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Markov Random Fields

A family of random variables $\Sigma$ is said to be a Markov Random Field on S with respect to a neighborhood system if and only if:
\begin{displaymath}P(\sigma)> 0,\forall \sigma \in \Sigma \end{displaymath}

\begin{displaymath}P(\sigma_i\vert\sigma_{S-\{i\}})=P(\sigma_i\vert f_{Ni}), \forall \sigma \in \Sigma \end{displaymath}

$f_{Ni}=\{\sigma_{i'}\vert i'\in N_i\}$  being the neighborhood system.

Neighborhood system for a Markov Random Field of order1 and corresponding cliques with cardinal lower or equal to 2:

Hammersley-Clifford theorem: A Markov Random Field can be written as a Gibbs distribution,

\begin{displaymath}P_T(\sigma )=Z_{T}^{-1}\times e^{-\frac {1}{T}U(\sigma)}\end{displaymath}

\begin{displaymath}U(\sigma)=\sum_CV_C \end{displaymath}

Christine Hivernat & Xavier Descombes

November 1998