# Markov random fields and their applications

## Ross Kindermann (Author of Markov Random Fields and Their Applications)

## Neural networks [3.8] : Conditional random fields - Markov network

## Markov random field

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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 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. 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.

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