Spatially Adaptive Bayesian Regression Splines

In this paper we study penalized regression splines (P-splines), which are low–order basis function splines with a penalty to avoid undersmoothing. Such P–splines are typically not spatially adaptive, and hence can have trouble when functions are varying rapidly. While frequentist methods are available to address this issue, no Bayesian techniques have been developed. Our approach is to model the penalty parameter inherent in the P–spline method as a heteroscedastic regression. We develop a full Bayesian hierarchical structure to do this. The method is extended to additive models with simultaneous spline based penalty functions for the unknown functions. In simulations, the method performs comparably to the current best frequentist P–spline method in terms of frequentist mean squared error, and better than some of the other Bayesian methods.