Comparing Maximum Likelihood and Pseudo-Maximum Likelihood Estimators for the Ising Model

The Ising model is one of the most popular models in network psychometrics. However, statistical analysis of the Ising model is difficult due to thepresence of its intractable normalizing constant in the probability function.As a result, maximum likelihood estimation using the exact likelihood is onlypossible for small graphs, and approximation methods are needed for largergraphs. Two popular approximations of the exact likelihood are the jointpseudolikelihood (JPL) and the disjoint pseudolikelihood (DPL). These approximations yield consistent estimators, but we do not know how well theyperform for finite data. In this paper, we investigate the relative performanceof parameter estimation methods based on the two approximations and compare them to maximum likelihood estimation using the exact likelihood. Weperform an extensive simulation study comparing the estimators in termsof bias and variance. We show that maximum pseudolikelihood estimationbased on the JPL is a stable estimation method that is able to accuratelyapproximate the maximum likelihood estimates, but that maximum pseudolikelihood estimation based on the DPL only works well for large samplesizes.