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{{short description|Probabilistic model}}

{{machine learning bar}}

{{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==

[[File:Graph model.svg|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, B, and C; and C depends on B and D; whereas A and B are each independent.]]

==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=[[Daphne Koller|Koller, D.]]

|author2=[[Nir Friedman|Friedman, N.]]

|title=Probabilistic Graphical Models

|url=http://pgm.stanford.edu/

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|___location=Massachusetts

|year=2009

|pages= 1208

|isbn=978-0-262-01319-2

|author2-link=Nir Friedman

|doi=

|author-link=Daphne Koller

|accessdate=

|archive-url=https://web.archive.org/web/20140427083249/http://pgm.stanford.edu/

}}</ref>

|archive-date=2014-04-27

|url-status=dead

}}</ref>

===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:

:<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]]

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

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

where <math>\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

:<math>P[AX_1,B\ldots,C,DX_n] = P[A]\cdot Pprod_{i=1}^nP[B]X_i|\cdot P[C,D|A,Btext{pa}(X_i)]</math>

where <math>\text{parentspa}(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, butshown an [[ancestral graph]]) consists ofin the randomFigure variablesthis <math>A,factorization B,would C, D</math>be

:<math>P[A,B,C,D] = P[A]\cdot P[B | A]\cdot P[C | A] \cdot P[D|A,C]</math>.

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]].

===Cyclic Directed Graphical Models===

[[File:Graph model.svg|thumb|right|alt=An example of a directed graphical model.| An example of a directed, cyclic graphical model. Each arrow indicates a dependency. In this example: D depends on A, B, and C; and C depends on B and D; whereas A and B are each independent.]]

The next figure depicts a graphical model with a cycle. This may be interpreted in terms of each variable 'depending' on the values of its parents in some manner.

withThe particular graph shown suggests a joint probability density that factors as

:<math>P[A,B,C,D] = P[A]\cdot P[B]\cdot P[C,D|A,B]</math>,

but other interpretations are possible.

<ref name=richardson95causal>{{cite book

|first=Thomas |last=Richardson

|chapter=A discovery algorithm for directed cyclic graphs

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

|year=1996

|publisher=Morgan Kaufmann Pub.

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

}}

</ref>

===Other types===

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

* 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]].

*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]].

* 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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| last = Bishop

| first = Christopher M.

| authorlinkauthor-link = Christopher Bishop

| title = Pattern Recognition and Machine Learning

| publisher = Springer

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| isbn=978-0-387-31073-2

| chapter= Chapter 8. Graphical Models

| chapterurlchapter-url=https://www.microsoft.com/en-us/research/wp-content/uploads/2016/05/Bishop-PRML-sample.pdf

| pages=359–422

| mr=2247587

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* {{cite book

|author=Cowell, Robert G.

|author2=[[Philip Dawid|Dawid, A. Philip]] |author3=Lauritzen, Steffen L. |author4-link=[[David Spiegelhalter |author4=Spiegelhalter, David J.]]

|title=Probabilistic networks and expert systems |publisher=Springer |___location=Berlin |year=1999 |pages= |isbn=978-0-387-98767-5 |doimr=1697175 |accessdateref=cowell |mrauthor2-link=1697175Philip |ref=cowellDawid }} A more advanced and statistically oriented book

* {{cite book |author=Jensen, Finn |title=An introduction to Bayesian networks |publisher=Springer |___location=Berlin |year=1996 |pages= |isbn=978-0-387-91502-9 |doi= |accessdate=}}

* {{Cite book

|first=Judea |last=Pearl |authorlinkauthor-link = Judea Pearl

| year = 1988

| title = Probabilistic Reasoning in Intelligent Systems

|url=https://archive.org/details/probabilisticrea00pear |url-access=registration | edition = 2nd revised

| ___location = San Mateo, CA

| publisher = [[Morgan Kaufmann]]

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===Journal articles===

* {{Cite journal

| author = Edoardo M. Airoldi |authorlink1author-link1=Edoardo Airoldi

| title = Getting Started in Probabilistic Graphical Models

| journal = [[PLoSPLOS Computational Biology]]

| volume = 3

| issue = 12

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| pmid = 18069887

| pmc = 2134967

|arxiv=0706.2040

}}

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

*{{Cite journal | last1 = Jordan | first1 = M. I. | authorlink=Michael I. Jordan| doi = 10.1214/088342304000000026 | title = Graphical Models | journal = Statistical Science | volume = 19 | pages = 140–155| year = 2004 | pmid = | pmc = }}

|doi-access=free

*{{Cite journal|last=Ghahramani|first=Zoubin|date=May 2015|title=Probabilistic machine learning and artificial intelligence|url=https://www.nature.com/articles/nature14541|journal=Nature|language=English|volume=521|issue=7553|pages= 452–459|doi=10.1038/nature14541|pmid=26017444|via=}}

}}

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

*{{Cite journal|last=Ghahramani|first=Zoubin|date=May 2015|title=Probabilistic machine learning and artificial intelligence|url=https://www.nature.com/articles/nature14541|journal=Nature|language=Englishen|volume=521|issue=7553|pages= 452–459|doi=10.1038/nature14541|pmid=26017444|viabibcode=2015Natur.521..452G|s2cid=216356|url=https://www.repository.cam.ac.uk/handle/1810/248538|doi-access=free}}

===Other===

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==External links==

* [http://sandeep-aparajit.blogspot.com/2013/06/how-does-conditional-random-field-crf.html Graphical models and Conditional Random Fields]

* [httphttps://www.cs.cmu.edu/~epxing/Class/10708/ Probabilistic Graphical Models taught by Eric Xing at CMU]

{{Statistics|analysis}}