Consistency of Bayes estimates for nonparametric regression: normal theory

Abstract

Performance characteristics of Bayes estimates are studied. More exactly, for each subject in a data set, let ξ be a vector of binary covariates and let Y be a normal response variable, with E{Y|ξ}=f(ξ) and var{Y|ξ}=1. Here, f is an unknown function to be estimated from the data; the subjects are independent and identically distributed. Define a prior distribution on f as ∑_kw_kπ_k/∑_kw_k, where π_k is standard normal on the set of f which only depend on the first k covariates and w_k>0 for infinitely many k. Bayes estimates are consistent for all f. On the other hand, if the π_k are flat, inconsistency is the rule.