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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. A Markov random field 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''), a set of random variables ''X'' = (''X''''v'')''v'' ∈ ''V'' indexed by ''V'' form a Markov random field with respect to ''G'' 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 ne(''v'') is the set of neighbors of ''v'', and cl(''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''. The above three Markov properties are not equivalent: The Local Markov property is stronger than the Pairwise one, while weaker than the Global one. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Markov random field」の詳細全文を読む スポンサード リンク
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