# Bayesian model selection for exponential random graph models via adjusted pseudolikelihoods

## TL;DR

This paper introduces an adjusted pseudolikelihood approach for Bayesian model selection in exponential random graph models, enabling more efficient and reliable inference in complex network analysis.

## Contribution

It proposes a novel adjustment method for pseudolikelihoods to approximate likelihoods accurately, facilitating Bayesian model comparison in doubly-intractable models.

## Key findings

- The adjusted pseudolikelihood provides evidence estimates comparable to existing methods.
- The proposed method reduces computational cost significantly.
- Empirical results demonstrate improved inference quality in network models.

## Abstract

Models with intractable likelihood functions arise in areas including network analysis and spatial statistics, especially those involving Gibbs random fields. Posterior parameter es timation 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.