Adaptive Bayesian Nonparametric Regression via Stationary Smoothness Priors

Abstract

A procedure for Bayesian nonparametric regression is described that automatically adjusts the degree of smoothing as the curvature of the underlying function changes. Relative to previous work adopting a similar approach that either employs a single global smoothing parameter or assumes that the smoothing process follows a random walk, the model considered here permits adaptive smoothing and imposes stationarity in the autoregressive smoothing process. An efficient Markov Chain Monte Carlo (MCMC) scheme for model estimation is fully described for this stationary case, and the performance of the method is illustrated in several generated data experiments. An application is also provided, analyzing the relationship between behavioral problems in students and academic achievement. Point estimates from the nonparametric methods suggest (a) expected achievement declines monotonically with a behavioral problems index (BPI) score and (b) the rate of decline is relatively flat at the left tail of the BPI distribution and then becomes sharply more negative.