants and generalizations to more than two variables.
为了适应不同的需要，已经提出了几种互信息的变形。其中包括变量归一化和对两个以上变量的泛化。
satisfies the properties of a metric (triangle inequality, non-negativity, indiscernability and symmetry). This distance metric is also known as the variation of information.
satisfies the properties of a metric (triangle inequality, non-negativity, indiscernability and symmetry). This distance metric is also known as the variation of information.
满足度量的性质（三角不等式、非负性、不可除性和对称性）。这种距离度量也称为信息的变化。
The metric 𝐷 is a universal metric, in that if any other distance measure places 𝑋 and 𝑌 close-by, then the 𝐷 will also judge them close.
The metric 𝐷 is a universal metric, in that if any other distance measure places 𝑋 and 𝑌 close-by, then the 𝐷 will also judge them close.
度量<math>D</math>是一种通用度量，即如果任何其他距离度量将<math>X</math>和<math>Y</math>认为是近的，则<math>D</math>也将判断它们接近。
度量<math>D</math>是一种通用度量，即如果任何其他距离度量将<math>X</math>和<math>Y</math>认为是近的，则<math>D</math>也将判断它们接近。<ref>{{cite journal|arxiv=q-bio/0311039|last1=Kraskov|first1=Alexander|title=Hierarchical Clustering Based on Mutual Information|last2=Stögbauer|first2=Harald|last3= Andrzejak|first3=Ralph G.|last4=Grassberger|first4=Peter|year=2003|bibcode=2003q.bio....11039K}}</ref>
Sometimes it is useful to express the mutual information of two random variables conditioned on a third.
Sometimes it is useful to express the mutual information of two random variables conditioned on a third.
有时，在以第三个随机变量为条件的情况下，表示两个随机变量的互信息也是有意义的。
For jointly discrete random variables this takes the form
For jointly discrete random variables this takes the form
对于联合离散随机变量，采用以下形式：
:<math>
:<math>
For jointly continuous random variables this takes the form
For jointly continuous random variables this takes the form
对于联合连续随机变量，其形式为：
:<math>
:<math>
Several generalizations of mutual information to more than two random variables have been proposed, such as total correlation (or multi-information) and interaction information. The expression and study of multivariate higher-degree mutual-information was achieved in two seemingly independent works: McGill (1954) who called these functions “interaction information”, and Hu Kuo Ting (1962) who also first proved the possible negativity of mutual-information for degrees higher than 2 and justified algebraically the intuitive correspondence to Venn diagrams
Several generalizations of mutual information to more than two random variables have been proposed, such as total correlation (or multi-information) and interaction information. The expression and study of multivariate higher-degree mutual-information was achieved in two seemingly independent works: McGill (1954) who called these functions “interaction information”, and Hu Kuo Ting (1962) who also first proved the possible negativity of mutual-information for degrees higher than 2 and justified algebraically the intuitive correspondence to Venn diagrams
目前提出了一些将互信息推广到两个以上随机变量的方法，如'''<font color="#ff8000">全相关 Total Correlation</font>'''（或'''<font color="#ff8000">多信息 Multi-Information</font>'''）以及'''<font color="#ff8000">交互信息 Interaction Information</font>'''。多元高阶互信息的表达和研究是在两部看似无关的著作中实现的：McGill 麦吉尔（1954年）<ref>{{cite journal | last1 = McGill| first1 = W. | year = 1954 | title = Multivariate information transmission | journal = Psychometrika | volume = 19 | issue = 1 | pages = 97–116 | doi = 10.1007/BF02289159 }}</ref>将这些函数统称为“交互信息”，胡国亭（1962年）也<ref>{{cite journal | last1 = Hu| first1 = K.T. | year = 1962 | title = On the Amount of Information | journal = Theory Probab. Appl. | volume = 7 | issue = 4 | pages = 439–447 | doi = 10.1137/1107041 }}</ref>首次证明了大于2度的互信息可能是负的，并在文献[10]中用代数的方法证明了互信息和维恩图的直观对应关系。
(This definition of multivariate mutual information is identical to that of interaction information except for a change in sign when the number of random variables is odd.)
(This definition of multivariate mutual information is identical to that of interaction information except for a change in sign when the number of random variables is odd.)
（这个多元互信息的定义与交互信息的定义相同，对于随机变量的数目为奇数时符号的变化除外。）
多元互信息函数将<math>I(X_1;X_2)=0</math>当且仅当<math>X_1,X_2</math>两两独立的情况推广到任意多变量。即当且仅当<math>2^n-n-1</math>的互信息函数为<math>I(X_1;...;X_k)=0</math>且math>n \ge k \ge 2</math>时，n个变量相互独立（定理2）。从这个意义上讲，<math>I(X_1;...;X_k)=0</math>可以用作一个精确的统计独立性标准。
多元互信息函数将<math>I(X_1;X_2)=0</math>当且仅当<math>X_1,X_2</math>两两独立的情况推广到任意多变量。当且仅当<math>2^n-n-1</math>的互信息函数为
<math>I(X_1;...;X_k)=0</math>且<math>n \ge k \ge 2</math>，n个变量相互独立（定理2<ref name=e21090869/>）。从这个意义上讲，<math>I(X_1;...;X_k)=0</math>可以用作一个精确的统计独立性标准。
--[[用户:flipped| flipped]]（[[用户讨论: flipped |第二句中的vanish不太理解]]）
==== 应用 Applications ====
==== 应用 Applications ====
For 3 variables, Brenner et al. applied multivariate mutual information to neural coding and called its negativity "synergy" and Watkinson et al. applied it to genetic expression . For arbitrary k variables, Tapia et al. applied multivariate mutual information to gene expression . '''<font color="#32CD32">The positivity corresponds to relations generalizing the pairwise correlations, nullity corresponds to a refined notion of independence, and negativity detects high dimensional "emergent" relations and clusterized datapoints </font>'''.
