Variational Bayesian inference for logistic regression models for group testing data: Comparing a new hybrid variational inference approach to traditional methods

Group testing is an efficient means of screening for infectious disease that combines specimens from multiple individuals and tests them as a single ‘pooled’ specimen. If a pool tests positive, subsequent tests are usually conducted to determine the disease status of the contributing individuals; if a pool tests negative, all samples in the pool are considered disease free. In recent years, Bayesian regression frameworks for group testing data have been developed, offering flexibility across group testing protocols and enabling the estimation of unknown assay accuracies. However, existing Bayesian methods rely heavily on Markov chain Monte Carlo (MCMC) sampling for estimation and inference, which becomes increasingly computationally inefficient as sample sizes grow, creating a bottleneck that limits the applicability of these models to large datasets. Variational Bayes (VB), an optimization-based algorithm that provides an analytical approximation to the posterior distribution, offers a more computationally efficient alternative to MCMC. To overcome the computational challenges of current Bayesian group testing methods, we develop a VB algorithm for group testing regression. Because VB methods are known to underestimate the posterior variance, we also propose an alternative inference method that exploits the asymptotic normality of VB estimators. We demonstrate with numerical studies that this hybrid inference approach achieves comparable inference to the MCMC approach while being more computationally scalable. We then apply the proposed VB method to COVID-19 group testing data from 43,185 individuals in the National Institutes of Health’s All of Us Research Program.