Rates of contraction with respect to $L_{2}$-distance for Bayesian nonparametric regression

Abstract

We systematically study the rates of contraction with respect to the integrated $L_{2}$-distance for Bayesian nonparametric regression in a generic framework, and, notably, without assuming the regression function space to be uniformly bounded. The generic framework is very flexible and can be applied to a wide class of nonparametric prior models. Three non-trivial applications of the framework are provided: The finite random series regression of an $\alpha$-Hölder function, with adaptive rates of contraction up to a logarithmic factor; The un-modified block prior regression of an $\alpha$-Sobolev function, with adaptive-and-exact rates of contraction; The Gaussian spline regression of an $\alpha$-Hölder function, with near optimal rates of contraction. These applications serve as generalization or complement of their respective results in the literature. Extensions to the fixed-design regression problem and sparse additive models in high dimensions are discussed as well.