For 3 variables, Brenner et al. applied multivariate mutual information to neural coding and called its negativity "synergy" and Watkinson et al. applied it to genetic expression . For arbitrary k variables, Tapia et al. applied multivariate mutual information to gene expression . '''<font color="#32CD32">The positivity corresponds to relations generalizing the pairwise correlations, nullity corresponds to a refined notion of independence, and negativity detects high dimensional "emergent" relations and clusterized datapoints </font>'''.
对于3个变量，Brenner 布伦纳等人<ref>{{cite journal | last1 = Brenner | first1 = N. | last2 = Strong | first2 = S. | last3 = Koberle | first3 = R. | last4 = Bialek | first4 = W. | year = 2000 | title = Synergy in a Neural Code | doi = 10.1162/089976600300015259 | pmid = 10935917 | journal = Neural Comput | volume = 12 | issue = 7 | pages = 1531–1552 }}</ref>将多元互信息应用到神经编码中，并将其称为'''<font color="#ff8000">负面“协同作用” Negativity "Synergy"</font>'''，接着Watkinson 沃特森等人<ref>{{cite journal | last1 = Watkinson | first1 = J. | last2 = Liang | first2 = K. | last3 = Wang | first3 = X. | last4 = Zheng | first4 = T.| last5 = Anastassiou | first5 = D. | year = 2009 | title = Inference of Regulatory Gene Interactions from Expression Data Using Three-Way Mutual Information | doi = 10.1111/j.1749-6632.2008.03757.x | pmid = 19348651 | journal = Chall. Syst. Biol. Ann. N. Y. Acad. Sci. | volume = 1158 | issue = 1 | pages = 302–313 | bibcode = 2009NYASA1158..302W | url = https://semanticscholar.org/paper/cb09223a34b08e6dcbf696385d9ab76fd9f37aa4 }}</ref>.将其应用到基因表达上。对于任意k个变量，Tapia 塔皮亚 等人<ref name=s41598>{{cite journal|last1=Tapia|first1=M.|last2=Baudot|first2=P.|last3=Formizano-Treziny|first3=C.|last4=Dufour|first4=M.|last5=Goaillard|first5=J.M.|year=2018|title=Neurotransmitter identity and electrophysiological phenotype are genetically coupled in midbrain dopaminergic neurons|doi= 10.1038/s41598-018-31765-z|pmid=30206240|pmc=6134142|journal=Sci. Rep.|volume=8|issue=1|pages=13637|bibcode=2018NatSR...813637T}}</ref> <ref name=e21090869/>将多元互信息应用于基因表达——它可以是0，正，或负。cite journal | last1 = Hu| first1 = K.T. | year = 1962 | title = On the Amount of Information | journal = Theory Probab. Appl. | volume = 7 | issue = 4 | pages = 439–447 | doi = 10.1137/1107041 }}</ref>'''<font color="#32CD32">正性对应于一般化成对相关性的关系，无效性对应于一个精确的独立性概念，负性检测高维“涌现”关系和聚合数据点</font>'''<ref name=s41598/>。
One high-dimensional generalization scheme which maximizes the mutual information between the joint distribution and other target variables is found to be useful in feature selection.
One high-dimensional generalization scheme which maximizes the mutual information between the joint distribution and other target variables is found to be useful in feature selection.
目前已经提出了一种能够最大化联合分布与其他目标变量之间的互信息的高维推广方案，该方法可用于'''<font color="#ff8000"> 特征选择 Feature Selection</font>'''。
目前已经提出了一种能够最大化联合分布与其他目标变量之间的互信息的高维推广方案，该方法可用于'''<font color="#ff8000"> 特征选择 Feature Selection</font>'''。<ref>{{cite book|author1=Christopher D. Manning |author2=Prabhakar Raghavan |author3=Hinrich Schütze | title = An Introduction to Information Retrieval| publisher = [[Cambridge University Press]]| year = 2008| isbn = 978-0-521-86571-5 }}</ref>
Mutual information is also used in the area of signal processing as a [[Similarity measure|measure of similarity]] between two signals. For example, FMI metric<ref>{{cite journal | last1 = Haghighat | first1 = M. B. A. | last2 = Aghagolzadeh | first2 = A. | last3 = Seyedarabi | first3 = H. | year = 2011 | title = A non-reference image fusion metric based on mutual information of image features | doi = 10.1016/j.compeleceng.2011.07.012 | journal = Computers & Electrical Engineering | volume = 37 | issue = 5| pages = 744–756 }}</ref> is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The [[Matlab]] code for this metric can be found at.<ref>{{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/45926-feature-mutual-information-fmi-image-fusion-metric|title=Feature Mutual Information (FMI) metric for non-reference image fusion - File Exchange - MATLAB Central|author=|date=|website=www.mathworks.com|accessdate=4 April 2018}}</ref>
Mutual information is also used in the area of signal processing as a [[Similarity measure|measure of similarity]] between two signals. For example, FMI metric<ref>{{cite journal | last1 = Haghighat | first1 = M. B. A. | last2 = Aghagolzadeh | first2 = A. | last3 = Seyedarabi | first3 = H. | year = 2011 | title = A non-reference image fusion metric based on mutual information of image features | doi = 10.1016/j.compeleceng.2011.07.012 | journal = Computers & Electrical Engineering | volume = 37 | issue = 5| pages = 744–756 }}</ref> is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The [[Matlab]] code for this metric can be found at.<ref>{{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/45926-feature-mutual-information-fmi-image-fusion-metric|title=Feature Mutual Information (FMI) metric for non-reference image fusion - File Exchange - MATLAB Central|author=|date=|website=www.mathworks.com|accessdate=4 April 2018}}</ref>
Mutual information is also used in the area of signal processing as a measure of similarity between two signals. For example, FMI metric is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The Matlab code for this metric can be found at.
