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{{more footnotes|date=February 2011}}

{{more footnotes|date=February 2011}}

{{Bayesian statistics}}

{{Bayesian statistics}}

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A Bayesian network, Bayes network, belief network, decision network, Bayes(ian) model or probabilistic directed acyclic graphical model is a probabilistic graphical model (a type of statistical model) that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor.  For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

A Bayesian network, Bayes network, belief network, decision network, Bayes(ian) model or probabilistic directed acyclic graphical model is a probabilistic graphical model (a type of statistical model) that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor.  For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

'''<font color="#ff8000"> 贝氏网路Bayesian network、贝叶斯网络 Bayes network</font>'''、信念网络、决策网络、贝叶斯模型或概率有向无环图模型是一种概率图模型(一种统计模型) ，它通过有向无环图无环图(DAG)表示一组变量及其条件依赖关系。贝叶斯网络是理想的事件发生和预测的可能性，任何一个几个可能的已知原因是影响因素。例如，'''<font color="#ff8000"> 贝氏网路Bayesian network</font>'''可以表示疾病和症状之间的概率关系。在给定症状的情况下，该网络可用于计算各种疾病出现的概率。

'''<font color="#ff8000"> 贝叶斯网络、信念网络belief network、决策网络decision network、贝叶斯模型Bayes(ian) modelBayesian network</font>'''或'''<font color="#ff8000">概率有向无环图模型probabilistic directed acyclic graphical modelBayesian network</font>'''是一种概率图模型(一种统计模型) ，它通过有向无环图无环图(DAG)表示一组随机变量及其条件依赖关系。贝叶斯网络是一种理想的分析工具，用来预测一个事件的发生是由已知原因中的哪一个（些）引起的。例如，贝叶斯网络可以表示疾病和症状之间的概率关系。在给定症状的情况下，该网络可用于计算各种疾病出现的概率。

Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (e.g. speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams.

Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (e.g. speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams.

有效的算法可以在'''<font color="#ff8000"> 贝叶斯网络 Bayes network</font>'''中进行推理和学习。'''<font color="#ff8000"> 贝叶斯网络 Bayes network</font>'''模型序列的变量(例如。语音信号或蛋白质序列)被称为动态贝叶斯网络。能够表示和解决不确定性决策问题的贝叶斯网络的推广称为影响图。

贝叶斯网络有多种变体。用来建模序列变量(例如，语音信号或蛋白质序列)的贝叶斯网络被称为动态贝叶斯网络。贝叶斯网络可以进一步扩展，用来表示和解决在不确定性因素下的决策问题，这种扩展称为影响图。现在已有高效的算法学习出贝叶斯网络的结构，并通过贝叶斯网络做推理。

==Graphical model图模型==

==图模型==

Formally, Bayesian networks are [[Directed acyclic graph|directed acyclic graphs]] (DAGs) whose nodes represent variables in the [[Bayesian probability|Bayesian]] sense: they may be observable quantities, [[latent variable]]s, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (no path connects one node to another) represent variables that are [[conditional independence|conditionally independent]] of each other. Each node is associated with a [[probability function]] that takes, as input, a particular set of values for the node's [[Glossary of graph theory#Directed acyclic graphs|parent]] variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if <math>m</math> parent nodes represent <math>m</math> [[Boolean data type|Boolean variables]], then the probability function could be represented by a table of <small><math>2^m</math></small> entries, one entry for each of the <small><math>2^m</math></small> possible parent combinations. Similar ideas may be applied to undirected, and possibly cyclic, graphs such as [[Markov network]]s.

Formally, Bayesian networks are [[Directed acyclic graph|directed acyclic graphs]] (DAGs) whose nodes represent variables in the [[Bayesian probability|Bayesian]] sense: they may be observable quantities, [[latent variable]]s, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (no path connects one node to another) represent variables that are [[conditional independence|conditionally independent]] of each other. Each node is associated with a [[probability function]] that takes, as input, a particular set of values for the node's [[Glossary of graph theory#Directed acyclic graphs|parent]] variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if <math>m</math> parent nodes represent <math>m</math> [[Boolean data type|Boolean variables]], then the probability function could be represented by a table of <small><math>2^m</math></small> entries, one entry for each of the <small><math>2^m</math></small> possible parent combinations. Similar ideas may be applied to undirected, and possibly cyclic, graphs such as [[Markov network]]s.

