Abstract: |
Bayesian modelling requires prior and likelihood models to be specified. In reservoir characterisation it is common practice to estimate the prior from a training image. We adopt a multi-grid approach for construction of prior models for binary variables. Thus, we first specify a parametric model class on a coarse lattice and thereafter sequentially specify parametric models on finer and finer grids given values on coarser levels. On each level we adopt a Markov random field (MRF) prior. Parameter estimation for MRFs is complicated by a computationally intractable normalising constant. To cope with this problem we first simulate from an approximation to the MRF, use the samples with importance sampling to define an estimated likelihood function, and finally maximise the estimated likelihood function. Unconditional simulation from the fitted model can easily be done, either by again adopting the MRF approximation or by Metropolis-Hastings simulation. Conditional simulation is complicated by the computationally intractable normalising constants. To cope with this we combine the submodels for all levels as one large model, this becomes an MRF with a complicated neighbourhood structure, and once more adopt the MRF approximation. The result is an MRF model with a simpler neighbourhood structure. Simulation conditioned to exact observations can then easily be done by once more adopting the MRF approximation. For conditional simulation with more complicated likelihood functions, the Metropolis-Hastings algorithm must be applied. Comparing our multi-grid MRF with popular multi-point priors, there are two important differences. First, the number of parameters to estimate is low in our model and very large for multi-point priors. Second, our procedure results in an explicit formula for the estimated prior, whereas multi-point priors are algorithmically defined. Thus, conditional simulation from multi-point models can not be done with the Metropolis--Hastings algorithm, in stead ad-hoc procedures must be invented for each observation type. |