Mutual information is also used in the area of signal processing as a measure of similarity between two signals. For example, FMI metric is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The Matlab code for this metric can be found at.
互信息也用于信号处理领域，用来进行两个信号之间的'''<font color="#ff8000">相似性度量 Similarity Measure</font>'''。例如，FMI 度量是一种利用互信息来度量融合图像包含的关于源图像的信息量的图像融合性能度量。这个度量的 Matlab 代码可以在参考文献[17]中找到。
互信息也用于信号处理领域，用来进行两个信号之间的'''<font color="#ff8000">相似性度量 Similarity Measure</font>'''。例如，FMI 度量<ref>{{cite journal | last1 = Haghighat | first1 = M. B. A. | last2 = Aghagolzadeh | first2 = A. | last3 = Seyedarabi | first3 = H. | year = 2011 | title = A non-reference image fusion metric based on mutual information of image features | doi = 10.1016/j.compeleceng.2011.07.012 | journal = Computers & Electrical Engineering | volume = 37 | issue = 5| pages = 744–756 }}</ref>是一种图像融合性能度量，它利用互信息来度量融合图像包含的关于源图像的信息量。这个度量的 Matlab 代码可以找到<ref>{{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/45926-feature-mutual-information-fmi-image-fusion-metric|title=Feature Mutual Information (FMI) metric for non-reference image fusion - File Exchange - MATLAB Central|author=|date=|website=www.mathworks.com|accessdate=4 April 2018}}</ref>。
=== 定向信息 Directed information ===
=== 定向信息 Directed information ===
Note that if 𝑛=1, the directed information becomes the mutual information. Directed information has many applications in problems where causality plays an important role, such as capacity of channel with feedback.
Note that if 𝑛=1, the directed information becomes the mutual information. Directed information has many applications in problems where causality plays an important role, such as capacity of channel with feedback.
注意，当<math>n=1</math>时，则定向信息成为互信息。定向信息在因果关系问题中有着广泛的应用，如反馈'''<font color="#ff8000">信道容量问题 Channel Capacity</font>'''。
注意，当<math>n=1</math>时，则定向信息成为互信息。定向信息在因果关系问题中有着广泛的应用，如反馈'''<font color="#ff8000">信道容量问题 Channel Capacity</font>'''。<ref>{{cite conference|last1=Massey|first1=James|title=Causality, Feedback And Directed Informatio|date=1990|book-title=Proc. 1990 Intl. Symp. on Info. Th. and its Applications, Waikiki, Hawaii, Nov. 27-30, 1990|citeseerx=10.1.1.36.5688}}</ref><ref>{{cite journal|last1=Permuter|first1=Haim Henry|last2=Weissman|first2=Tsachy|last3=Goldsmith|first3=Andrea J.|title=Finite State Channels With Time-Invariant Deterministic Feedback|journal=IEEE Transactions on Information Theory|date=February 2009|volume=55|issue=2|pages=644–662|doi=10.1109/TIT.2008.2009849|arxiv=cs/0608070}}</ref>
=== 标准化变形 Normalized variants ===
=== 归一化变量 Normalized variants ===
Normalized variants of the mutual information are provided by the ''coefficients of constraint'',{{sfn|Coombs|Dawes|Tversky|1970}} [[uncertainty coefficient]]<ref name=pressflannery>{{Cite book|last1=Press|first1=WH|last2=Teukolsky |first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press|location=New York|isbn=978-0-521-88068-8|chapter=Section 14.7.3. Conditional Entropy and Mutual Information|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=758}}</ref> or proficiency:<ref name=JimWhite>{{Cite conference| last1= White |first1= Jim | last2= Steingold | first2=Sam | last3= Fournelle | first3=Connie | title = Performance Metrics for Group-Detection Algorithms | conference = Interface 2004 | url = http://www.interfacesymposia.org/I04/I2004Proceedings/WhiteJim/WhiteJim.paper.pdf}}</ref>
Normalized variants of the mutual information are provided by the ''coefficients of constraint'',{{sfn|Coombs|Dawes|Tversky|1970}} [[uncertainty coefficient]]<ref name=pressflannery>{{Cite book|last1=Press|first1=WH|last2=Teukolsky |first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press|location=New York|isbn=978-0-521-88068-8|chapter=Section 14.7.3. Conditional Entropy and Mutual Information|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=758}}</ref> or proficiency:<ref name=JimWhite>{{Cite conference| last1= White |first1= Jim | last2= Steingold | first2=Sam | last3= Fournelle | first3=Connie | title = Performance Metrics for Group-Detection Algorithms | conference = Interface 2004 | url = http://www.interfacesymposia.org/I04/I2004Proceedings/WhiteJim/WhiteJim.paper.pdf}}</ref>
Normalized variants of the mutual information are provided by the coefficients of constraint, uncertainty coefficient or proficiency:
Normalized variants of the mutual information are provided by the coefficients of constraint, uncertainty coefficient or proficiency:
互信息的归一化变量由约束系数、不确定系数<ref name=pressflannery>{{Cite book|last1=Press|first1=WH|last2=Teukolsky |first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press|location=New York|isbn=978-0-521-88068-8|chapter=Section 14.7.3. Conditional Entropy and Mutual Information|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=758}}</ref>或熟练程度组成<ref name=JimWhite>{{Cite conference| last1= White |first1= Jim | last2= Steingold | first2=Sam | last3= Fournelle | first3=Connie | title = Performance Metrics for Group-Detection Algorithms | conference = Interface 2004 | url = http://www.interfacesymposia.org/I04/I2004Proceedings/WhiteJim/WhiteJim.paper.pdf}}</ref>:
:<math>
:<math>
which represents the harmonic mean of the two uncertainty coefficients <math>C_{XY}, C_{YX}</math>.
which represents the harmonic mean of the two uncertainty coefficients <math>C_{XY}, C_{YX}</math>.
