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.

For an autologistic binary-image model on a finite undirected graph, let $x_v\in \{0,1\}$ and give the configuration prior mass proportional toSuppose $Y_v\mid x$ are independent Gaussian distribution observations with expected value $x_v$ and unit variance. Their local log-likelihood ratio for $x_v=1$ versus $0$ isWriting $z_v={\sum }_{w\sim v}{x}_w$, the full posterior conditional probability is thereforeThus the local evidence changes the one-site field while retaining the same neighbor interaction. For a noise variance ${\sigma }^2>0$, replace $y_v-1/2$ by $(y_v-1/2)/{\sigma }^2$.

The graphical conditional independence survives this update: vertices in any independent color class are conditionally independent given the other vertices and the observations. On a nearest-neighbor square grid, the two checkerboard colors give such classes, so one color can be updated in parallel in a Gibbs sampler. An arbitrary graph need not be bipartite and may require more colors. Updating every vertex simultaneously from its old neighbors is not the same as this exact conditional block update.

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 $S$.