Adaptive Priors Based on Splines with Random Knots

Abstract

Splines are useful building blocks when constructing priors on nonparametric models indexed by functions. Recently it has been established in the literature that hierarchical adaptive priors based on splines with a random number of equally spaced knots and random coefficients in the B-spline basis corresponding to those knots lead, under some conditions, to optimal posterior contraction rates, over certain smoothness functional classes. In this paper we extend these results for when the location of the knots is also endowed with a prior. This has already been a common practice in Markov chain Monte Carlo applications, but a theoretical basis in terms of adaptive contraction rates was missing. Under some mild assumptions, we establish a result that provides sufficient conditions for adaptive contraction rates in a range of models, over certain functional classes of smoothness up to the order of the splines that are used. We also present some numerical results illustrating how such a prior adapts to inhomogeneous variability (smoothness) of the function in the context of nonparametric regression.