  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.
