  In this paper, we consider Bayesian methods for non-parametric quantile
regressions with multiple continuous predictors ranging values in the unit
interval. In the first method, the quantile function is assumed to be smooth
over the explanatory variable and is expanded in tensor product of B-spline
basis functions. While in the second method, the distribution function is
assumed to be smooth over the explanatory variable and is expanded in tensor
product of B-spline basis functions. Unlike other existing methods of
non-parametric quantile regressions, the proposed methods estimate the whole
quantile function instead of estimating on a grid of quantiles. Priors on the
B-spline coefficients are put in such a way that the monotonicity of the
estimated quantile levels are maintained unlike local polynomial quantile
regression methods. The proposed methods have also been modified for quantile
grid data where only the percentile range of each response observations are
known. Simulations studies have been provided for both complete and quantile
grid data. The proposed method has been used to estimate the quantiles of US
household income data and North Atlantic hurricane intensity data.