它表示两个不确定系数<math>C_{XY}, C_{YX}</math>的'''<font color="#ff8000">调和平均数 Harmonic Mean</font>'''<ref name=pressflannery />。
This normalized version also known as Information Quality Ratio (IQR) which quantifies the amount of information of a variable based on another variable against total uncertainty:
This normalized version also known as Information Quality Ratio (IQR) which quantifies the amount of information of a variable based on another variable against total uncertainty:
这个标准化版本也被称为'''<font color="#ff8000">信息质量比率 Information Quality Ratio, IQR</font>''' ，它根据另一个变量量化了一个变量的信息量，来对抗总的不确定性:
这个标准化版本也被称为'''<font color="#ff8000">信息质量比率 Information Quality Ratio（IQR）</font>''' ，它根据另一个变量，相对于总的不确定性来量化另一个变量的信息量: <ref name=DRWijaya>{{Cite journal| last1= Wijaya |first1= Dedy Rahman | last2= Sarno| first2=Riyanarto| last3= Zulaika | first3=Enny| title = Information Quality Ratio as a novel metric for mother wavelet selection| journal = Chemometrics and Intelligent Laboratory Systems| journal = Chemometrics and Intelligent Laboratory Systems| volume = 160| pages = 59–71| doi = 10.1016/j.chemolab.2016.11.012|year= 2017 }}</ref>
:<math>IQR(X, Y) = \operatorname{E}[\operatorname{I}(X;Y)]
:<math>IQR(X, Y) = \operatorname{E}[\operatorname{I}(X;Y)]
There's a normalization which derives from first thinking of mutual information as an analogue to [[covariance]] (thus [[Entropy (information theory)|Shannon entropy]] is analogous to [[variance]]). Then the normalized mutual information is calculated akin to the [[Pearson product-moment correlation coefficient|Pearson correlation coefficient]],
There's a normalization which derives from first thinking of mutual information as an analogue to [[covariance]] (thus [[Entropy (information theory)|Shannon entropy]] is analogous to [[variance]]). Then the normalized mutual information is calculated akin to the [[Pearson product-moment correlation coefficient|Pearson correlation coefficient]],
有一种归一化<ref name="strehl-jmlr02">{{cite journal| title = Cluster Ensembles – A Knowledge Reuse Framework for Combining Multiple Partitions| journal = The Journal of Machine Learning Research| pages = 583–617 | volume = 3 | year = 2003| last1 = Strehl | first1 = Alexander | last2 = Ghosh | first2 = Joydeep| doi=10.1162/153244303321897735| url=http://www.jmlr.org/papers/volume3/strehl02a/strehl02a.pdf}}</ref>起源于互信息的最初思想，看作是'''<font color="#ff8000">协方差 Covariance</font>'''的类比（因此香农熵类似于方差）。然后计算归一化互信息，类似于'''<font color="#ff8000">皮尔森相关系数 Pearson Product-moment</font>'''：
</math>
</math>
=== 加权变量 Weighted variants ===
=== 加权变量 Weighted variants ===
each event or object specified by <math>(x, y)</math> is weighted by the corresponding probability <math>p(x, y)</math>. This assumes that all objects or events are equivalent apart from their probability of occurrence. However, in some applications it may be the case that certain objects or events are more significant than others, or that certain patterns of association are more semantically important than others.
each event or object specified by <math>(x, y)</math> is weighted by the corresponding probability <math>p(x, y)</math>. This assumes that all objects or events are equivalent apart from their probability of occurrence. However, in some applications it may be the case that certain objects or events are more significant than others, or that certain patterns of association are more semantically important than others.
<math>(x, y)</math> 指定的每个事件或对象都由相应的概率<math>p(x, y)</math>进行加权。这假设所有的物体或事件除了发生的概率外都是相等的。然而，在某些应用场景中，某些对象或事件可能比其他对象或事件更重要，或者某些关联模式在语义上比其他模式更重要。
<math>(x, y)</math> 指定的每个事件或对象都由相应的概率<math>p(x, y)</math>进行加权。这假设所有的物体或事件除了发生的概率外都是相等的。然而，在某些应用场景中，某些特定的对象或事件可能比其他对象或事件更重要，或者某些特定的关联模式在语义上比其他模式更重要。
For example, the deterministic mapping {(1,1),(2,2),(3,3)} may be viewed as stronger than the deterministic mapping {(1,3),(2,1),(3,2)}, although these relationships would yield the same mutual information. This is because the mutual information is not sensitive at all to any inherent ordering in the variable values, and is therefore not sensitive at all to the form of the relational mapping between the associated variables. If it is desired that the former relation—showing agreement on all variable values—be judged stronger than the later relation, then it is possible to use the following weighted mutual information.
