Posterior Inference for Quantile Regression: Adaptation to Sparsity

TL;DR

This paper introduces a Bayesian approach for quantile regression that adapts to sparsity and heterogeneity, providing valid and efficient inference without explicit variable selection.

Contribution

It develops a Bayesian framework coupling asymmetric Laplace likelihood with shrinkage priors for adaptive, stable, and efficient quantile regression inference.

Findings

Achieves asymptotic validity after variance adjustment.

Attains oracle asymptotic efficiency for active coefficients.

Demonstrates super-efficiency for non-active coefficients.

Abstract

Quantile regression is a powerful data analysis tool that accommodates heterogeneous covariate-response relationships. We find that by coupling the asymmetric Laplace working likelihood with appropriate shrinkage priors, we can deliver posterior inference that automatically adapts to possible sparsity in quantile regression analysis. After a suitable adjustment on the posterior variance, the posterior inference provides asymptotically valid inference under heterogeneity. Furthermore, the proposed approach leads to oracle asymptotic efficiency for the active (nonzero) quantile regression coefficients and super-efficiency for the non-active ones. By avoiding the need to pursue dichotomous variable selection, the Bayesian computational framework demonstrates desirable inference stability with respect to tuning parameter selection. Our work helps to uncloak the value of Bayesian computational methods in frequentist inference for quantile regression.