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An analysis of Bayesian inference for nonparametric regression. (English) Zbl 0778.62003
Summary: The observation model $y\sb i=\beta(i/n)+\varepsilon\sb i$, $1\le i\le n$, is considered, where the $\varepsilon$’s are i.i.d. with mean zero and variance $\sigma\sp 2$ and $\beta$ is an unknown smooth function. A Gaussian prior distribution is specified by assuming $\beta$ is the solution of a high order stochastic differential equation. The estimation error $\delta=\beta-\hat\beta$ is analyzed, where $\hat\beta$ is the posterior expectation of $\beta$. Asymptotic posterior and sampling distributional approximations are given for $\Vert\delta\Vert\sp 2$ when $\Vert\cdot\Vert$ is one of a family of norms natural to the problem. It is shown that the frequentist coverage probability of a variety of $(1-\alpha)$ posterior probability regions tends to be larger than $1- \alpha$, but will be infinitely often less than any $\varepsilon>0$ as $n\to\infty$ with prior probability 1. A related continuous time signal estimation problem is also studied.

62A01Foundations and philosophical topics in statistics
62G07Density estimation
62G15Nonparametric tolerance and confidence regions
62E20Asymptotic distribution theory in statistics
62M99Inference from stochastic processes
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