Using an asymmetric Laplace distribution, which provides a mechanism forBayesian inference of quantile regression models, we develop a fully Bayesianapproach to fitting single-index models in conditional quantile regression. Inthis work, we use a Gaussian process prior for the unknown nonparametric linkfunction and a Laplace distribution on the index vector, with the lattermotivated by the recent popularity of the Bayesian lasso idea. We design aMarkov chain Monte Carlo algorithm for posterior inference. Carefulconsideration of the singularity of the kernel matrix, and tractability of someof the full conditional distributions leads to a partially collapsed approachwhere the nonparametric link function is integrated out in some of the samplingsteps. Our simulations demonstrate the superior performance of the Bayesianmethod versus the frequentist approach. The method is further illustrated by anapplication to the hurricane data.
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