  Using an asymmetric Laplace distribution, which provides a mechanism for
Bayesian inference of quantile regression models, we develop a fully Bayesian
approach to fitting single-index models in conditional quantile regression. In
this work, we use a Gaussian process prior for the unknown nonparametric link
function and a Laplace distribution on the index vector, with the latter
motivated by the recent popularity of the Bayesian lasso idea. We design a
Markov chain Monte Carlo algorithm for posterior inference. Careful
consideration of the singularity of the kernel matrix, and tractability of some
of the full conditional distributions leads to a partially collapsed approach
where the nonparametric link function is integrated out in some of the sampling
steps. Our simulations demonstrate the superior performance of the Bayesian
method versus the frequentist approach. The method is further illustrated by an
application to the hurricane data.
