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Several equivalent definitions of a Bayesian network have been offered. For the following, let G = (V,E) be a directed acyclic graph (DAG) and let X = (X_v), v ∈ V be a set of random variables indexed by V.

Several equivalent definitions of a Bayesian network have been offered. For the following, let G = (V,E) be a directed acyclic graph (DAG) and let X = (X_v), v ∈ V be a set of random variables indexed by V.

贝氏网路的几个等价定义已经被提出。设 g (v，e)是有向无环图(DAG) ，x (x 子 v / sub) ，v ∈ v 是 v 指示的一组随机变量。

'''<font color="#ff8000"> 贝叶斯网络Bayesian network</font>'''的几个等价定义已经被提出。设 g (v，e)是有向无环图(DAG) ，x (x 子 v / sub) ，v ∈ v 是 v 指示的一组随机变量。

X is a Bayesian network with respect to G if its joint probability density function (with respect to a product measure) can be written as a product of the individual density functions, conditional on their parent variables:

X is a Bayesian network with respect to G if its joint probability density function (with respect to a product measure) can be written as a product of the individual density functions, conditional on their parent variables:

X 是 g 的贝氏网路，如果它的联合概率密度函数(关于乘积测度)可以写成单个密度函数的乘积，条件是它们的父变量:

X 是 g 的'''<font color="#ff8000"> 贝叶斯网络Bayesian network</font>'''，如果它的联合概率密度函数(关于乘积测度)可以写成单个密度函数的乘积，条件是它们的父变量:

X is a Bayesian network with respect to G if it satisfies the local Markov property: each variable is conditionally independent of its non-descendants given its parent variables:

X is a Bayesian network with respect to G if it satisfies the local Markov property: each variable is conditionally independent of its non-descendants given its parent variables:

如果满足局部马尔可夫性，则 x 关于 g 是一个贝氏网路: 给定父变量，每个变量有条件地独立于其非后代变量:

如果满足局部马尔可夫性，则 x 关于 g 是一个'''<font color="#ff8000"> 贝叶斯网络Bayesian network</font>''': 给定父变量，每个变量有条件地独立于其非后代变量:

===Developing Bayesian networks发展贝叶斯网络===

===Developing Bayesian networks生成贝叶斯网络===

Developing a Bayesian network often begins with creating a DAG ''G'' such that ''X'' satisfies the local Markov property with respect to ''G''. Sometimes this is a [[Causal graph|causal]] DAG. The conditional probability distributions of each variable given its parents in ''G'' are assessed. In many cases, in particular in the case where the variables are discrete, if the joint distribution of ''X'' is the product of these conditional distributions, then ''X'' is a Bayesian network with respect to ''G''.<ref>{{cite book |first=Richard E. |last=Neapolitan | name-list-format = vanc |title=Learning Bayesian networks |url={{google books |plainurl=y |id=OlMZAQAAIAAJ}} |year=2004 |publisher=Prentice Hall |isbn=978-0-13-012534-7 }}</ref>

Developing a Bayesian network often begins with creating a DAG ''G'' such that ''X'' satisfies the local Markov property with respect to ''G''. Sometimes this is a [[Causal graph|causal]] DAG. The conditional probability distributions of each variable given its parents in ''G'' are assessed. In many cases, in particular in the case where the variables are discrete, if the joint distribution of ''X'' is the product of these conditional distributions, then ''X'' is a Bayesian network with respect to ''G''.<ref>{{cite book |first=Richard E. |last=Neapolitan | name-list-format = vanc |title=Learning Bayesian networks |url={{google books |plainurl=y |id=OlMZAQAAIAAJ}} |year=2004 |publisher=Prentice Hall |isbn=978-0-13-012534-7 }}</ref>

Developing a Bayesian network often begins with creating a DAG G such that X satisfies the local Markov property with respect to G. Sometimes this is a causal DAG. The conditional probability distributions of each variable given its parents in G are assessed. In many cases, in particular in the case where the variables are discrete, if the joint distribution of X is the product of these conditional distributions, then X is a Bayesian network with respect to G.

Developing a Bayesian network often begins with creating a DAG G such that X satisfies the local Markov property with respect to G. Sometimes this is a causal DAG. The conditional probability distributions of each variable given its parents in G are assessed. In many cases, in particular in the case where the variables are discrete, if the joint distribution of X is the product of these conditional distributions, then X is a Bayesian network with respect to G.

