Marginal maximum likelihood estimation of conditional autoregressive models with missing data

Maximum likelihood (ML) estimation of spatial autocorrelation models is well established for the case where each node in the graph is directly observed. When one or more nodes are not observed, the user has a variety of computational tools at her or his disposal ranging from the expectation-maximization algorithm, which has become a standard for missing-data problems, to marginal likelihood estimation methods and to fully Bayesian approaches. In this article, we give a comprehensive overview of likelihood-based computational frameworks for parameter estimation of the conditional autoregressive model, and we establish connections with several algorithms in the literature that are iterative and often computationally suboptimal. We show that a vanilla marginal ML approach, which we provide computational details for, is still generally orders of magnitude faster than the iterative approaches, even on large data sets and especially so when the number of unobserved units is relatively large.