Title: Posterior Asymptotic Normality for an Individual Coordinate in High-dimensional Linear Regression,
Abstract: We consider the sparse high-dimensional linear regression model
$Y=Xb+\epsilon$ where $b$ is a sparse vector. For the Bayesian approach to this
problem, many authors have considered the behavior of the posterior
distribution when, in truth, $Y=X\beta+\epsilon$ for some given $\beta$. There
have been numerous results about the rate at which the posterior distribution
concentrates around $\beta$, but few results about the shape of that posterior
distribution. We propose a prior distribution for $b$ such that the marginal
posterior distribution of an individual coordinate $b_i$ is asymptotically
normal centered around an asymptotically efficient estimator, under the truth.
Such a result gives Bayesian credible intervals that match with the confidence
intervals obtained from an asymptotically efficient estimator for $b_i$. We
also discuss ways of obtaining such asymptotically efficient estimators on
individual coordinates. We compare the two-step procedure proposed by Zhang and
Zhang (2014) and a one-step modified penalization method.