Graphical model: Difference between revisions

Revision as of 01:43, 9 April 2004

Revision as of 08:37, 11 May 2004

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In [[probability theory]] and [[statistics]], a '''graphical model''' (GM) represents [[statistical independence|dependencies]] among [[random variable|random variables]] by a [[graph (mathematics)|graph]] in which each random variable is a node.

In the simplest case, the network structure of the model is a [[directed acyclic graph]]. Then the GM represents a factorization of the joint [[probability]] of all random variables. More precisely, if the random variables are X_1 through X_n, then the joint probability P(X_1,...,X_n) is equal to the product of the [[conditional probability|conditional probabilities]] P(X_i | parents of X_i) for all i=1,...,n. 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 conditionally [[statistical independence|independent]] given the values of their parents.

:''X''₁, ..., ''X''_''n'',

then the joint probability

:''P''(''X''₁, ..., ''X''_''n''),

is equal to the product of the [[conditional probability|conditional probabilities]]

:P(''X_i''| parents of ''X_i'') for ''i'' = 1,...,''n''.

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 conditionally [[statistical independence|independent]] given the values of their parents.

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