Bayesian Multivariate Quantile Regression Using Dependent Dirichlet Process Prior

TL;DR

This paper introduces a flexible Bayesian multivariate quantile regression method using a Dependent Dirichlet Process prior, enabling dependence modeling across covariates and providing theoretical guarantees.

Contribution

It develops a novel non-parametric Bayesian approach with a DDP prior for multivariate quantile regression, including posterior computation and consistency analysis.

Findings

Method performs well in simulations.

Real data applications demonstrate practical utility.

Posterior consistency is theoretically established.

Abstract

In this article, we consider a non-parametric Bayesian approach to multivariate quantile regression. The collection of related conditional distributions of a response vector Y given a univariate covariate X is modeled using a Dependent Dirichlet Process (DDP) prior. The DDP is used to introduce dependence across x. As the realizations from a Dirichlet process prior are almost surely discrete, we need to convolve it with a kernel. To model the error distribution as flexibly as possible, we use a countable mixture of multidimensional normal distributions as our kernel. For posterior computations, we use a truncated stick-breaking representation of the DDP. This approximation enables us to deal with only a finitely number of parameters. We use a Block Gibbs sampler for estimating the model parameters. We illustrate our method with simulation studies and real data applications. Finally, we provide a theoretical justification for the proposed method through posterior consistency. Our proposed procedure is new even when the response is univariate.