370
个编辑
更改
→Definitions and concepts定义与概念
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<sub>v</sub>), 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<sub>v</sub>), v ∈ V be a set of random variables indexed by V.
'''<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。
贝叶斯网络现在有好几个等价的定义，在下文的介绍中我们用到两个符号设 ''G'' = (''V'',''E'')是一个有向无环图(DAG)，再设''X'' = (''X''<sub>''v''</sub>)为一组随机变量。
===因子分解定义===
===Factorization definition因子分解定义===
''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:{{sfn|Russell|Norvig|2003|p=496}}
''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:{{sfn|Russell|Norvig|2003|p=496}}
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'''<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:
若一组随机变量''X''的联合概率分布函数可以写成由几个单独的条件概率函数的乘积，则''X''是关于图''G''的贝叶斯网络:
: <math> p (x) = \prod_{v \in V} p \left(x_v \,\big|\, x_{\operatorname{pa}(v)} \right) </math>
: <math> p (x) = \prod_{v \in V} p \left(x_v \,\big|\, x_{\operatorname{pa}(v)} \right) </math>
where pa(''v'') is the set of parents of ''v'' (i.e. those vertices pointing directly to ''v'' via a single edge).
where pa(''v'') is the set of parents of ''v'' (i.e. those vertices pointing directly to ''v'' via a single edge).
where pa(v) is the set of parents of v (i.e. those vertices pointing directly to v via a single edge).
where pa(v) is the set of parents of v (i.e. those vertices pointing directly to v via a single edge).
其中 pa (v)是 v 的父母的集合(即。这些顶点通过一条边直接指向 v)。
其中 pa(''v'')是 v 的父变量的集合(即，这些顶点通过一条边直接指向 v)。
For any set of random variables, the probability of any member of a joint distribution can be calculated from conditional probabilities using the chain rule (given a topological ordering of X) as follows:
For any set of random variables, the probability of any member of a joint distribution can be calculated from conditional probabilities using the chain rule (given a topological ordering of X) as follows:
对于任何一组随机变量，他们各种取值组合的联合概率都可以通过使用链式规则(给定一个 ''X'' 的拓扑排序)从条件概率中计算出来，如下所示:
: <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P \left(X_v=x_v \mid X_{v+1}=x_{v+1}, \ldots, X_n=x_n \right)</math>
: <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P \left(X_v=x_v \mid X_{v+1}=x_{v+1}, \ldots, X_n=x_n \right)</math>
Using the definition above, this can be written as:
Using the definition above, this can be written as:
使用上面贝叶斯网络的定义，还可以写成这样:
: <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P (X_v=x_v \mid X_j=x_j \text{ for each } X_j\, \text{ that is a parent of } X_v\, )</math>
: <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P (X_v=x_v \mid X_j=x_j \text{ for each } X_j\, \text{ that is a parent of } X_v\, )</math>
The difference between the two expressions is the conditional independence of the variables from any of their non-descendants, given the values of their parent variables.
The difference between the two expressions is the conditional independence of the variables from any of their non-descendants, given the values of their parent variables.
这两个表达式之间的区别是：给定父变量值，所有变量与他们的非后代变量条件独立。
==='''<font color="#ff8000">Local Markov property局部马尔可夫性质</font>'''===
===局部马尔可夫性===
''X'' is a Bayesian network with respect to ''G'' if it satisfies the ''local Markov property'': each variable is [[Conditional independence|conditionally independent]] of its non-descendants given its parent variables:{{sfn|Russell|Norvig|2003|p=499}}
''X'' is a Bayesian network with respect to ''G'' if it satisfies the ''local Markov property'': each variable is [[Conditional independence|conditionally independent]] of its non-descendants given its parent variables:{{sfn|Russell|Norvig|2003|p=499}}
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:
若一组随机变量''X''满足局部马尔可夫性，即每个变量在给定父变量的情况下，条件独立于所有非后代变量，则称 ''X''是关于 ''G'' 的一个贝叶斯网络: :
:<math> X_v \perp\!\!\!\perp X_{V \,\smallsetminus\, \operatorname{de}(v)} \mid X_{\operatorname{pa}(v)} \quad\text{for all }v \in V</math>
:<math> X_v \perp\!\!\!\perp X_{V \,\smallsetminus\, \operatorname{de}(v)} \mid X_{\operatorname{pa}(v)} \quad\text{for all }v \in V</math>
where de(''v'') is the set of descendants and ''V'' \ de(''v'') is the set of non-descendants of ''v''.
where de(''v'') is the set of descendants and ''V'' \ de(''v'') is the set of non-descendants of ''v''.
where de(v) is the set of descendants and V \ de(v) is the set of non-descendants of v.
where de(v) is the set of descendants and V \ de(v) is the set of non-descendants of v.
