Statistical inférence methods for Gibbs random fields

Due to the Markovian dependence structure, the normalizing constant of Markov random fields cannot be computed with standard analytical or numerical methods. This forms a central issue in terms of parameter inference or model selection as the computation of the likelihood is an integral part of the procedure. When the Markov random field is directly observed, we propose to estimate the posterior distribution of model parameters by replacing the likelihood with a composite likelihood, that is a product of marginal or conditional distributions of the model easy to compute. Our first contribution is to correct the posterior distribution resulting from using a misspecified likelihood function by modifying the curvature at the mode in order to avoid overly precise posterior parameters. In a second part we suggest to perform model selection between hidden Markov random fields with approximate Bayesian computation (ABC) algorithms that compare the observed data and many Monte-Carlo simulations through summary statistics. To make up for the absence of sufficient statistics with regard to this model choice, we introduce summary statistics based on the connected components of the dependency graph of each model in competition. We assess their efficiency using a novel conditional misclassification rate that evaluates their local power to discriminate between models. We set up an efficient procedure that reduces the computational cost while improving the quality of decision and using this local error rate we build up an ABC procedure that adapts the summary statistics to the observed data. In a last part, in order to circumvent the computation of the intractable likelihood in the Bayesian Information Criterion (BIC), we extend the mean field approaches by replacing the likelihood with a product of distributions of random vectors, namely blocks of the lattice. On that basis, we derive BLIC (Block Likelihood Information Criterion) that answers model choice questions of a wider scope than ABC, such as the joint selection of the dependency structure and the number of latent states. We study the performances of BLIC in terms of image segmentation.