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We propose a variational Bayesian (VB) procedure for high-dimensional linearmodel inferences with heavy tail shrinkage priors, such as student-t prior.Theoretically, we establish the consistency of the proposed VB method and provethat under the proper choice of prior specifications, the contraction rate ofthe VB posterior is nearly optimal. It justifies the validity of VB inferenceas an alternative of Markov Chain Monte Carlo (MCMC) sampling. Meanwhile,comparing to conventional MCMC methods, the VB procedure achieves much highercomputational efficiency, which greatly alleviates the computing burden formodern machine learning applications such as massive data analysis. Throughnumerical studies, we demonstrate that the proposed VB method leads to shortercomputing time, higher estimation accuracy, and lower variable selection errorthan competitive sparse Bayesian methods.

Nearly Optimal Variational Inference for High Dimensional Regression with Shrinkage Priors