Graphical model: Difference between revisions

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A '''graphical model''' or '''probabilistic graphical model''' ('''PGM''') or '''structured probabilistic model''' is a [[probabilistic model]] for which a [[Graph (discrete mathematics)|graph]] expresses the [[conditional dependence]] structure between [[random variable]]s. Graphical models are commonly used in [[probability theory]], [[statistics]]—particularly [[Bayesian statistics]]—and [[machine learning]].

==Types of graphical models==

Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a distribution over a multi-dimensional space and a graph that is a compact or [[Factor graph|factorized]] representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, [[Bayesian network]]s and [[Markov random field]]s. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce.<ref name=koller09>{{cite book

|author=Koller, D.

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===Undirected Graphical Model===

[[File:Examples of an Undirected Graph.svg|thumb|alt=An undirected graph with four vertices.|An undirected graph with four vertices.]]

The undirected graph shown may have one of several interpretations; the common feature is that the presence of an edge implies some sort of dependence between the corresponding random variables. From this graph, we might deduce that B, C, and D are all [[Conditional independence|conditionally independent]] given A. This means that if the value of A is known, then the values of B, C, and D provide no further information about each other. Equivalently (in this case), the joint probability distribution can be factorized as:

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{{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.]]

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*[[Dependency network (graphical model)|Dependency network]] where cycles are allowed

*Tree-augmented classifier or '''TAN model'''

[[File:Tan corral.png|thumb| TAN model for "corral dataset".]]

*Targeted Bayesian network learning (TBNL) [[File:Tbnl corral.jpg|thumb|TBNL model for "corral dataset"]]

*A [[factor graph]] is an undirected [[bipartite graph]] connecting variables and factors. Each factor represents a function over the variables it is connected to. This is a helpful representation for understanding and implementing [[belief propagation]].