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: $$P(\sigma)> 0,\forall \sigma \in \Sigma$$

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

$f_{Ni}=\{\sigma_{i'}\vert i'\in N_i\}$  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,

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

$$U(\sigma)=\sum_CV_C$$

November 1998