Bayesian Model Selection for Exponential Random Graph Models via Adjusted Pseudolikelihoods

Abstract

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

Bayesian Variational Inference for Exponential Random Graph Models

“…Properties of the pseudolikelihood are not well understood (van Duijn et al, 2009) and its use may also lead to biased estimates. Bouranis et al (2018) proposed an adjusted pseudolikelihood which attempts to correct the mode, curvature and magnitude of the pseudolikelihood. It is given bỹ…”

“…The normalizing constant z(θ ML ) is intractable and Bouranis et al (2018) propose an importance sampling procedure for estimating it. Introducing a sequence of temperatures 0 = t 0 < t 1 < · · · < t L = 1, they write…”

“…) and s(y) −1 denotes the vector of sufficient statistics excluding the first element. Bouranis et al (2018) used the fully adjusted pseudolikelihood in place of the true likelihood in MCMC algorithms and demonstrated that the Bayes factor for performing model selection can be estimated accurately at greatly reduced computation times. In the next section, we describe a variety of variational inference methods for the ERGM, one of which utilizes this adjusted pseudolikelihood as a plug-in for the true likelihood.…”

“…The posterior distributions estimated using the exchange algorithm are regarded as the "ground truth" and we use it as a basis for evaluating the accuracy of the posterior distributions estimated using the variational methods. An estimate of the marginal likelihood using Chib and Jeliazkov's method (Chib and Jeliazkov, 2001) based on the adjusted pseudolikelihood is also obtained using the evidence CJ function (Bouranis et al, 2018).…”

Characterising group-level brain connectivity: a framework using Bayesian exponential random graph models

“…We used the summary statistics suggested in Simpson et al [34], which were selected via a graphical goodness-of-fit approach, in which the 'best' metrics are chosen from a prespecified set of potential metrics. While it is possible to perform Bayesian model selection for an ERGM on a single network [6,3], further work is needed to extend these approaches to perform model selection for a group of networks.…”