Graphical model: Difference between revisions

Revision as of 09:25, 16 March 2019

Revision as of 00:08, 2 May 2019

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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|pa_i\text{parents}(X_i)]</math>

where <math>pa_i\text{parents}(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, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an [[ancestral graph]]) consists of the random variables <math>A, B, C, D</math>

with a joint probability density that factors as