be a finite undirected graph.
is said to be a neighbor
is called a clique
We denote the set of all cliques by
be a common state space.
Given a parameter
we can define a Gibbs distribution
with respect to
is the normalizing constant.
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
depends only on those coordinates
is a energy function
and the model abhors to retain a high energy when the temperature
Markov random field.
Let be a -valued random variable.
is an MRF (Markov random field) with respect to if
for every and
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)
In the Gibbs distribution,
the conditional probability for site update
is given by
the normalizing constant is canceled
is only the partial sum over neighboring cliques of
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