Pseudolikelihood estimators for graphical models: existence and uniqueness

Summary Graphical and sparse (inverse) covariance models have found widespread use in modern sample-starved high-dimensional applications. A part of their wide appeal stems from the significantly low sample sizes required for existence of the estimators, especially in comparison with the classical full covariance model. For undirected Gaussian graphical models, the minimum sample size required for the existence of maximum likelihood estimators had been an open question for almost half a century, and has recently been addressed in a series of works (Uhler, 2012; Ben-David, 2015; Gross & Sullivant, 2018). The very same question for pseudolikelihood estimators has remained unsolved ever since their introduction in the 1970s. Pseudolikelihood estimators have recently received renewed attention as they impose fewer restrictive assumptions and have better computational tractability, improved statistical performance and appropriateness in high-dimensional applications, thus renewing interest in this longstanding problem. In this paper, we undertake a comprehensive study of this open problem within the context of the two classes of pseudolikelihood methods proposed in the literature. We provide a precise answer to this question for both pseudolikelihood approaches and relate the corresponding solutions to their Gaussian counterparts.