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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 $\alpha$-Holder 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 $\alpha$-Holder 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.
Highlights
Consider the standard nonparametric regression problem yi = f (xi) + ei, i = 1, · · ·, n, where the set of predictors (xi)ni=1 are referred to as design and take values in [0, 1]p ⊂ Rp, ei’s are independent and identically distributed (i.i.d.) mean-zero Gaussian noise with var(ei) = σ2, and yi’s are the responses
A closely related reference is [26], where the authors discuss the rates of contraction of the rescaled-Gaussian process prior for the nonparametric random-design regression problem with respect to the integrated L1-distance, which is weaker than the integrated L2-distance
It is natural to ask the following fundamental question: for Bayesian nonparametric regression, can one systematically study rates of contraction for various priors with respect to the integrated L2-distance without assuming the uniform boundedness of the regression function space? In this paper we provide a positive answer to this question
Summary
A closely related reference is [26], where the authors discuss the rates of contraction of the rescaled-Gaussian process prior for the nonparametric random-design regression problem with respect to the integrated L1-distance, which is weaker than the integrated L2-distance. It is natural to ask the following fundamental question: for Bayesian nonparametric regression, can one systematically study rates of contraction for various priors with respect to the integrated L2-distance without assuming the uniform boundedness of the regression function space? The major contribution of this work is that we prove the existence of an ad-hoc test function that is required in the generic rates of contraction framework in [13] by leveraging Bernstein’s inequality and imposing certain structural assumption on the sieves with large prior probabilities For a metric space (F , d), for any > 0, the -covering number of (F, d), denoted by N ( , F, d), is defined to be the minimum number of -balls of the form B(f, ) := {g ∈ F : d(f, g) < } that are needed to cover F
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