# 联合熵

In information theory, joint entropy is a measure of the uncertainty associated with a set of variables.[1]

## Definition 定义

The joint Shannon entropy (in bits) of two discrete random variables $\displaystyle{ X }$ and $\displaystyle{ Y }$ with images $\displaystyle{ \mathcal X }$ and $\displaystyle{ \mathcal Y }$ is defined as[2]:16

$\displaystyle{ \Eta(X,Y) = -\sum_{x\in\mathcal X} \sum_{y\in\mathcal Y} P(x,y) \log_2[P(x,y)] }$

(Eq.1)

where $\displaystyle{ x }$ and $\displaystyle{ y }$ are particular values of $\displaystyle{ X }$ and $\displaystyle{ Y }$, respectively, $\displaystyle{ P(x,y) }$ is the joint probability of these values occurring together, and $\displaystyle{ P(x,y) \log_2[P(x,y)] }$ is defined to be 0 if $\displaystyle{ P(x,y)=0 }$.

For more than two random variables $\displaystyle{ X_1, ..., X_n }$ this expands to

$\displaystyle{ \Eta(X_1, ..., X_n) = -\sum_{x_1 \in\mathcal X_1} ... \sum_{x_n \in\mathcal X_n} P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)] }$

(Eq.2)

where $\displaystyle{ x_1,...,x_n }$ are particular values of $\displaystyle{ X_1,...,X_n }$, respectively, $\displaystyle{ P(x_1, ..., x_n) }$ is the probability of these values occurring together, and $\displaystyle{ P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)] }$ is defined to be 0 if $\displaystyle{ P(x_1, ..., x_n)=0 }$.

## Properties 属性

### Nonnegativity 非负性

The joint entropy of a set of random variables is a nonnegative number.

$\displaystyle{ H(X,Y) \geq 0 }$
$\displaystyle{ H(X_1,\ldots, X_n) \geq 0 }$

### Greater than individual entropies 大于单个熵

The joint entropy of a set of variables is greater than or equal to the maximum of all of the individual entropies of the variables in the set.

$\displaystyle{ H(X,Y) \geq \max \left[H(X),H(Y) \right] }$
$\displaystyle{ H \bigl(X_1,\ldots, X_n \bigr) \geq \max_{1 \le i \le n} \Bigl\{H\bigl(X_i\bigr) \Bigr\} }$

### Less than or equal to the sum of individual entropies 小于或等于单个熵的总和

The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. This is an example of subadditivity. This inequality is an equality if and only if $\displaystyle{ X }$ and $\displaystyle{ Y }$ are statistically independent.[2]:30

$\displaystyle{ H(X,Y) \leq H(X) + H(Y) }$
$\displaystyle{ H(X_1,\ldots, X_n) \leq H(X_1) + \ldots + H(X_n) }$

## Relations to other entropy measures 与其他熵测度的关系

Joint entropy is used in the definition of conditional entropy[2]:22

$\displaystyle{ H(X|Y) = H(X,Y) - H(Y)\, }$,

and $\displaystyle{ H(X_1,\dots,X_n) = \sum_{k=1}^n H(X_k|X_{k-1},\dots, X_1) }$

It is also used in the definition of mutual information[2]:21 它也用于定义 交互信息Mutual information

$\displaystyle{ \operatorname{I}(X;Y) = H(X) + H(Y) - H(X,Y)\, }$

In quantum information theory, the joint entropy is generalized into the joint quantum entropy.

### Applications 应用

A python package for computing all multivariate joint entropies, mutual informations, conditional mutual information, total correlations, information distance in a dataset of n variables is available.[3]

## Joint differential entropy 联合微分熵

### Definition 定义

The above definition is for discrete random variables and just as valid in the case of continuous random variables. The continuous version of discrete joint entropy is called joint differential (or continuous) entropy. Let $\displaystyle{ X }$ and $\displaystyle{ Y }$ be a continuous random variables with a joint probability density function $\displaystyle{ f(x,y) }$. The differential joint entropy $\displaystyle{ h(X,Y) }$ is defined as[2]:249

$\displaystyle{ h(X,Y) = -\int_{\mathcal X , \mathcal Y} f(x,y)\log f(x,y)\,dx dy }$

(Eq.3)

For more than two continuous random variables $\displaystyle{ X_1, ..., X_n }$ the definition is generalized to:

$\displaystyle{ h(X_1, \ldots,X_n) = -\int f(x_1, \ldots,x_n)\log f(x_1, \ldots,x_n)\,dx_1 \ldots dx_n }$

(Eq.4)

The integral is taken over the support of $\displaystyle{ f }$. It is possible that the integral does not exist in which case we say that the differential entropy is not defined.

### Properties 属性

As in the discrete case the joint differential entropy of a set of random variables is smaller or equal than the sum of the entropies of the individual random variables:

$\displaystyle{ h(X_1,X_2, \ldots,X_n) \le \sum_{i=1}^n h(X_i) }$[2]:253

The following chain rule holds for two random variables:

$\displaystyle{ h(X,Y) = h(X|Y) + h(Y) }$

In the case of more than two random variables this generalizes to:[2]:253

$\displaystyle{ h(X_1,X_2, \ldots,X_n) = \sum_{i=1}^n h(X_i|X_1,X_2, \ldots,X_{i-1}) }$

Joint differential entropy is also used in the definition of the mutual information between continuous random variables:

$\displaystyle{ \operatorname{I}(X,Y)=h(X)+h(Y)-h(X,Y) }$

## References 参考文献

1. Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. ISBN 0-486-41147-8.
2. Thomas M. Cover; Joy A. Thomas. Elements of Information Theory. Hoboken, New Jersey: Wiley. ISBN 0-471-24195-4.
3. "InfoTopo: Topological Information Data Analysis. Deep statistical unsupervised and supervised learning - File Exchange - Github". github.com/pierrebaudot/infotopopy/. Retrieved 26 September 2020.