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## 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

where

is the normalizing constant.

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

where each depends only on those coordinates , . The Hamiltonian is a energy function, and the model abhors to retain a high energy when the temperature is low.

Markov random field. Let be a -valued random variable. is an MRF (Markov random field) with respect to if for every and

1. ;
2. .

Hammersley-Clifford theorem. is an MRF with respect to if and only if is a Gibbs distribution with respect to . In this sense a Gibbs distribution is also known as GRF (Gibbs random field).

Site update. In the Gibbs distribution, the conditional probability for site update is given by

Here the normalizing constant is canceled and

is only the partial sum over neighboring cliques of .