Editing Graphical model (section)

===Bayesian network===
{{main|Bayesian network}}

[[File:Example of a Directed Graph.svg|thumb|alt=Example of a directed acyclic graph on four vertices.|Example of a directed acyclic graph on four vertices.]]


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

:<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, in the directed acyclic graph shown in the Figure this factorization would be
:<math>P[A,B,C,D] = P[A]\cdot P[B | A]\cdot P[C | A] \cdot P[D|A,C]</math>.

Any two nodes are [[Conditional independence|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|''d''-separation]] holds in the graph.  Local independences and global independences are equivalent in 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]], [[Artificial neural network|neural networks]] and newer models such as [[variable-order Markov model]]s can be considered special cases of Bayesian networks.

One of the simplest Bayesian Networks is the [[Naive Bayes classifier]].