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| A '''graphical model''' is a [[probabilistic model]] for which a [[graph (mathematics)|graph]] denotes the [[conditional dependence]] structure between [[random variable]]s. They are commonly used in [[probability theory]], [[statistics]]—particularly [[Bayesian statistics]]—and [[machine learning]].
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| [[File:graphical_model_example.png|thumb|right|alt=An example of a graphical model.| An example of a graphical model. Each arrow indicates a dependency. In this example: D depends on A, D depends on B, D depends on C, C depends on B, and C depends on D. ]]
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| ==Types of graphical models==
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| Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a complete distribution over a multi-dimensional space and a graph that is a compact or [[Graph factorization|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 network]]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>Koller; Friedman (2009). Probabilistic Graphical Models. Massachusetts: MIT Press. ISBN 0-262-01319-3.</ref>
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| ===Bayesian network===
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| {{main|Bayesian network}}
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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 | |
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| :<math>P[X_1,\ldots,X_n]=\prod_{i=1}^nP[X_i|pa_i]</math>
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| where <math>pa_i</math> is the set of parents of node <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 consists of the random variables <math>A, B, C, D</math>
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| with a joint probability density that factors as
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| :<math>P[A,B,C,D] = P[A] P[B] P[C|B,D] P[D|A,B,C].</math>
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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.
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| 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]], [[neural networks]] and newer models such as [[variable-order Markov model]]s can be considered special cases of Bayesian networks.
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| ===Markov random field===
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| {{main|Markov random field}}
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| A Markov random field, also known as a Markov network, is a model over an [[undirected graph]]. A graphical model with many repeated subunits can be represented with [[plate notation]].
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| ===Other types===
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| * A [[factor graph]] is an undirected [[bipartite graph]] connecting variables and factors. Each factor represents a probability distribution over the variables it is connected to. Graphs are converted into factor graph form to perform [[belief propagation]].
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| * 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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| * A [[chain graph]] is a graph which may have both directed and undirected edges, but without any directed cycles (i.e. if we start at any vertex and move along the graph respecting the directions of any arrows, we cannot return to the vertex we started from if we have passed an arrow). Both directed acyclic graphs and undirected graphs are special cases of chain graphs, which can therefore provide a way of unifying and generalizing Bayesian and Markov networks.<ref>{{cite journal|last=Frydenberg|first=Morten|year=1990|title=The Chain Graph Markov Property|journal=Scandinavian Journal of Statistics|volume=17|issue=4|pages=333–353|mr=1096723|jstor=4616181 }}
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| </ref> | |
| * An [[ancestral graph]] is a further extension, having directed, bidirected and undirected edges.<ref>{{cite journal
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| |first1=Thomas |last1=Richardson |first2=Peter |last2=Spirtes
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| |title=Ancestral graph Markov models
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| |journal=[[Annals of Statistics]]
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| |volume=30 |issue=4 |year=2002 |pages=962–1030
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| |doi=10.1214/aos/1031689015
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| |mr=1926166 | zbl = 1033.60008
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| }}</ref>
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| * A [[conditional random field]] is a [[discriminative model]] specified over an undirected graph.
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| * A [[restricted Boltzmann machine]] is a [[generative model]] specified over an undirected graph.
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| ==Applications==
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| The framework of the models, which provides algorithms for discovering and analyzing structure in complex distributions to describe them succinctly and extract the unstructured information, allows them to be constructed and utilized effectively.<ref name=koller09/> Applications of graphical models include [[information extraction]], [[speech recognition]], [[computer vision]], decoding of [[low-density parity-check codes]], modeling of [[gene regulatory network]]s, gene finding and diagnosis of diseases, and [[graphical models for protein structure]].
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| ==See also==
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| * [[Belief propagation]]
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| * [[Structural equation model]]
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| ==Notes==
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| {{reflist}}
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| ==References and further reading==
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| ===Books and book chapters===
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| * {{cite book
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| | last = Bishop
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| | first = Christopher M.
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| | authorlink = Christopher Bishop
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| | title = Pattern Recognition and Machine Learning
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| | publisher = Springer
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| | year = 2006
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| | url = http://research.microsoft.com/~cmbishop/PRML/
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| | isbn=0-387-31073-8
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| | chapter= Chapter 8. Graphical Models
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| | chapterurl=http://research.microsoft.com/~cmbishop/PRML/Bishop-PRML-sample.pdf
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| | pages=359–422
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| | mr=2247587
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| }}
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| * {{cite book
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| |author=Cowell, Robert G.
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| |coauthors=[[Philip Dawid|Dawid, A. Philip]]; Lauritzen, Steffen L.; [[David Spiegelhalter|Spiegelhalter, David J.]]
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| |title=Probabilistic networks and expert systems |publisher=Springer |location=Berlin |year=1999 |pages= |isbn=0-387-98767-3 |doi= |accessdate= |mr=1697175 |ref=cowell }} A more advanced and statistically oriented book
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| * {{cite book |author=Jensen, Finn |title=An introduction to Bayesian networks |publisher=Springer |location=Berlin |year=1996 |pages= |isbn=0-387-91502-8 |doi= |accessdate=}}
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| * {{cite book
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| |author=[[Daphne_Koller|Koller, D.]]
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| |coauthors=Friedman, N.
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| |title=[http://pgm.stanford.edu/ Probabilistic Graphical Models]
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| |publisher=MIT Press
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| |location=Massachusetts
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| |year=2009
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| |pages= 1208
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| |isbn=0-262-01319-3
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| |doi=
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| |accessdate=
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| }}
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| * {{Cite book
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| |first=Judea |last=Pearl |authorlink = Judea Pearl
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| | year = 1988
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| | title = Probabilistic Reasoning in Intelligent Systems
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| | edition = 2nd revised
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| | location = San Mateo, CA
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| | publisher = [[Morgan Kaufmann]]
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| | mr = 0965765
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| |isbn = 1-55860-479-0
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| }} A computational reasoning approach, where the relationships between graphs and probabilities were formally introduced.
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| ===Journal articles===
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| * {{Cite journal
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| | author = Edoardo M. Airoldi
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| | title = Getting Started in Probabilistic Graphical Models
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| | journal = [[PLoS Computational Biology]]
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| | volume = 3
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| | issue = 12
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| | pages = e252
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| | year = 2007
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| | doi = 10.1371/journal.pcbi.0030252
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| | pmid = 18069887
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| | pmc = 2134967
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| }}
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| *{{cite doi|10.1214/088342304000000026}}
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| ===Other===
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| *[ftp://ftp.research.microsoft.com/pub/tr/tr-95-06.pdf Heckerman's Bayes Net Learning Tutorial]
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| *[http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html A Brief Introduction to Graphical Models and Bayesian Networks]
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| *[http://www.cedar.buffalo.edu/~srihari/CSE574 Sargur Srihari's lecture slides on probabilistic graphical models]
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| {{Statistics|analysis}}
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| [[Category:Bayesian statistics]]
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| [[Category:Statistical models]]
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| [[Category:Graphical models| ]]
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