Editing Markov random field (section)

{{Short description|Set of random variables}}
[[File:markov random field example.png|thumb|alt=An example of a Markov random field.|An example of a Markov random field. Each edge represents dependency. In this example: A depends on B and D. B depends on A and D. D depends on A, B, and E. E depends on D and C. C depends on E.]]

In the domain of [[physics]] and [[probability]], a '''Markov random field''' ('''MRF'''), '''Markov network''' or '''undirected [[graphical model]]''' is a set of [[random variable]]s having a [[Markov property]] described by an [[undirected graph]]. In other words, a [[random field]] is said to be a [[Andrey Markov|Markov]] random field if it satisfies Markov properties. The concept originates from the [[Spin glass#Sherrington–Kirkpatrick model|Sherrington–Kirkpatrick model]].<ref>{{citation|title=Solvable Model of a Spin-Glass|number=35|year=1975|author1= Sherrington, David|author2=Kirkpatrick, Scott|journal=Physical Review Letters|volume=35|pages=1792–1796|doi=10.1103/PhysRevLett.35.1792|bibcode=1975PhRvL..35.1792S}}</ref>

A Markov network or MRF is similar to a [[Bayesian network]] in its representation of dependencies; the differences being that Bayesian networks are [[directed acyclic graph|directed and acyclic]], whereas Markov networks are undirected and may be cyclic. Thus, a Markov network can represent certain dependencies that a Bayesian network cannot (such as cyclic dependencies {{Explain|date=July 2018}}); on the other hand, it can't represent certain dependencies that a Bayesian network can (such as induced dependencies {{Explain|date=July 2018}}). The underlying graph of a Markov random field may be finite or infinite.

When the [[joint probability distribution|joint probability density]] of the random variables is strictly positive, it is also referred to as a '''Gibbs random field''', because, according to the [[Hammersley–Clifford theorem]], it can then be represented by a [[Gibbs measure]] for an appropriate (locally defined) energy function. The prototypical Markov random field is the [[Ising model]]; indeed, the Markov random field was introduced as the general setting for the Ising model.<ref name="Kindermann-Snell80">{{cite book
 |first1=Ross
 |last1=Kindermann
 |first2=J. Laurie
 |last2=Snell
 |url=http://www.cmap.polytechnique.fr/~rama/ehess/mrfbook.pdf
 |title=Markov Random Fields and Their Applications
 |year=1980
 |publisher=American Mathematical Society
 |isbn=978-0-8218-5001-5
 |mr=0620955
 |access-date=2012-04-09
 |archive-date=2017-08-10
 |archive-url=https://web.archive.org/web/20170810092327/http://www.cmap.polytechnique.fr/%7Erama/ehess/mrfbook.pdf
 |url-status=dead
 }}</ref> In the domain of [[artificial intelligence]], a Markov random field is used to model various low- to mid-level tasks in [[image processing]] and [[computer vision]].<ref>{{cite book
 |first1=S. Z. |last1=Li 
 |title=Markov Random Field Modeling in Image Analysis
 |year=2009
 |publisher=Springer
 |url=https://books.google.com/books?id=rDsObhDkCIAC
|isbn=9781848002791 
 }}</ref>