AN ANALYSIS OF BAYESIAN-INFERENCE FOR NONPARAMETRIC REGRESSION

The observation model y(i) = beta(i/n) + epsilon(i), 1 less-than-or-equal-to i less-than-or-equal-to n, is considered, where the epsilon's are i.i.d. with mean zero and variance sigma2 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 is analyzed, where beta is the posterior expectation of beta. Asymptotic posterior and sampling distributional approximations are given for \\delta\\2 when \\ . \\ 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 --> infinity with prior probability 1. A related continuous time signal estimation problem is also studied.