Construction of binary multi-grid Markov random field models from training images

We define the class of multi-grid discrete Markov random field (MRF) models and discuss how to estimate associated model parameters from a given training image. The intention is to use the resulting model as a prior for the spatial facies distribution in a Bayesian model. The multi-grid MRF model includes normalisation constants which it is computationally infeasible to compute. To cope with this complication we use a partially order Markov model (POMM) approximation to each MRF included in the multi-grid MRF model. We thereby get an explicit expression for the resulting estimated (approximate) multi-grid MRF model. This enables direct unconditional simulation from the model. Moreover, used as a prior model in a Bayesian context, we also have an explicit expression, up to an normalising constant, of the corresponding posterior. The Metropolis-Hastings algorithm can thereby be used to generate samples from the posterior. It is also possible to adopt once more the POMM approximation to MRF idea and generate realisations from a POMM approximation to the posterior distribution without resorting to iterative algorithms.