Nearly optimal Bayesian Shrinkage for High Dimensional Regression
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During the past decade, shrinkage priors have received much attention in Bayesian analysis of high-dimensional data. In this paper, we study the problem for high-dimensional linear regression models. We show that if the shrinkage prior has a heavy and flat tail, and allocates a sufficiently large probability mass in a very small neighborhood of zero, then its posterior properties are as good as those of the spike-and-slab prior. While enjoying its efficiency in Bayesian computation, the shrinkage prior can lead to a nearly optimal contraction rate and selection consistency as the spike-and-slab prior. Our numerical results show that under posterior consistency, Bayesian methods can yield much better results in variable selection than the regularization methods, such as Lasso and SCAD. We also establish a Bernstein von-Mises type results comparable to Castillo et al (2015), this result leads to a convenient way to quantify uncertainties of the regression coefficient estimates, which has been beyond the ability of regularization methods.
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