The Bayesian approach to variable selection in regression is a powerful tool for tackling many scientific problems. Inference for variable selection models is usually implemented using Markov chain Monte Carlo (MCMC). Because MCMC can impose a high computational cost in studies with a large number of variables, we assess an alternative to MCMC based on a simple variational approximation. Our aim is to retain useful features of Bayesian variable selection at a reduced cost. Using simulations designed to mimic genetic association studies, we show that this simple variational approximation yields posterior inferences in some settings that closely match exact values. In less restrictive (and more realistic) conditions, we show that posterior probabilities of inclusion for individual variables are often incorrect, but variational estimates of other useful quantities|including posterior distributions of the hyperparameters|are remarkably accurate. We illustrate how these results guide the use of variational inference for a genome-wide association study with thousands of samples and hundreds of thousands of variables.
"Scalable Variational Inference for Bayesian Variable Selection in Regression, and Its Accuracy in Genetic Association Studies." Bayesian Anal. 7 (1) 73 - 108, March 2012. https://doi.org/10.1214/12-BA703