A Bayesian approach to nonparametric bivariate regression

This article outlines a general Bayesian approach to estimating a bivariate regression function in a nonparametric manner. It models the function using a bivariate regression spline basis with many terms. Binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression. The regression function is estimated using an estimate of its posterior mean, smoothing over the distribution of these binary indicator variables. To make the computations tractable, all estimates are obtained using Markov chain Monte Carlo sampling. Extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data-driven bivariate surface estimators prominent in the literature. It is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings. The first case allows for outlying values in the dependent variable. The second case considers data collected in time order with the errors potentially autocorrelated. Simulated and real data examples illustrate the effectiveness of the methodology in tackling such difficult problems.