An Analysis of Bayesian Inference for Nonparametric Regression

The observation model yᵢ = (i/n) + ᵢ, 1 i n, is considered, where the 's are i. i. d. with mean zero and variance ² and is an unknown smooth function. A Gaussian prior distribution is specified by assuming is the solution of a high order stochastic differential equation. The estimation error = - is analyzed, where is the posterior expectation of. Asymptotic posterior and sampling distributional approximations are given for \|\|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 -) posterior probability regions tends to be larger than 1 -, but will be infinitely often less than any > 0 as n with prior probability 1. A related continuous time signal estimation problem is also studied.