其中 de (v)是后裔集合，v de (v)是 v 的非后裔集合。
其中 de(''v'')是节点v的后代集合，''V'' \ de(''v'')是节点v的非后代集合。
这可以用类似于第一个定义的术语来表示，如
这可以用类似于第一个定义的术语来表示，如
:<math>
:<math>
\begin{align}
\begin{align}
& \operatorname P(X_v=x_v \mid X_i=x_i \text{ for each } X_i \text{ that is not a descendant of } X_v\, ) \\[6pt]
& \operatorname P(X_v=x_v \mid X_i=x_i \text{ for each } X_i \text{ that is not a descendant of } X_v\, ) \\[6pt]
= {} & P(X_v=x_v \mid X_j=x_j \text{ for each } X_j \text{ that is a parent of } X_v\, )
= {} & P(X_v=x_v \mid X_j=x_j \text{ for each } X_j \text{ that is a parent of } X_v\, )
\end{align}
\end{align}
</math>
</math>
The set of parents is a subset of the set of non-descendants because the graph is acyclic.
The set of parents is a subset of the set of non-descendants because the graph is acyclic.
因为贝叶斯网络是无环图，所以每个节点的父节点集同时也是该节点的是非后代节点集的一个子集。
===Developing Bayesian networks生成贝叶斯网络===
===生成一个贝叶斯网络===
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.
生成一个贝叶斯网络通常从创建一个有向无环图''G''开始，然后使图''G''里的随机变量''X''满足的局部马尔可夫性。有时这个图同时还是一个因果图。接下来，图里的每一个随机变量的概率分布都要被估计出来。在许多情况下，特别是在变量都是离散的情况下，如果 ''X'' 的联合分布是这些单个随机变量的条件分布的乘积，那么 ''X'' 就是 ''G'' 的贝叶斯网络。
==='''<font color="#ff8000"> Markov blanket马尔科夫毯</font>'''===
===马尔科夫毯===
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毯是由其父节点、其子节点和其子节点的所有其他父节点组成的节点集。一个节点的马尔可夫毯可以使该节点独立于网络的其余部分。一个节点的马尔可夫毯中所有变量的联合分布是计算该节点分布的一个充分条件。如果网络中的每个节点再给定其马尔可夫毯的情况下，条件地独立于网络中的所有其他节点，那么 ''X'' 就是 ''G'' 的贝叶斯网络。
===={{anchor|d-separation}}''d''-separation{{|d-分离}“d”-分离====
===''d''-分隔===
This definition can be made more general by defining the "d"-separation of two nodes, where d stands for directional. We first define the "d"-separation of a trail and then we will define the "d"-separation of two nodes in terms of that.
This definition can be made more general by defining the "d"-separation of two nodes, where d stands for directional. We first define the "d"-separation of a trail and then we will define the "d"-separation of two nodes in terms of that.
使用''d''-分隔的概念，贝叶斯网络的定义可以更加通用，其中 d 代表有向(directed)。我们首先定义一条轨迹的''d''-分隔的线索，然后我们再定义两个节点间的''d''-分隔。
Let P be a trail from node u to v. A trail is a loop-free, undirected (i.e. all edge directions are ignored) path between two nodes. Then P is said to be d-separated by a set of nodes Z if any of the following conditions holds:
Let P be a trail from node u to v. A trail is a loop-free, undirected (i.e. all edge directions are ignored) path between two nodes. Then P is said to be d-separated by a set of nodes Z if any of the following conditions holds:
设 p 是从节点 u 到节点 v 的路径，路径是无循环的、无向的(例如:。所有边方向被忽略)路径之间的两个节点。如果下列任何一个条件成立，则 p 被一组节点 z 分开:
设 ''P'' 是从节点 ''u'' 到节点 ''v'' 的轨迹。轨迹是一条两个节点之间的无环、无向的(即所有边方向被忽略)的路径。如果下列任何一个条件成立，则称 ''P'' 被一组节点 ''Z'' 分隔:
*''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”包含（但不必完全是）有向链，<math> u \cdots \leftarrow m \leftarrow \cdots v</math> or <math> u \cdots \rightarrow m \rightarrow \cdots v</math>, 使中间节点“m”位于''Z''，
*“P”包含一条有向链<math> u \cdots \leftarrow m \leftarrow \cdots v</math> or <math> u \cdots \rightarrow m \rightarrow \cdots v</math>, 其中间节点''m''属于点集''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
*“P”包含一个分支，<math>u\cdots\leftarrow m\rightarrow\cdots v</math>, 使中间节点“m”位于“z”中，或
*“P”包含一个分叉<math> u \cdots \leftarrow m \rightarrow \cdots v</math>, 其中间节点''m''位于“Z”中，或
*''P'' contains an inverted fork (or collider), <math> u \cdots \rightarrow m \leftarrow \cdots v</math>, such that the middle node ''m'' is not in ''Z'' and no descendant of ''m'' is in ''Z''.
*''P'' contains an inverted fork (or collider), <math> u \cdots \rightarrow m \leftarrow \cdots v</math>, such that the middle node ''m'' is not in ''Z'' and no descendant of ''m'' is in ''Z''.
