Bayesian Non-parametric Quantile Process Regression and Estimation of Marginal Quantile Effects

We propose a non-parametric method to simultaneously estimate non-crossing, non-linear quantile curves. We expand the conditional distribution function of the response in $\mathcal{I}$-spline basis functions where the coefficients are further modeled as functions of the covariates using feed-forward neural networks. By leveraging the approximation power of splines and neural networks, our model can approximate any continuous quantile function. Compared to existing methods, our method estimates all rather than a finite subset of quantiles, scales well to high dimensions and accounts for estimation uncertainty. While the model is arbitrarily flexible, interpretable marginal quantile effects are estimated using accumulative local effect plots and variable importance measures. A simulation study shows that compared to existing methods, our model can better recover quantiles of the response distribution when the sample size is small, and illustrative applications to birth weight and tropical cyclone intensity are presented.