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→Relations to other entropy measures
:<math>H(X_1,\ldots, X_n) \leq H(X_1) + \ldots + H(X_n)</math>
:<math>H(X_1,\ldots, X_n) \leq H(X_1) + \ldots + H(X_n)</math>
==Relations to other entropy measures==
== Relations to other entropy measures 与其他熵测度的关系 ==
Joint entropy is used in the definition of [[conditional entropy]]<ref name=cover1991 />{{rp|22}}
Joint entropy is used in the definition of [[conditional entropy]]<ref name=cover1991 />{{rp|22}}
联合熵用于定义'''<font color="#ff8000"> 条件熵Conditional entropy </font>'''：
:<math>\Eta(X|Y) = \Eta(X,Y) - \Eta(Y)\,</math>,
:<math>\Eta(X|Y) = \Eta(X,Y) - \Eta(Y)\,</math>,
and <math display="block">\Eta(X_1,\dots,X_n) = \sum_{k=1}^n \Eta(X_k|X_{k-1},\dots, X_1)</math>It is also used in the definition of [[mutual information]]<ref name=cover1991 />{{rp|21}}
and <math display="block">\Eta(X_1,\dots,X_n) = \sum_{k=1}^n \Eta(X_k|X_{k-1},\dots, X_1)</math>
It is also used in the definition of [[mutual information]]<ref name=cover1991 />{{rp|21}}
它也用于定义交互信息：
:<math>\operatorname{I}(X;Y) = \Eta(X) + \Eta(Y) - \Eta(X,Y)\,</math>
:<math>\operatorname{I}(X;Y) = \Eta(X) + \Eta(Y) - \Eta(X,Y)\,</math>
In [[quantum information theory]], the joint entropy is generalized into the [[joint quantum entropy]].
In [[quantum information theory]], the joint entropy is generalized into the [[joint quantum entropy]].
=== Applications ===
在'''<font color="#ff8000"> 量子信息论Quantum information theory</font>'''中，联合熵被广义化为'''<font color="#ff8000"> 联合量子熵Joint quantum entropy</font>'''。
=== 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.<ref>{{cite web|url=https://infotopo.readthedocs.io/en/latest/index.html|title=InfoTopo: Topological Information Data Analysis. Deep statistical unsupervised and supervised learning - File Exchange - Github|author=|date=|website=github.com/pierrebaudot/infotopopy/|accessdate=26 September 2020}}</ref>
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.<ref>{{cite web|url=https://infotopo.readthedocs.io/en/latest/index.html|title=InfoTopo: Topological Information Data Analysis. Deep statistical unsupervised and supervised learning - File Exchange - Github|author=|date=|website=github.com/pierrebaudot/infotopopy/|accessdate=26 September 2020}}</ref>
提供了一个python软件包，用于计算n个变量的数据集中的所有多元联合熵，交互信息，条件交互信息，总相关性，信息距离。
==Joint differential entropy==
==Joint differential entropy==