Formally, Bayesian networks are directed acyclic graphs (DAGs) whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (no path connects one node to another) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if <math>m</math> parent nodes represent <math>m</math> Boolean variables, then the probability function could be represented by a table of <small><math>2^m</math></small> entries, one entry for each of the <small><math>2^m</math></small> possible parent combinations. Similar ideas may be applied to undirected, and possibly cyclic, graphs such as Markov networks.

Formally, Bayesian networks are directed acyclic graphs (DAGs) whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected (no path connects one node to another) represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if <math>m</math> parent nodes represent <math>m</math> Boolean variables, then the probability function could be represented by a table of <small><math>2^m</math></small> entries, one entry for each of the <small><math>2^m</math></small> possible parent combinations. Similar ideas may be applied to undirected, and possibly cyclic, graphs such as Markov networks.

在形式上，'''<font color="#ff8000"> 贝叶斯网络 Bayes network</font>'''是有向无环图(DAGs) ，其节点表示贝叶斯意义上的变量: 它们可以是可观测量、潜在变量、未知参数或假设。边表示条件依赖; 未连接(没有路径连接一个节点到另一个节点)的节点表示彼此有条件独立的变量。每个节点都与一个'''<font color="#ff8000"> 概率密度函数Probability function </font>'''节点相关联，该节点接受一组特定的父变量值作为输入，并给出(作为输出)该节点表示的变量的概率(或概率分布，如果适用的话)。例如，如果 math m / math 父节点表示 math m / math 布尔变量，那么概率密度函数可以用一个小 math 2 ^ m / math / small 条目表示，每个小 math 2 ^ m / math / small 可能的父节点都有一个条目。类似的想法可以应用于无向图，也可能是循环图，如马尔可夫网络。

在形式上，贝叶斯网络是有向无环图(DAG) ，其节点表示随机变量(其概率为贝叶斯概率): 它们可以是可观测量变量、隐变量、未知参数或假设。图中的边表示条件依赖; 未连接(没有路径连接一个节点到另一个节点)的节点表示彼此之间条件独立。每个节点都与一个'''<font color="#ff8000"> 概率函数Probability function </font>'''节点相关联，该函数把所有父节点代表的变量值作为输入，并给出该节点表示的随机变量的概率(或概率分布话)。例如，如果 <math> m </math> 父节点表示 <math>m</math> 布尔变量，那么概率函数可以用一个包含<small><math>2^m</math></small>行的表格表示，每一行代表一种（父节点）变量值的组合，以及对应的子节点变量的概率值。类似的想法可以应用于有环无向图，如马尔可夫网络。

==Example举例==

==举例==

[[Image:SimpleBayesNet.svg|400px|thumb|right|A simple Bayesian network with [[conditional probability table]]s ]]

[[Image:SimpleBayesNet.svg|400px|thumb|right|A simple Bayesian network with [[conditional probability table]]s ]]

Two events can cause grass to be wet: an active sprinkler or rain. Rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler usually is not active). This situation can be modeled with a Bayesian network (shown to the right). Each variable has two possible values, T (for true) and F (for false).

Two events can cause grass to be wet: an active sprinkler or rain. Rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler usually is not active). This situation can be modeled with a Bayesian network (shown to the right). Each variable has two possible values, T (for true) and F (for false).

有两种情况会导致草地湿润: 主动洒水或者下雨。雨对洒水车的使用有直接的影响(也就是说，当下雨时，洒水车通常是不活跃的)。这种情况可以用贝氏网路来模拟(如右图所示)。每个变量有两个可能的值，t (表示真)和 f (表示假)。

草地变得湿润，可能有两种原因: 主动洒水或者下雨。雨对洒水车的使用有直接的影响(也就是说，当下雨时，洒水车通常是不工作的)。这种情况可以用贝叶斯网络来模拟(如右图所示)。每个变量有两个可能的值，t (表示真)和 f (表示假)。

 

 

The [[Joint probability distribution|joint probability function]] is:

The [[Joint probability distribution|joint probability function]] is:

The joint probability function is:

