An Analysis of Bayesian Inference for Non-Parametric Regression.

The observation model Y sub i Betain epsilon sub i, 1 or n, is considered, where the epsilons are i.i.d. mean zero and variance sigma-sq 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 - beta-average is analyzed, where beta-average is the posterior expectation of beta. Asymptotic posterior and sampling distributional approximations are given for abs. val delsquare when abs. valsquare 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 epsilon 0 as n approaches infinity with prior probability 1. A related continuous time signal estimation problem is also studied. Keywords Bayesian inference Nonparametric regression Confidence regions Signal extraction Smoothing splices.