A Theoretical Framework for Bayesian Nonparametric Regression: Orthonormal Random Series and Rates of Contraction
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We develop a unifying framework for Bayesian nonparametric regression to study the rates of contraction with respect to the integrated L_2-distance without assuming the regression function space to be uniformly bounded. The framework is built upon orthonormal random series in a flexible manner. A general theorem for deriving rates of contraction for Bayesian nonparametric regression is provided under the proposed framework. As specific applications, we obtain the near-parametric rate of contraction for the squared-exponential Gaussian process when the true function is analytic, the adaptive rates of contraction for the sieve prior, and the adaptive-and-exact rates of contraction for the un-modified block prior when the true function is α-smooth. Extensions to wavelet series priors and fixed-design regression problems are also discussed.
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