Local conditional and marginal approach to parameter estimation in discrete graphical models

Abstract

Discrete graphical models are an essential tool in the identification of the relationship between variables in complex high-dimensional problems. When the number of variables p is large, computing the maximum likelihood estimate (henceforth abbreviated MLE) of the parameter is difficult. A popular approach is to estimate the composite MLE (abbreviated MCLE) rather than the MLE, i.e., the value of the parameter that maximizes the product of local conditional or local marginal likelihoods, centered around each vertex v of the graph underlying the model. The purpose of this paper is to first show that, when all the neighbors of v are linked to other nodes in the graph, the estimates obtained through local conditional and marginal likelihoods are identical. Thus the two MCLE are usually very close. Second, we study the asymptotic properties of the composite MLE obtained by averaging of the estimates from the local conditional likelihoods: this is done under the double asymptotic regime when both p and N go to infinity.