  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.
