Adaptive Empirical Bayesian Smoothing Splines

In this paper we develop and study adaptive empirical Bayesian smoothing splines. These are smoothing splines with both smoothing parameter and penalty order determined via the empirical Bayes method from the marginal likelihood of the model. The selected order and smoothing parameter are used to construct adaptive credible sets with good frequentist coverage for the underlying regression function. We use these credible sets as a proxy to show the superior performance of adaptive empirical Bayesian smoothing splines compared to frequentist smoothing splines.