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In the ___domain of [[physics]] and [[probability]], a '''Markov random field''' (often abbreviated as '''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, Jr.|Markov]] random field if it satisfies Markov properties.
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>{{cite book
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