Graphical model: Difference between revisions

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{{short description|Probabilistic model}}

{{about|the representation of probability distributions using graphs|the computer graphics journal|Graphical Models{{!}}''Graphical Models''}}

{{Machine learning|Structured prediction}}

{{More footnotes|date=May 2017}}

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. TheyGraphical 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 <math>B, C, and D</math> are all mutually[[Conditional independence|conditionally independent,]] oncegiven A. This means that if the value of <math>A</math> is known, orthen (equivalentlythe values of B, C, and D provide no further information about each other. Equivalently (in this case), thatthe joint probability distribution can be factorized as:

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 <math>B,C,D</math> are all mutually independent, once <math>A</math> is known, or (equivalently in this case) that

:<math>P[A,B,C,D] = f_{AB}[A,B] \cdot f_{AC}[A,C] \cdot f_{AD}[A,D]</math>

for some non-negative functions <math>f_{AB}, f_{AC}, f_{AD}</math>.

===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.]]

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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]].

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|title=Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence

|year=1996

|publisher=Morgan Kaufmann Pub.

|isbn=978-1-55860-412-4

}}

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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]].

* A [[clique tree]] or junction tree is a [[tree (graph theory)|tree]] of [[clique (graph theory)|cliques]], used in the [[junction tree algorithm]].

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|arxiv=0706.2040

|bibcode=2007PLSCB...3..252A

|doi-access=free

}}

*{{Cite journal | last1 = Jordan | first1 = M. I. | author-link=Michael I. Jordan| doi = 10.1214/088342304000000026 | title = Graphical Models | journal = Statistical Science | volume = 19 | pages = 140–155| year = 2004 | doi-access = free }}