Markov Random Fields

Network description

Network : set S of sites s_i , i.e., the nodes.

Neighboring system : , and : $V=\{V_s, s\in S\}$

Clique : singleton or set of neighbor sites.

Markov field

X is a Markov Random Field (MRF) relatively to V

if and only if

$$\forall s \in S, P(X_s=x_s \vert X_t=x_t, t\in S-\{s\}) = P(X_s=x_s \vert X_t=x_t, t\in V_s)$$

Gibbs field

Field whose probability is a Gibbs measure :

$$P(X=x)=\frac{1}{Z}\exp(-U(x))$$

    with the associated energy,
              Z normalizing constant (partition fonction).

Hammersley-Clifford Theorem

X is a MRF relatively to V and

X is a Gibbs field whose energy U is associated to V.