Markov random field

A Markov random field is an undirected graphical model whose graph encodes conditional independence by vertex separation.

On a graph, a binary configuration has probability proportional to $\exp (\alpha {\sum }_v{x}_v+\beta {\sum }_{\{v,w\}}{x}_v{x}_w)$, with each edge counted once. The full conditional probability of a one is logistic with linear predictor $\alpha +\beta {\sum }_{w\sim v}{x}_w$. Local neighbor sums eliminate the need to evaluate the normalizing constant.

Independent unit-variance Gaussian observations with mean $x_v$ add $y_v-1/2$ to the local binary field. The posterior distribution conditional probability is $\mathrm{logistic}(\alpha +\beta z_v+y_v-1/2)$. The neighbor interaction is unchanged, so checkerboard conditional independence and the same Gibbs schemes remain valid.

A clique potential is a nonnegative function of the variables belonging to one clique. Products of clique potentials define the unnormalized density of a Markov random field.

The global Markov property says that graph separation of vertex sets $A$ and $B$ by $S$ implies conditional independence of the corresponding random vectors given the variables at $S$.