Abstract
We study Bayes procedures for nonparametric regression problems with Gaussian errors, giving conditions under which a Bernstein-von Mises result holds for the marginal posterior distribution of the error standard deviation. We apply our general results to show that a single Bayes procedure using a hierarchical spline-based prior on the regression function and an independent prior on the error variance, can simultaneously achieve adaptive, rate-optimal estimation of a smooth, multivariate regression function and efficient, n−√-consistent estimation of the error standard deviation.
Highlights
We study Bayes procedures for nonparametric regression problems with Gaussian errors, giving conditions under which a Bernstein–von Mises result holds for the marginal posterior distribution of the error standard deviation
We apply our general results to show that a single Bayes procedure using a hierarchical spline-based prior on the regression function and an independent prior on the error variance, can simultaneously achieve adaptive, rate-optimal estimation of a smooth, multivariate regression function and efficient, √n-consistent estimation of the error standard deviation
In this paper we study the asymptotic behavior of the marginal posterior for the error standard deviation in a nonparametric, fixed design regression model with Gaussian errors
Summary
In this paper we study the asymptotic behavior of the marginal posterior for the error standard deviation in a nonparametric, fixed design regression model with Gaussian errors. We give details for two specific popular families of Gaussian priors: multiply integrated Brownian motions and the class of Matern processes In both cases we find that BvM holds if the prior is rough enough relative to the degree of smoothness of the true regression function f0. In De Jonge and Van Zanten (2012) it was shown that when properly constructed, such a prior yields adaptive, nearly rate-optimal estimation of a smooth regression function f. We investigate this prior in this paper because we are interested in the question whether or not we can have adaptive estimation of f and BvM for σ at the same time. This result is essentially known, but a proof has never been published
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