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Concept
Markov random field
Summary
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.
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 ); on the other hand, it can't represent certain dependencies that a Bayesian network can (such as induced dependencies ). 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. In the domain of artificial intelligence, a Markov random field is used to model various low- to mid-level tasks in and computer vision.
Given an undirected graph , a set of random variables indexed by form a Markov random field with respect to if they satisfy the local Markov properties:
Pairwise Markov property: Any two non-adjacent variables are conditionally independent given all other variables:
Local Markov property: A variable is conditionally independent of all other variables given its neighbors:
where is the set of neighbors of , and is the closed neighbourhood of .
Global Markov property: Any two subsets of variables are conditionally independent given a separating subset:
where every path from a node in to a node in passes through .
Official source
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