Variational Bayes for high-dimensional linear regression with sparse priors
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We study a mean-field variational Bayes (VB) approximation to Bayesian model selection priors, which include the popular spike-and-slab prior, in the sparse high-dimensional linear regression model. Under suitable conditions on the design matrix, the mean-field VB approximation is shown to converge to the sparse truth at the optimal rate for ℓ_2-recovery and to give optimal prediction of the response vector. The empirical performance of our algorithm is studied, showing that it works comparably well as other state-of-the-art Bayesian variable selection methods. We also numerically demonstrate that the widely used coordinate-ascent variational inference (CAVI) algorithm can be highly sensitive to the updating order of the parameters leading to potentially poor performance. To counteract this we propose a novel prioritized updating scheme that uses a data-driven updating order and performs better in simulations.
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