开发一个贝氏网路通常从创建一个 DAG g 开始，这样 x 就满足了 g 的局部马尔可夫性，有时这就是一个因果 DAG。评估了 g 中每个变量给定其父变量的条件概率分布。在许多情况下，特别是在变量是离散的情况下，如果 x 的联合分布是这些条件分布的乘积，那么 x 就是 g 的贝氏网路。

开发一个'''<font color="#ff8000"> 贝叶斯网络Bayesian network</font>'''通常从创建一个 DAG g 开始，这样 x 就满足了 g 的局部马尔可夫性，有时这就是一个因果 DAG。评估了 g 中每个变量给定其父变量的条件概率分布。在许多情况下，特别是在变量是离散的情况下，如果 x 的联合分布是这些条件分布的乘积，那么 x 就是 g 的贝氏网路。

===Markov blanket马尔科夫毯===

==='''<font color="#ff8000"> Markov blanket马尔科夫毯</font>'''===

The [[Markov blanket]] of a node is the set of nodes consisting of its parents, its children, and any other parents of its children. The Markov blanket renders the node independent of the rest of the network; the joint distribution of the variables in the Markov blanket of a node is sufficient knowledge for calculating the distribution of the node. ''X'' is a Bayesian network with respect to ''G'' if every node is conditionally independent of all other nodes in the network, given its [[Markov blanket]].{{sfn|Russell|Norvig|2003|p=499}}

The [[Markov blanket]] of a node is the set of nodes consisting of its parents, its children, and any other parents of its children. The Markov blanket renders the node independent of the rest of the network; the joint distribution of the variables in the Markov blanket of a node is sufficient knowledge for calculating the distribution of the node. ''X'' is a Bayesian network with respect to ''G'' if every node is conditionally independent of all other nodes in the network, given its [[Markov blanket]].{{sfn|Russell|Norvig|2003|p=499}}

The Markov blanket of a node is the set of nodes consisting of its parents, its children, and any other parents of its children. The Markov blanket renders the node independent of the rest of the network; the joint distribution of the variables in the Markov blanket of a node is sufficient knowledge for calculating the distribution of the node. X is a Bayesian network with respect to G if every node is conditionally independent of all other nodes in the network, given its Markov blanket.

The Markov blanket of a node is the set of nodes consisting of its parents, its children, and any other parents of its children. The Markov blanket renders the node independent of the rest of the network; the joint distribution of the variables in the Markov blanket of a node is sufficient knowledge for calculating the distribution of the node. X is a Bayesian network with respect to G if every node is conditionally independent of all other nodes in the network, given its Markov blanket.

一个节点的马尔可夫覆盖层是由其父节点、其子节点和其子节点的任何其他父节点组成的节点集。马尔可夫包络使节点独立于网络的其余部分，节点的马尔可夫包络中变量的联合分布是计算节点分布的充分知识。如果网络中的每个节点都有条件地独立于网络中的所有其他节点，那么 x 就是 g 的贝氏网路。

一个节点的马尔可夫覆盖层是由其父节点、其子节点和其子节点的任何其他父节点组成的节点集。马尔可夫包络使节点独立于网络的其余部分，节点的马尔可夫包络中变量的联合分布是计算节点分布的充分知识。如果网络中的每个节点都有条件地独立于网络中的所有其他节点，那么 x 就是 g 的'''<font color="#ff8000"> 贝叶斯网络Bayesian network</font>'''。

*''P'' contains (but does not need to be entirely) a directed chain, <math> u \cdots \leftarrow m \leftarrow \cdots v</math> or <math> u \cdots \rightarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'',

*''P'' contains (but does not need to be entirely) a directed chain, <math> u \cdots \leftarrow m \leftarrow \cdots v</math> or <math> u \cdots \rightarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'',

 

*''P'' contains a fork, <math> u \cdots \leftarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'', or

*''P'' contains a fork, <math> u \cdots \leftarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'', or

20世纪80年代后期，皮尔的《智能系统中的概率推理》和那不勒斯的《专家系统中的概率推理》总结了它们的性质，并将它们确立为一个研究领域。

20世纪80年代后期，皮尔的《智能系统中的概率推理》和那不勒斯的《专家系统中的概率推理》总结了它们的性质，并将它们确立为一个研究领域。

== See also又及 ==

== See also又及 ==