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Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a  distribution over a multi-dimensional space and a graph that is a compact or factorized representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov random fields. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce.<ref name=koller09>{{cite book

Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a  distribution over a multi-dimensional space and a graph that is a compact or factorized representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov random fields. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce.<ref name=koller09>{{cite book

一般来说，概率图模型使用将图的表示方法作为对多维空间上的分布进行编码的基础，而图是一组独立分布的紧凑或分解表示。常用的概率图模型大致分为两类，即贝叶斯网络和马尔可夫随机场。这两个族都包含因子分解和独立性的性质，但是它们在它们可以编码的独立性集合和它们所诱导的分布的因子分解上有所不同。 09{ cite book

一般来说，概率图模型中图的表示方法常常作为对多维空间上的分布进行编码的基础，而图是一组独立分布的紧凑或分解表示。常用的概率图模型大致分为两类：贝叶斯网络和马尔可夫随机场。这两种都包含因子分解和独立性的性质，但是它们在它们可以编码的一系列独立性和它们所诱导的分布的因子分解上有所不同。 09{ cite book

  |author=Koller, D.

  |author=Koller, D.

===Bayesian network===

===Bayesian network===

 

{{main|Bayesian network}}

{{main|Bayesian network}}

If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables.  More precisely, if the events are <math>X_1,\ldots,X_n</math> then the joint probability satisfies

If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables.  More precisely, if the events are <math>X_1,\ldots,X_n</math> then the joint probability satisfies

如果模型的网络结构是有向无环图，则模型表示所有随机变量的联合概率的因子分解。更确切地说，如果事件是数学 x1， ldots，xn / math，那么联合概率满足

如果模型的网络结构是有向无环图，则模型表示所有随机变量的联合概率的因子分解。更确切地说，如果事件是<math>X_1,\ldots,X_n</math>，那么联合概率满足

<math>P[X_1,\ldots,X_n]=\prod_{i=1}^nP[X_i|\text{pa}(X_i)]</math>

<math>P[X_1,\ldots,X_n]=\prod_{i=1}^nP[X_i|\text{pa}(X_i)]</math>

 

where <math>\text{pa}(X_i)</math> is the set of parents of node <math>X_i</math> (nodes with edges directed towards <math>X_i</math>).  In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables <math>A, B, C, D</math>

where <math>\text{pa}(X_i)</math> is the set of parents of node <math>X_i</math> (nodes with edges directed towards <math>X_i</math>).  In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables <math>A, B, C, D</math>

其中 math text { pa }(xi) / math 是节点 math x i / math 的父节点集(边指向 math x i / math 的节点)。换句话说，联合分布因子成为条件分布的乘积。例如，上面图中的图形模型(实际上不是有向无环图，而是祖先的图形)由随机变量数学 a，b，c，d / math 组成

其中 <math>\text{pa}(X_i)</math> 是节点 <math>X_i</math> 的父节点集(节点的边际指向 <math>X_i</math> )。换句话说，联合分布因子可以表示为条件分布的乘积。例如，上图中的图模型(实际上不是有向无环图，而是原始图)是由随机变量<math>A, B, C, D</math> 组成

with a joint probability density that factors as

with a joint probability density that factors as

with a joint probability density that factors as

with a joint probability density that factors as

<math>P[A,B,C,D] = P[A]\cdot P[B]\cdot P[C,D|A,B]</math>

<math>P[A,B,C,D] = P[A]\cdot P[B]\cdot P[C,D|A,B]</math>

 

Any two nodes are conditionally independent given the values of their parents.  In general, any two sets of nodes are conditionally independent given a third set if a criterion called d-separation holds in the graph.  Local independences and global independences are equivalent in Bayesian networks.

Any two nodes are conditionally independent given the values of their parents.  In general, any two sets of nodes are conditionally independent given a third set if a criterion called d-separation holds in the graph.  Local independences and global independences are equivalent in Bayesian networks.

任何两个节点都是根据其父节点的值条件独立的。一般来说，如果一个称为 d- 分离的准则在图中成立，那么给定第三个集合的任意两组节点都是条件独立的。贝叶斯网络中的局部独立性和全局独立性是等价的。

任何两个节点都是和其父节点的值是条件独立的。一般来说，如果 d- 分离准则在图中成立，那么任意两组节点和给定第三个节点集都是条件独立的。在贝叶斯网络中，局部独立性和全局独立性是等价的。

This type of graphical model is known as a directed graphical model, Bayesian network, or belief network. Classic machine learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks.

This type of graphical model is known as a directed graphical model, Bayesian network, or belief network. Classic machine learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks.

===Other types===

===Other types===

*[[Naive Bayes classifier]] where we use a tree with a single root

*[[Naive Bayes classifier]] where we use a tree with a single root

 

*[[Dependency network (graphical model)|Dependency network]]  where cycles are allowed

*[[Dependency network (graphical model)|Dependency network]]  where cycles are allowed

 

*Tree-augmented classifier or '''TAN model'''

*Tree-augmented classifier or '''TAN model'''

 

*A [[factor graph]] is an undirected [[bipartite graph]] connecting variables and factors. Each factor represents a function over the variables it is connected to. This is a helpful representation for understanding and implementing [[belief propagation]].

*A [[factor graph]] is an undirected [[bipartite graph]] connecting variables and factors. Each factor represents a function over the variables it is connected to. This is a helpful representation for understanding and implementing [[belief propagation]].

 

* A [[clique tree]] or junction tree is a [[tree (graph theory)|tree]] of [[clique (graph theory)|cliques]], used in the [[junction tree algorithm]].

* A [[clique tree]] or junction tree is a [[tree (graph theory)|tree]] of [[clique (graph theory)|cliques]], used in the [[junction tree algorithm]].

</ref>

</ref>

* [[Belief propagation]]

* [[Belief propagation]]

 

* [[Structural equation model]]

* [[Structural equation model]]