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A Markov network, or Markov random field, is a model of the (full) joint probability distribution of a set of random variables. A Markov network is similar to a Bayesian network in its representation of dependencies, but a Markov network can represent dependencies that a Bayesian network cannot, such as cyclic dependencies. Formally, a Markov network consists of: The joint distribution represented by a Markov network is given by: where is the state of the random variables in the kth clique, and the normalizing constant (also called a partition function), where . The Markov blanket of a node in a Markov network is defined to be every node with an edge to , i.e. all such that . Every node in a Markov network is conditionally independent of every other node given the Markov blanket of . As in a Bayesian network, one may calculate the conditional distribution of a set of nodes given values to another set of nodes in the Markov network by summing over all possible assignments to ; this is called exact inference. However, exact inference is in general a #P-complete problem, and thus computationally intractable. Approximation techniques such as Markov chain Monte Carlo and loopy belief propagation are more feasible in practice. (Though note that some particular subclasses of MRF have polynomial algorithms; discovering such subclasses is an active research topic.) One notable variant of a Markov network is a conditional random field, in which each random variable may also be conditioned upon a set of global observations . In this model, each function is a mapping from all assignments to both the clique k and the observations to the nonnegative real numbers. This form of the Markov network may be more appropriate for producing discriminative classifiers, which do not model the distribution over the observations.
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