Bayesian Sparse Linear Regression with Unknown Symmetric Error

TL;DR

This paper develops a Bayesian approach for sparse linear regression with unknown symmetric error distributions, providing theoretical guarantees for posterior consistency, convergence rates, and model selection.

Contribution

It introduces a Bayesian method using a symmetrized Dirichlet process mixture for unknown errors and establishes theoretical properties including consistency and convergence rates.

Findings

Posterior consistency in mean Hellinger distance.

Minimax convergence rates for regression coefficients.

Strong model selection consistency and Bernstein-von Mises theorem.

Abstract

We study full Bayesian procedures for sparse linear regression when errors have a symmetric but otherwise unknown distribution. The unknown error distribution is endowed with a symmetrized Dirichlet process mixture of Gaussians. For the prior on regression coefficients, a mixture of point masses at zero and continuous distributions is considered. We study behavior of the posterior with diverging number of predictors. Conditions are provided for consistency in the mean Hellinger distance. The compatibility and restricted eigenvalue conditions yield the minimax convergence rate of the regression coefficients in $\ell_1$- and $\ell_2$-norms, respectively. The convergence rate is adaptive to both the unknown sparsity level and the unknown symmetric error density under compatibility conditions. In addition, strong model selection consistency and a semi-parametric Bernstein-von Mises theorem are proven under slightly stronger conditions.