Graphical model: Difference between revisions

Revision as of 00:38, 9 July 2007

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In [[probability theory]], [[statistics]], and [[statisticsmachine learning]], a '''graphical model (GM)''' is a graph that represents [[statistical independence|dependencies]] among [[random variable]]s by a [[graph (mathematics)|graph]] in which each random variablenode is a noderandom variable, and the edges between the nodes represent conditional dependencies.

InTwo thecommon simplesttypes case,of GMs correspond to graphs with directed and undirected edges. If the network structure of the model is a [[directed acyclic graph]] (DAG). Then, the GM represents a factorization of the joint [[probability]] of all random variables. More precisely, if the events are

:''X''₁, ..., ''X''_''n'',

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In other words, the [[probability distribution|joint distribution]] factors into a product of conditional distributions. The graph structure indicates direct dependencies among random variables. Any two nodes that are not in a descendant/ancestor relationship are [[Conditional independence|conditionally independent]] given the values of their parents.

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

Graphical models with undirected edges are generally called [[Markov random field]]s or [[Markov network]]s. It can be shown that they have the same representational capacity as directed graphical models. However, while directed models are better at explicitly representing the joint probability, undirected models are better for representing conditional independences.

Applications of graphical models include modellingmodeling of [[gene regulatory network]]s, [[speech recognition]], gene finding, [[computer vision]] and diagnosis of diseases.

A good reference for learning the basics of graphical models is written by Neapolitan, Learning Bayesian networks (2004). A more advanced and statistically oriented book is by Cowell, Dawid, Lauritzen and Spiegelhalter, Probabilistic networks and expert systems (1999).