Gibbs Distribution
Let be a finite undirected graph. The site is said to be a neighbor of if . is called a clique if whenever . We denote the set of all cliques by . Let be a common state space. Given a parameter , we can define a Gibbs distribution with respect to by
Hamiltonian. In the language of statistical mechanics, a vertex is called a site. The parameter and are respectively called a ``inverse temperature'' and ``partition function.'' And , called a ``Hamiltonian,'' has the form
Markov random field. Let be a valued random variable. is an MRF (Markov random field) with respect to if for every and
 ;

.
Site update. In the Gibbs distribution, the conditional probability for site update is given by
Here the normalizing constant is canceled and
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