For example, the deterministic mapping {(1,1),(2,2),(3,3)} may be viewed as stronger than the deterministic mapping {(1,3),(2,1),(3,2)}, although these relationships would yield the same mutual information. This is because the mutual information is not sensitive at all to any inherent ordering in the variable values, and is therefore not sensitive at all to the form of the relational mapping between the associated variables. If it is desired that the former relation—showing agreement on all variable values—be judged stronger than the later relation, then it is possible to use the following weighted mutual information.
例如，确定性映射<math>\{(1,1),(2,2),(3,3)\}</math>可能被视为比确定性映射<math>\{(1,3),(2,1),(3,2)\}</math>更强，尽管这些关系产生的互信息是相同的。这是因为互信息对变量值的任何内在顺序都不敏感，因此对关联变量之间的关系映射形式一点也不敏感。如果希望对所有变量值表示一致的前一个关系比后一个关系强，则可以使用以下加权互信息的方法：
例如，确定性映射<math>\{(1,1),(2,2),(3,3)\}</math>可能被视为比确定性映射<math>\{(1,3),(2,1),(3,2)\}</math>更强，尽管这些关系产生的互信息是相同的。这是因为互信息对变量值的任何内在顺序都不敏感，因此对关联变量之间的关系映射形式一点也不敏感。如果希望对所有变量值的前一个关系比后一个关系强，则可以使用以下加权互信息的方法：
which places a weight 𝑤(𝑥,𝑦) on the probability of each variable value co-occurrence, 𝑝(𝑥,𝑦). This allows that certain probabilities may carry more or less significance than others, thereby allowing the quantification of relevant holistic or Prägnanz factors. In the above example, using larger relative weights for 𝑤(1,1), 𝑤(2,2), and 𝑤(3,3) would have the effect of assessing greater informativeness for the relation {(1,1),(2,2),(3,3)} than for the relation {(1,3),(2,1),(3,2)}, which may be desirable in some cases of pattern recognition, and the like. This weighted mutual information is a form of weighted KL-Divergence, which is known to take negative values for some inputs,[24] and there are examples where the weighted mutual information also takes negative values.[25]
which places a weight 𝑤(𝑥,𝑦) on the probability of each variable value co-occurrence, 𝑝(𝑥,𝑦). This allows that certain probabilities may carry more or less significance than others, thereby allowing the quantification of relevant holistic or Prägnanz factors. In the above example, using larger relative weights for 𝑤(1,1), 𝑤(2,2), and 𝑤(3,3) would have the effect of assessing greater informativeness for the relation {(1,1),(2,2),(3,3)} than for the relation {(1,3),(2,1),(3,2)}, which may be desirable in some cases of pattern recognition, and the like. This weighted mutual information is a form of weighted KL-Divergence, which is known to take negative values for some inputs,[24] and there are examples where the weighted mutual information also takes negative values.[25]
设每个变量值同时出现的概率<math>p(x,y)</math>的权重为<math>w(x,y)</math>。这使得某些概率可能比其他概率具有更多（或更少）的重要性，从而可以量化整个相关的整体或其中的Prägnanz因素。在上面的例子中，对<math>w(1,1)</math>、<math>w(2,2)</math>和<math>w(3,3)</math>使用更大的相对权重与评估关系<math>\{(1,1),(2,2),(3,3)\}</math>比关系<math>\{(1,3),(2,1),(3,2)\}</math>更大的信息性的效果是一样的，这在一些模式识别等情况下是可行的。这种加权互信息是加权KL散度的一种形式，已知它对某些输入取负值，并且在一些例子中加权互信息也取负值。
##
设每个变量值同时出现的概率<math>p(x,y)</math>的权重为<math>w(x,y)</math>。这使得某些特定概率可能比其他概率具有更多（或更少）的重要性，从而可以量化相关的整体或Prägnanz因素。在上面的例子中，对<math>w(1,1)</math>、<math>w(2,2)</math>和<math>w(3,3)</math>使用更大的相对权重，评估关系<math>\{(1,1),(2,2),(3,3)\}</math>比关系<math>\{(1,3),(2,1),(3,2)\}</math>具有更大的信息性，这在一些模式识别等情况下是可行的。这种加权互信息是加权KL散度的一种形式，通常对某些输入取负值，<ref name="weighted-kl">{{cite journal | last1 = Kvålseth | first1 = T. O. | year = 1991 | title = The relative useful information measure: some comments | url = | journal = Information Sciences | volume = 56 | issue = 1| pages = 35–38 | doi=10.1016/0020-0255(91)90022-m}}</ref>并且在一些例子中加权互信息也取负值。<ref>{{cite dissertation|title=Feature Selection Via Joint Likelihood|first=A. |last=Pocock|year=2012|url=http://www.cs.man.ac.uk/~gbrown/publications/pocockPhDthesis.pdf}}</ref>
=== 调整后的互信息 Adjusted mutual information ===
=== 调整后的互信息 Adjusted mutual information ===
A probability distribution can be viewed as a partition of a set. One may then ask: if a set were partitioned randomly, what would the distribution of probabilities be? What would the expectation value of the mutual information be? The adjusted mutual information or AMI subtracts the expectation value of the MI, so that the AMI is zero when two different distributions are random, and one when two distributions are identical. The AMI is defined in analogy to the adjusted Rand index of two different partitions of a set.
A probability distribution can be viewed as a partition of a set. One may then ask: if a set were partitioned randomly, what would the distribution of probabilities be? What would the expectation value of the mutual information be? The adjusted mutual information or AMI subtracts the expectation value of the MI, so that the AMI is zero when two different distributions are random, and one when two distributions are identical. The AMI is defined in analogy to the adjusted Rand index of two different partitions of a set.
