We develop methodology and theory for a mean field variational Bayes approximation to a linear model with a spike and slab prior on the regression coefficients. In particular we show how our method forces a subset of regression coefficients to be numerically indistinguishable from zero; under mild regularity conditions estimators based on our method consistently estimate the model parameters with easily obtainable and (asymptotically) appropriately sized standard error estimates; and select the true model at an exponential rate in the sample size. We also develop a practical method for simultaneously choosing reasonable initial parameter values and tuning the main tuning parameter of our algorithms which is both computationally efficient and empirically performs as well or better than some popular variable selection approaches. Our method is also faster and highly accurate when compared to MCMC.
John T. Ormerod. Chong You. Samuel Müller. "A variational Bayes approach to variable selection." Electron. J. Statist. 11 (2) 3549 - 3594, 2017. https://doi.org/10.1214/17-EJS1332