Fully Bayesian binary Markov random field models: Prior specification and posterior simulation

We propose a flexible prior model for the parameters of a binary Markov random field (MRF) defined on a rectangular lattice and with k ×l cliques. The prior model allows higher-order interactions to be included in the MRF. We also define a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm to sample from the associated posterior distribution. The number of possible parameters for an MRF with k × l cliques becomes high even for small values of k and l. To get a flexible model which may adapt to the structure of a particular observed image we do not put any absolute restrictions on the parametrization. Instead we define a parametric form for the MRF where the parameters have interpretation as potentials for the various clique configurations, and limit the effective number of parameters by assigning apriori discrete probabilities for events where groups of parameter values are equal. To run our RJMCMC algorithm we have to cope with the computationally intractable normalizing constant of MRFs. For this we adopt a previously defined approximation for binary MRFs, but we also briefly discuss other alternatives. We demonstrate the flexibility of our prior formulation in two examples with simulated data and in one real data example.