Bayesian Sparse Linear Regression with Unknown Symmetric Error

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.

35 pages

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Cited by in corpus (5)

• Variational Bayes for high-dimensional linear regression with sparse priors
• Concentration of posterior probabilities and normalized L0 criteria
• Adaptive variational Bayes: Optimality, computation and applications
• Bayesian High-dimensional Semi-parametric Inference beyond sub-Gaussian Errors
• Unified Bayesian theory of sparse linear regression with nuisance parameters