概率分布可以被看作是集合划分。可能有人会问: 如果一个集合被随机分割，概率的分布会是什么？互信息的期望值是什么？我们用'''<font color="#ff8000">调整后的互信息 Adjusted Mutual Information</font>'''或 AMI 减去 MI 的期望值，这样当两个不同的分布是随机的时候 AMI 为零，当两个分布是相同的时候 AMI 也为零。AMI的定义类似于一个集合的两个不同分区的'''<font color="#ff8000">调整后的Rand指数 Adjusted Rand Index</font>'''。
=== 绝对互信息 Absolute mutual information ===<!-- This section is linked from Kolmogorov complexity -->
=== 绝对互信息 Absolute mutual information ===<!-- This section is linked from Kolmogorov complexity -->
When 𝑋 and 𝑌 are limited to be in a discrete number of states, observation data is summarized in a contingency table, with row variable 𝑋 (or 𝑖) and column variable 𝑌 (or 𝑗). Mutual information is one of the measures of association or correlation between the row and column variables. Other measures of association include Pearson's chi-squared test statistics, G-test statistics, etc. In fact, mutual information is equal to G-test statistics divided by 2𝑁, where 𝑁 is the sample size.
When 𝑋 and 𝑌 are limited to be in a discrete number of states, observation data is summarized in a contingency table, with row variable 𝑋 (or 𝑖) and column variable 𝑌 (or 𝑗). Mutual information is one of the measures of association or correlation between the row and column variables. Other measures of association include Pearson's chi-squared test statistics, G-test statistics, etc. In fact, mutual information is equal to G-test statistics divided by 2𝑁, where 𝑁 is the sample size.
当<math>X</math>和<math>Y</math>被限制为离散状态时，观测数据汇总在'''<font color="#ff8000">列联表 Contingency Table</font>'''中，列变量为行变量<math>X</math>（或<math>i</math>）和列变量<math>Y</math>（或<math>j</math>）。互信息是行和列变量之间关联或相关性的度量之一。其他关联度量包括Pearson卡方检验统计量、'''<font color="#ff8000">G检验 G-Test</font>'''统计量等。事实上，互信息等于G检验统计量除以<math>2N</math>，其中<math>N</math>为样本量。
当<math>X</math>和<math>Y</math>被限制为离散状态时，观测数据汇总在'''<font color="#ff8000">列联表 Contingency Table</font>'''中，其中行变量<math>X</math>（或<math>i</math>）和列变量<math>Y</math>（或<math>j</math>）。互信息是行和列变量之间关联或相关性的度量之一。其他关联度量包括Pearson卡方检验统计量、'''<font color="#ff8000">G检验 G-Test</font>'''统计量等。事实上，互信息等于G检验统计量除以<math>2N</math>，其中<math>N</math>为样本量。
== 应用 Applications ==
== 应用 Applications ==
* In [[search engine technology]], mutual information between phrases and contexts is used as a feature for [[k-means clustering]] to discover semantic clusters (concepts).<ref name=magerman>[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.78.4178&rep=rep1&type=pdf Parsing a Natural Language Using Mutual Information Statistics] by David M. Magerman and Mitchell P. Marcus</ref> For example, the mutual information of a bigram might be calculated as:
* In [[search engine technology]], mutual information between phrases and contexts is used as a feature for [[k-means clustering]] to discover semantic clusters (concepts).<ref name=magerman>[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.78.4178&rep=rep1&type=pdf Parsing a Natural Language Using Mutual Information Statistics] by David M. Magerman and Mitchell P. Marcus</ref> For example, the mutual information of a bigram might be calculated as:
在搜索引擎技术中，短语和上下文之间的互信息用作k均值聚类的功能，以发现语义聚类（概念）。<ref name=magerman>[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.78.4178&rep=rep1&type=pdf Parsing a Natural Language Using Mutual Information Statistics] by David M. Magerman and Mitchell P. Marcus</ref> 例如，一个二元组的互信息可以计算为：
{{Equation box 1
{{Equation box 1
where <math>f_{XY}</math> is the number of times the bigram xy appears in the corpus, <math>f_{X}</math> is the number of times the unigram x appears in the corpus, B is the total number of bigrams, and U is the total number of unigrams.
where <math>f_{XY}</math> is the number of times the bigram xy appears in the corpus, <math>f_{X}</math> is the number of times the unigram x appears in the corpus, B is the total number of bigrams, and U is the total number of unigrams.
其中<math>f_{XY}</math>是 二元语法 XY 在语料库中出现的次数，<math>f_{X}</math>是一元模型x在语料库中出现的次数，b 是二元语法的总数，u 是一元模型的总数。
其中<math>f_{XY}</math>是 二元语法 XY 在语料库中出现的次数，<math>f_{X}</math>是一元模型x在语料库中出现的次数，B 是二元语法的总数，U 是一元模型的总数。<ref name=magerman/>
* In [[telecommunications]], the [[channel capacity]] is equal to the mutual information, maximized over all input distributions.
* In [[telecommunications]], the [[channel capacity]] is equal to the mutual information, maximized over all input distributions.
In telecommunications, the channel capacity is equal to the mutual information, maximized over all input distributions.
In telecommunications, the channel capacity is equal to the mutual information, maximized over all input distributions.
在电信中，信道容量等于互信息，在所有输入分配中最大化。
* [[Discriminative model|Discriminative training]] procedures for [[hidden Markov model]]s have been proposed based on the [[maximum mutual information]] (MMI) criterion.
* [[Discriminative model|Discriminative training]] procedures for [[hidden Markov model]]s have been proposed based on the [[maximum mutual information]] (MMI) criterion.
Mutual information is used in determining the similarity of two different clusterings of a dataset. As such, it provides some advantages over the traditional Rand index.
