Bayesian Model Selection for Exponential Random Graph Models via Adjusted Pseudolikelihoods

Models with intractable likelihood functions arise in areas including network analysisand spatial statistics, especially those involving Gibbs random fields. Posterior parameter estimationin these settings is termed a doubly-intractable problem because both the likelihoodfunction and the posterior distribution are intractable. The comparison of Bayesian models isoften based on the statistical evidence, the integral of the un-normalised posterior distributionover the model parameters which is rarely available in closed form. For doubly-intractablemodels, estimating the evidence adds another layer of difficulty. Consequently, the selectionof the model that best describes an observed network among a collection of exponentialrandom graph models for network analysis is a daunting task. Pseudolikelihoods offer atractable approximation to the likelihood but should be treated with caution because they canlead to an unreasonable inference. This paper specifies a method to adjust pseudolikelihoodsin order to obtain a reasonable, yet tractable, approximation to the likelihood. This allowsimplementation of widely used computational methods for evidence estimation and pursuitof Bayesian model selection of exponential random graph models for the analysis of socialnetworks. Empirical comparisons to existing methods show that our procedure yields similarevidence estimates, but at a lower computational cost.