The joint probability function is:

'''<font color="#ff8000"> 联合概率密度函数Joint probability function</font>'''是:

对应的'''<font color="#ff8000"> 联合概率函数Joint probability function</font>'''是:

 

: <math>\Pr(G,S,R)=\Pr(G\mid S,R) \Pr(S\mid R)\Pr(R)</math>

: <math>\Pr(G,S,R)=\Pr(G\mid S,R) \Pr(S\mid R)\Pr(R)</math>

where G = "Grass wet (true/false)", S = "Sprinkler turned on (true/false)", and R = "Raining (true/false)".

where G = "Grass wet (true/false)", S = "Sprinkler turned on (true/false)", and R = "Raining (true/false)".

 

The model can answer questions about the presence of a cause given the presence of an effect (so-called inverse probability) like "What is the probability that it is raining, given the grass is wet?" by using the conditional probability formula and summing over all nuisance variables:

The model can answer questions about the presence of a cause given the presence of an effect (so-called inverse probability) like "What is the probability that it is raining, given the grass is wet?" by using the conditional probability formula and summing over all nuisance variables:

这个模型可以在给定一个效应(所谓的逆概率)的情况下回答关于一个原因是否存在的问题，比如“给定草是湿的，下雨的概率是多少? ”通过使用条件概率公式和对所有滋扰变量的求和:

这个模型可以回答在给定一个结果的情况下一个原因是否存在的问题，比如“给定草是湿的，下雨的概率是多少? ”通过使用条件概率公式并对所有干扰变量的求和:

 

 

: <math>\Pr(R=T\mid G=T) =\frac{\Pr(G=T,R=T)}{\Pr(G=T)} = \frac{\sum_{S \in \{T, F\}}\Pr(G=T, S,R=T)}{\sum_{S, R \in \{T, F\}} \Pr(G=T,S,R)}</math>

: <math>\Pr(R=T\mid G=T) =\frac{\Pr(G=T,R=T)}{\Pr(G=T)} = \frac{\sum_{S \in \{T, F\}}\Pr(G=T, S,R=T)}{\sum_{S, R \in \{T, F\}} \Pr(G=T,S,R)}</math>

Using the expansion for the joint probability function <math>\Pr(G,S,R)</math> and the conditional probabilities from the [[Conditional probability table|conditional probability tables (CPTs)]] stated in the diagram, one can evaluate each term in the sums in the numerator and denominator. For example,

Using the expansion for the joint probability function <math>\Pr(G,S,R)</math> and the conditional probabilities from the [[Conditional probability table|conditional probability tables (CPTs)]] stated in the diagram, one can evaluate each term in the sums in the numerator and denominator. For example,

Using the expansion for the joint probability function <math>\Pr(G,S,R)</math> and the conditional probabilities from the conditional probability tables (CPTs) stated in the diagram, one can evaluate each term in the sums in the numerator and denominator. For example,

Using the expansion for the joint probability function <math>\Pr(G,S,R)</math> and the conditional probabilities from the conditional probability tables (CPTs) stated in the diagram, one can evaluate each term in the sums in the numerator and denominator. For example,

使用联合的概率密度函数数学 Pr (g，s，r) / 数学的展开式和图中列出的条件概率表的条件概率，我们可以用分子和分母的和来计算每个项。比如说,

展开概率密度函数<math>\Pr(G,S,R)</math> ，并使用图中列出条件概率，我们可以算出分子和分母中的各个项。比如说,

 

: <math>\begin{align}

: <math>\begin{align}

\Pr(G=T, S=T,R=T) & = \Pr(G=T\mid S=T,R=T)\Pr(S=T\mid R=T)\Pr(R=T) \\

\Pr(G=T, S=T,R=T) & = \Pr(G=T\mid S=T,R=T)\Pr(S=T\mid R=T)\Pr(R=T) \\

   & = 0.99 \times 0.01 \times 0.2 \\

   & = 0.99 \times 0.01 \times 0.2 \\

   & = 0.00198.

   & = 0.00198.