Mutual information is used in determining the similarity of two different clusterings of a dataset. As such, it provides some advantages over the traditional Rand index.
互信息用于确定数据集中两个不同聚类的相似性。因此，它与传统的Rand指数相比具有一定的优势。
* Mutual information of words is often used as a significance function for the computation of [[collocation]]s in [[corpus linguistics]]. This has the added complexity that no word-instance is an instance to two different words; rather, one counts instances where 2 words occur adjacent or in close proximity; this slightly complicates the calculation, since the expected probability of one word occurring within <math>N</math> words of another, goes up with <math>N</math>.
* Mutual information of words is often used as a significance function for the computation of [[collocation]]s in [[corpus linguistics]]. This has the added complexity that no word-instance is an instance to two different words; rather, one counts instances where 2 words occur adjacent or in close proximity; this slightly complicates the calculation, since the expected probability of one word occurring within <math>N</math> words of another, goes up with <math>N</math>.
Mutual information of words is often used as a significance function for the computation of collocations in corpus linguistics. This has the added complexity that no word-instance is an instance to two different words; rather, one counts instances where 2 words occur adjacent or in close proximity; this slightly complicates the calculation, since the expected probability of one word occurring within 𝑁 words of another, goes up with 𝑁.
Mutual information of words is often used as a significance function for the computation of collocations in corpus linguistics. This has the added complexity that no word-instance is an instance to two different words; rather, one counts instances where 2 words occur adjacent or in close proximity; this slightly complicates the calculation, since the expected probability of one word occurring within 𝑁 words of another, goes up with 𝑁.
在'''<font color="#ff8000">语料库语言学 Corpus Linguistics</font>'''中，词的互信息常常被用作计算搭配的意义函数。这增加了复杂性，即没有一个单词实例是两个不同单词的实例；相反，我们统计两个单词相邻或非常接近的实例；这稍微使计算复杂化，因为一个单词出现在另一个单词的<math>N</math>单词中的预期概率会增加。
在'''<font color="#ff8000">语料库语言学 Corpus Linguistics</font>'''中，单词的互信息常常被用作搭配运算的重要函数。这增加了复杂性，即没有一个单词实例是两个不同单词的实例；相反，我们统计两个单词相邻或非常接近的实例；这稍微使计算复杂化，因为一个单词出现在另一个单词的<math>N</math>单词内的预期概率会增加。
* Mutual information is used in [[medical imaging]] for [[image registration]]. Given a reference image (for example, a brain scan), and a second image which needs to be put into the same [[coordinate system]] as the reference image, this image is deformed until the mutual information between it and the reference image is maximized.
* Mutual information is used in [[medical imaging]] for [[image registration]]. Given a reference image (for example, a brain scan), and a second image which needs to be put into the same [[coordinate system]] as the reference image, this image is deformed until the mutual information between it and the reference image is maximized.
In statistical mechanics, Loschmidt's paradox may be expressed in terms of mutual information.[27][28] Loschmidt noted that it must be impossible to determine a physical law which lacks time reversal symmetry (e.g. the second law of thermodynamics) only from physical laws which have this symmetry. He pointed out that the H-theorem of Boltzmann made the assumption that the velocities of particles in a gas were permanently uncorrelated, which removed the time symmetry inherent in the H-theorem. It can be shown that if a system is described by a probability density in phase space, then Liouville's theorem implies that the joint information (negative of the joint entropy) of the distribution remains constant in time. The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by Boltzmann's constant).
In statistical mechanics, Loschmidt's paradox may be expressed in terms of mutual information.[27][28] Loschmidt noted that it must be impossible to determine a physical law which lacks time reversal symmetry (e.g. the second law of thermodynamics) only from physical laws which have this symmetry. He pointed out that the H-theorem of Boltzmann made the assumption that the velocities of particles in a gas were permanently uncorrelated, which removed the time symmetry inherent in the H-theorem. It can be shown that if a system is described by a probability density in phase space, then Liouville's theorem implies that the joint information (negative of the joint entropy) of the distribution remains constant in time. The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by Boltzmann's constant).
在统计力学中，'''<font color="#ff8000">洛施密特悖论 Loschmidt's Paradox</font>'''可以用互信息来表示。洛施密特指出，只有从具有这种对称性的物理定律中确定缺乏时间反转对称性的物理定律（例如'''<font color="#ff8000">热力学第二定律 Second Law of Thermodynamics</font>'''）是不可能的。他指出，玻尔兹曼的H-定理假设气体中粒子的速度是永久不相关的，这就消除了H-定理固有的时间对称性。可以证明，如果系统在相空间中用概率密度来描述，那么'''<font color="#ff8000">刘维尔定理 Liouville's Theorem</font>'''意味着分布的联合信息（联合熵的负）在时间上保持不变。关节信息等于互信息加上每个粒子坐标的所有边缘信息（负的边缘熵）之和。玻尔兹曼的假设相当于在熵的计算中忽略了相互信息，从而得到了热力学熵（除以玻尔兹曼常数）。
在统计力学中，'''<font color="#ff8000">洛施密特悖论 Loschmidt's Paradox</font>'''可以用互信息来表示。<ref name=everett56>[[Hugh Everett]] [https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf Theory of the Universal Wavefunction], Thesis, Princeton University, (1956, 1973), pp 1–140 (page 30)</ref><ref name=everett57>{{cite journal | last1 = Everett | first1 = Hugh | authorlink = Hugh Everett | year = 1957 | title = Relative State Formulation of Quantum Mechanics | url = http://www.univer.omsk.su/omsk/Sci/Everett/paper1957.html | journal = Reviews of Modern Physics | volume = 29 | issue = 3 | pages = 454–462 | doi = 10.1103/revmodphys.29.454 | bibcode = 1957RvMP...29..454E | access-date = 2012-07-16 | archive-url = https://web.archive.org/web/20111027191052/http://www.univer.omsk.su/omsk/Sci/Everett/paper1957.html | archive-date = 2011-10-27 | url-status = dead }}</ref>洛施密特指出，只从具有这种对称性的物理定律中确定缺乏时间反转对称性的物理定律（例如'''<font color="#ff8000">热力学第二定律 Second Law of Thermodynamics</font>'''）是不可能的。他指出，Boltzmann 玻尔兹曼的H-定理假设气体中粒子的速度是永久不相关的，这就消除了H-定理固有的时间对称性。可以证明，如果系统在相空间中用概率密度来描述，那么'''<font color="#ff8000">刘维尔定理 Liouville's Theorem</font>'''意味着分布的联合信息（联合熵的负）在时间上保持不变。联合信息等于互信息加上每个粒子坐标的所有边缘信息（负的边缘熵）之和。玻尔兹曼的假设相当于在熵的计算中忽略了互信息，从而得到了热力学熵（除以玻尔兹曼常数）。
The mutual information is used to learn the structure of Bayesian networks/dynamic Bayesian networks, which is thought to explain the causal relationship between random variables, as exemplified by the GlobalMIT toolkit:[29] learning the globally optimal dynamic Bayesian network with the Mutual Information Test criterion.