*“P”包含一个倒叉（或对撞机），<math>u\cdots\rightarrow m \leftarrow \cdots v</math>, 中间节点“m”不在“z”中，在''Z''中也没有“m”i的后代。
*“P”包含一个倒叉（或称对撞），<math> u \cdots \rightarrow m \leftarrow \cdots v</math>，其中间节点''m''不在''Z''中，在''Z''中也没有''m''的后代节点。
The nodes u and v are d-separated by Z if all trails between them are d-separated. If u and v are not d-separated, they are d-connected.
The nodes u and v are d-separated by Z if all trails between them are d-separated. If u and v are not d-separated, they are d-connected.
如果节点 u 和 v 之间的所有轨迹都是 d 分开的，则节点 u 和 v 被 z 分开。如果 u 和 v 不是 d- 分离的，则它们是 d- 连通的。
如果节点 ''u'' 和 ''v'' 之间的所有轨迹都是''d''-分隔的，则称节点 ''u'' 和 ''v'' 被 ''Z'' 分开。如果 '''u'' 和 ''v'' 不是''d''-分隔的，则称它们是 ‘’d‘’-连通的。
X is a Bayesian network with respect to G if, for any two nodes u, v:
X is a Bayesian network with respect to G if, for any two nodes u, v:
我们称''X''是''G''的贝叶斯网络，当对于图''G''中任意两个节点 ''u''，''v''满足:
: <math>X_u \perp\!\!\!\perp X_v \mid X_Z</math>
: <math>X_u \perp\!\!\!\perp X_v \mid X_Z</math>
where Z is a set which d-separates u and v. (The Markov blanket is the minimal set of nodes which d-separates node v from all other nodes.)
where Z is a set which d-separates u and v. (The Markov blanket is the minimal set of nodes which d-separates node v from all other nodes.)
其中 z 是一个将 u 和 v 分离的集合(马尔可夫覆盖层是将节点 v 与其他所有节点分离的最小节点集合)
其中 ''Z'' 是一个将 ''u'' 和 ''v''进行了 ''d''-分隔的集合(马尔可夫毯其实就是将节点 ''v'' 与其他所有节点''d''-分隔的最小节点集合)。
===Causal networks因果网络===
===因果网络===
Although Bayesian networks are often used to represent [[causality|causal]] relationships, this need not be the case: a directed edge from ''u'' to ''v'' does not require that ''X<sub>v</sub>'' be causally dependent on ''X<sub>u</sub>''. This is demonstrated by the fact that Bayesian networks on the graphs:
Although Bayesian networks are often used to represent [[causality|causal]] relationships, this need not be the case: a directed edge from ''u'' to ''v'' does not require that ''X<sub>v</sub>'' be causally dependent on ''X<sub>u</sub>''. This is demonstrated by the fact that Bayesian networks on the graphs:
Although Bayesian networks are often used to represent causal relationships, this need not be the case: a directed edge from u to v does not require that X<sub>v</sub> be causally dependent on X<sub>u</sub>. This is demonstrated by the fact that Bayesian networks on the graphs:
Although Bayesian networks are often used to represent causal relationships, this need not be the case: a directed edge from u to v does not require that X<sub>v</sub> be causally dependent on X<sub>u</sub>. This is demonstrated by the fact that Bayesian networks on the graphs:
虽然贝叶斯网络经常被用来表示因果关系，但这种情形并不是必须的: 从 ''u'' 到 ''v'' 的有向边并不意味着 X<sub>v</sub>一定是X<sub>u</sub>导致的结果。下面这两个贝叶斯网络是等价的:
:<math> a \rightarrow b \rightarrow c \qquad \text{and} \qquad a \leftarrow b \leftarrow c </math>
:<math> a \rightarrow b \rightarrow c \qquad \text{and} \qquad a \leftarrow b \leftarrow c </math>
are equivalent: that is they impose exactly the same conditional independence requirements.
are equivalent: that is they impose exactly the same conditional independence requirements.
也就是说，它们表示的条件独立要求完全相同。
A causal network is a Bayesian network with the requirement that the relationships be causal. The additional semantics of causal networks specify that if a node X is actively caused to be in a given state x (an action written as do(X = x)), then the probability density function changes to that of the network obtained by cutting the links from the parents of X to X, and setting X to the caused value x. Using these semantics, the impact of external interventions from data obtained prior to intervention can be predicted.
A causal network is a Bayesian network with the requirement that the relationships be causal. The additional semantics of causal networks specify that if a node X is actively caused to be in a given state x (an action written as do(X = x)), then the probability density function changes to that of the network obtained by cutting the links from the parents of X to X, and setting X to the caused value x. Using these semantics, the impact of external interventions from data obtained prior to intervention can be predicted.
当一个贝叶斯网络中的节点关系全都是因果关系时，这网络才能被称为时因果网络。因果网络还规定，如果一个节点 ''X''再被干预的情况下变成了状态 x (写作do(''X'' = ''x'')) ，那么概率密度函数就会发生改变，表示成一个从 ''X'' 的父节点到 ''X''的链接被切断的网络，并将变量''X''的值设置为''x''。利用这些规则，我们可以在真正实施外部干预前，就能从数据中预测到实施干预后会产生的影响。