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 variables having a Markov property described by an undirected graph. In other words, a random field is said to be a Markov random field if it satisfies Markov properties. The concept originates from the Sherrington–Kirkpatrick model.[1]
A Markov network or MRF is similar to a Bayesian network in its representation of dependencies; the differences being that Bayesian networks are 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 [further explanation needed]); on the other hand, it can't represent certain dependencies that a Bayesian network can (such as induced dependencies [further explanation needed]). The underlying graph of a Markov random field may be finite or infinite.
When the 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.[2] 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.[3]
^Sherrington, David; Kirkpatrick, Scott (1975), "Solvable Model of a Spin-Glass", Physical Review Letters, 35 (35): 1792–1796, Bibcode:1975PhRvL..35.1792S, doi:10.1103/PhysRevLett.35.1792
^Kindermann, Ross; Snell, J. Laurie (1980). Markov Random Fields and Their Applications(PDF). American Mathematical Society. ISBN 978-0-8218-5001-5. MR 0620955. Archived from the original (PDF) on 2017-08-10. Retrieved 2012-04-09.
^Li, S. Z. (2009). Markov Random Field Modeling in Image Analysis. Springer. ISBN 9781848002791.
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