\end{align}

\end{align}

</math>

</math>

Then the numerical results (subscripted by the associated variable values) are

Then the numerical results (subscripted by the associated variable values) are

 

 

: <math>\Pr(R=T\mid G=T) = \frac{ 0.00198_{TTT} + 0.1584_{TFT} }{ 0.00198_{TTT} + 0.288_{TTF} + 0.1584_{TFT} + 0.0_{TFF} } = \frac{891}{2491}\approx 35.77 \%.</math>

: <math>\Pr(R=T\mid G=T) = \frac{ 0.00198_{TTT} + 0.1584_{TFT} }{ 0.00198_{TTT} + 0.288_{TTF} + 0.1584_{TFT} + 0.0_{TFF} } = \frac{891}{2491}\approx 35.77 \%.</math>

To answer an interventional question, such as "What is the probability that it would rain, given that we wet the grass?" the answer is governed by the post-intervention joint distribution function

To answer an interventional question, such as "What is the probability that it would rain, given that we wet the grass?" the answer is governed by the post-intervention joint distribution function

回答一个介入性的问题，比如“既然我们把草弄湿了，那么下雨的可能性有多大? ”答案取决于干预后的'''<font color="#ff8000">联合分布函数 Joint distribution function</font>'''

现在回答一个干预性的问题，比如“现在我们把草弄湿了，那么下雨的可能性有多大? ”答案取决于干预后的'''<font color="#ff8000">联合分布函数 Joint distribution function</font>'''

 

: <math>\Pr(S,R\mid\text{do}(G=T)) = \Pr(S\mid R) \Pr(R)</math>

: <math>\Pr(S,R\mid\text{do}(G=T)) = \Pr(S\mid R) \Pr(R)</math>

obtained by removing the factor <math>\Pr(G\mid S,R)</math> from the pre-intervention distribution. The do operator forces the value of G to be true. The probability of rain is unaffected by the action:

obtained by removing the factor <math>\Pr(G\mid S,R)</math> from the pre-intervention distribution. The do operator forces the value of G to be true. The probability of rain is unaffected by the action:

通过从干预前的分布中去除因子 math Pr (g  mid s，r) / math，得到干预前的分布。Do 运算符强制 g 的值为真。下雨的可能性不受行动的影响:

该分布通过从干预前的分布中去除因子<math>\Pr(G\mid S,R)</math>得到，其中do算子强行使 G 的值为真。下雨的可能性不受行动的影响:

 

 

: <math>\Pr(R\mid\text{do}(G=T)) = \Pr(R).</math>

: <math>\Pr(R\mid\text{do}(G=T)) = \Pr(R).</math>

To predict the impact of turning the sprinkler on:

To predict the impact of turning the sprinkler on:

预测开启洒水装置的影响:

预测开启洒水装置的影响:

: <math>\Pr(R,G\mid\text{do}(S=T)) = \Pr(R)\Pr(G\mid R,S=T)</math>

: <math>\Pr(R,G\mid\text{do}(S=T)) = \Pr(R)\Pr(G\mid R,S=T)</math>

with the term <math>\Pr(S=T\mid R)</math> removed, showing that the action affects the grass but not the rain.

with the term <math>\Pr(S=T\mid R)</math> removed, showing that the action affects the grass but not the rain.

移除了 math Pr (s t  mid r) / math 这个术语，表明这种行为影响的是草，而不是雨。

移除<math>\Pr(S=T\mid R)</math> 这个项，表明这种行为影响的是草，而不是雨。

 

这些预测可能不可行的给予未观测的变量，因为在大多数政策评估问题。但是，只要满足后门准则，仍然可以预测操作 math  text { do }(x) / math 的效果。它指出，如果一组 z 节点可以观察到 d-分隔(或阻塞)从 x 到 y 的所有后门路径

这些预测可能不可行的给予未观测的变量，因为在大多数政策评估问题。但是，只要满足后门准则，仍然可以预测操作 math  text { do }(x) / math 的效果。它指出，如果一组 z 节点可以观察到 d-分隔(或阻塞)从 x 到 y 的所有后门路径

: <math>\Pr(Y,Z\mid\text{do}(x)) = \frac{\Pr(Y,Z,X=x)}{\Pr(X=x\mid Z)}.</math>

: <math>\Pr(Y,Z\mid\text{do}(x)) = \frac{\Pr(Y,Z,X=x)}{\Pr(X=x\mid Z)}.</math>