Comparing Gaussian Graphical Models with the Posterior Predictive Distribution and Bayesian Model Selection

Gaussian graphical models are commonly used to characterize conditional independence structures (i.e., networks) of psychological constructs. Recently attention has shifted from estimating single networks to those from various sub-populations. The focus is primarily to detect differences or demonstrate replicability. We introduce two novel Bayesian methods for comparing networks that explicitly address these aims. The first is based on the posterior predictive distribution, with Kullback-Leibler divergence as the discrepancy measure, that tests differences between two multivariate normal distributions. The second approach makes use of Bayesian model selection, with the Bayes factor, and allows for gaining evidence for invariant network structures. This overcomes limitations of current approaches in the literature that use classical hypothesis testing, where it is only possible to determine whether groups are significantly different from each other. With simulation we show the posterior predictive method is approximately calibrated under the null hypothesis ($\alpha = 0.05$) and has more power to detect differences than alternative approaches. We then examine the necessary sample sizes for detecting invariant network structures with Bayesian hypothesis testing, in addition to how this is influenced by the choice of prior distribution. The methods are applied to post-traumatic stress disorder symptoms that were measured in four groups. We end by summarizing our major contribution, that is proposing two novel methods for comparing GGMs, which extends beyond the social-behavioral sciences. The methods have been implemented in the R package BGGM.