Strong structure recovery for partially observed discrete Markov random fields on graphs

We propose a penalized maximum likelihood criterion to estimate the graph of conditional dependencies in a discrete Markov random field, that can be partially observed. We prove the almost sure convergence of the estimator in the case of a finite or countable infinite set of variables. In the finite case, the underlying graph can be recovered with probability one, while in the countable infinite case we can recover any finite subgraph with probability one, by allowing the candidate neighborhoods to grow with the sample size n. Our method requires minimal assumptions on the probability distribution and contrary to other approaches in the literature, the usual positivity condition is not needed.