Structure recovery for partially observed discrete Markov random fields on graphs under not necessarily positive distributions

TL;DR

This paper introduces a new penalized pseudo-likelihood method for recovering the structure of partially observed discrete Markov random fields on graphs, applicable to both finite and infinite node sets without requiring positivity conditions.

Contribution

It develops a novel estimation approach that works under minimal assumptions and proves convergence and recovery guarantees for finite and countably infinite graphs.

Findings

Estimator converges with probability one for finite graphs

Finite sub-graph recovery with probability one in infinite graphs

Method performs well on simulated and real stock market data

Abstract

We propose a penalized pseudo-likelihood criterion to estimate the graph of conditional dependencies in a discrete Markov random field that can be partially observed. We prove the convergence of the estimator in the case of a finite or countable infinite set of nodes. In the finite case, the underlying graph can be recovered with probability one, while in the countable infinite case, we can recover any finite sub-graph with probability one by allowing the candidate neighborhoods to grow as a function o(log n), with n the sample size. Our method requires minimal assumptions on the probability distribution, and contrary to other approaches in the literature, the usual positivity condition is not needed. We evaluate the performance of the estimator on simulated data, and we apply the methodology to a real dataset of stock index markets in different countries.