A set of random variables having a Markov property described by an undirected graph. A Markov random field is similar to a Bayesian network in its representation of dependencies. The differences are listed below.
Bayesian networks are directed and acyclic.
Markov networks are undirected and may be cyclic.
* Markov property: A stochastic process has the Markov property if the conditional probability distribution of future states of the process depends only upon the present state, not on the sequence of events that preceded it.
- Pairwise Markov property(不相鄰則獨立): Any two non-adjacent variables are conditionally independent given all other variables:
- Local Markov property(Markov blanket): A variable is conditionally independent of all other variables given its neighbours:
where ne(v) is the set of neighbours of v, and cl(v) = {v} ∪ ne(v) is the closed neighbourhood of v.
- Global Markov property: Any two subsets of variables are conditionally independent given a separating subset:
where every path from a node in A to a node in B passes through S.
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