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 image processing and computer vision.
Definition Given an undirected graph G = ( V , E ) {\displaystyle G=(V,E)} , a set of random variables X = ( X v ) v ∈ V {\displaystyle X=(X_{v})_{v\in V}} indexed by V {\displaystyle V} form a Markov random field with respect to G {\displaystyle G} if they satisfy the local Markov properties:
Pairwise Markov property: Any two non-adjacent variables are conditionally independent given all other variables:
X u ⊥ ⊥ X v ∣ X V ∖ { u , v } {\displaystyle X_{u}\perp \!\!\!\perp X_{v}\mid X_{V\smallsetminus \{u,v\}}}
Local Markov property: A variable is conditionally independent of all other variables given its neighbors:
X v ⊥ ⊥ X V ∖ N [ v ] ∣ X N ( v ) {\displaystyle X_{v}\perp \!\!\!\perp X_{V\smallsetminus \operatorname {N} [v]}\mid X_{\operatorname {N} (v)}}
where N ( v ) {\textstyle \operatorname {N} (v)} is the set of neighbors of v {\displaystyle v} , and N [ v ] = v ∪ N ( v ) {\displaystyle \operatorname {N} [v]=v\cup \operatorname {N} (v)} is the closed neighbourhood of v {\displaystyle v} . Global Markov property: Any two subsets of variables are conditionally independent given a separating subset:
X A ⊥ ⊥ X B ∣ X S {\displaystyle X_{A}\perp \!\!\!\perp X_{B}\mid X_{S}}
where every path from a node in A {\displaystyle A} to a node in B {\displaystyle B} passes through S {\displaystyle S} . The Global Markov property is stronger than the Local Markov property, which in turn is stronger than the Pairwise one. However, the above three Markov properties are equivalent for positive distributions (those that assign only nonzero probabilities to the associated variables). The relation between the three Markov properties is particularly clear in the following formulation:
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