Finite sample posterior concentration in high-dimensional regression

We study the behavior of the posterior distribution in high-dimensional Bayesian Gaussian linear regression models having p ≫ n, where p is the number of predictors and n is the sample size. Our focus is on obtaining quantitative finite sample bounds ensuring sufficient posterior probability assigned in neighborhoods of the true regression coefficient vector (β0) with high probability. We assume that β0 is approximately S-sparse and also obtain universal bounds, which provide insight into the role of the prior in controlling concentration of the posterior. Based on these finite sample bounds, we examine the implied asymptotic contraction rates for several examples, showing that sparsely structured and heavy-tail shrinkage priors exhibit rapid contraction rates. We also demonstrate that a stronger result holds for the sparsity(S)-Gaussian1 prior. These types of finite sample bounds provide guidelines for designing and evaluating priors for high-dimensional problems.