Adaptive Priors based on Splines with Random Knots

TL;DR

This paper extends the theoretical understanding of spline-based hierarchical priors by incorporating priors on knot locations, demonstrating conditions for adaptive posterior contraction rates in nonparametric models.

Contribution

It introduces a theoretical framework for adaptive contraction rates when priors are placed on both spline coefficients and knot locations, enhancing the spatial adaptivity of Bayesian nonparametric models.

Findings

Establishes sufficient conditions for adaptive contraction rates.

Extends previous results to include priors on knot locations.

Provides theoretical support for spatial adaptivity in spline-based priors.

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 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 certain conditions, to adaptive 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 MCMC applications, where the resulting posterior is expected to be more "spatially adaptive", 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.