The mutual information is used to learn the structure of Bayesian networks/dynamic Bayesian networks, which is thought to explain the causal relationship between random variables, as exemplified by the GlobalMIT toolkit:[29] learning the globally optimal dynamic Bayesian network with the Mutual Information Test criterion.
互信息用于学习'''<font color="#ff8000">贝叶斯网络 Bayesian Network</font>'''/'''<font color="#ff8000">动态贝叶斯网络 Dynamic Bayesian Network</font>'''的结构，被认为是用来解释随机变量之间的因果关系，如GlobalMIT工具包[29]用互信息检验准则学习全局最优动态贝叶斯网络。
互信息用于学习'''<font color="#ff8000">贝叶斯网络 Bayesian Network</font>'''/'''<font color="#ff8000">动态贝叶斯网络 Dynamic Bayesian Network</font>'''的结构，被认为是用来解释随机变量之间的因果关系，如GlobalMIT工具包<ref>{{Google Code|globalmit|GlobalMIT}}</ref>用互信息检验准则学习全局最优动态贝叶斯网络。
The mutual information is used in cosmology to test the influence of large-scale environments on galaxy properties in the Galaxy Zoo.
The mutual information is used in cosmology to test the influence of large-scale environments on galaxy properties in the Galaxy Zoo.
在'''<font color="#32CD32">星系 Galaxy Zoo</font>'''中，利用互信息在'''<font color="#ff8000">宇宙学 Cosmology</font>'''中测试大尺度环境对星系性质的影响。
在'''<font color="# ff8000">星系 Galaxy Zoo</font>'''中，利用互信息在'''<font color="#ff8000">宇宙学 Cosmology</font>'''中测试大尺度环境对星系性质的影响。
The mutual information was used in Solar Physics to derive the solar differential rotation profile, a travel-time deviation map for sunspots, and a time–distance diagram from quiet-Sun measurements.
The mutual information was used in Solar Physics to derive the solar differential rotation profile, a travel-time deviation map for sunspots, and a time–distance diagram from quiet-Sun measurements.
在'''<font color="#ff8000">太阳物理学 Solar Physics</font>'''中，相互信息被用于推导太阳差分自转剖面图、太阳黑子的旅行时间偏差图和安静太阳测量的时间-距离图。
在'''<font color="#ff8000">太阳物理学 Solar Physics</font>'''中，互信息被用于推导太阳差分自转剖面图、太阳黑子的旅行时间偏差图和从安静太阳测量的时间-距离图。<ref>{{cite journal|last1=Keys|first1=Dustin|last2=Kholikov|first2=Shukur|last3=Pevtsov|first3=Alexei A.|title=Application of Mutual Information Methods in Time Distance Helioseismology|journal=Solar Physics|date=February 2015|volume=290|issue=3|pages=659–671|doi=10.1007/s11207-015-0650-y|arxiv=1501.05597|bibcode=2015SoPh..290..659K}}</ref>
* Used in Invariant Information Clustering to automatically train neural network classifiers and image segmenters given no labelled data.<ref name=iic>[https://arxiv.org/abs/1807.06653 Invariant Information Clustering for Unsupervised Image Classification and Segmentation] by Xu Ji, Joao Henriques and Andrea Vedaldi</ref>
* Used in Invariant Information Clustering to automatically train neural network classifiers and image segmenters given no labelled data.<ref name=iic>[https://arxiv.org/abs/1807.06653 Invariant Information Clustering for Unsupervised Image Classification and Segmentation] by Xu Ji, Joao Henriques and Andrea Vedaldi</ref>
Used in Invariant Information Clustering to automatically train neural network classifiers and image segmenters given no labelled data.
Used in Invariant Information Clustering to automatically train neural network classifiers and image segmenters given no labelled data.
用于不变信息聚类，在没有标记数据的情况下自动训练神经网络分类器和图像分割器。
用于不变信息聚类，在没有标记数据的情况下自动训练神经网络分类器和图像分割器。<ref name=iic>[https://arxiv.org/abs/1807.06653 Invariant Information Clustering for Unsupervised Image Classification and Segmentation] by Xu Ji, Joao Henriques and Andrea Vedaldi</ref>
== 参见 See also ==
== 参见 See also ==