Nonparametric Binary Regression: A Bayesian Approach

Abstract

The performance of Bayes estimates are studied, under an assumption of conditional exchangeability. More exactly, for each subject in a data set, let ξ be a vector of binary covariates and let η be a binary response variable, with P{η = 1∣ ξ} = f(ξ). Here, f is an unknown function to be estimated from the data; the subjects are independent, and satisfy a natural "balance" condition. Define a prior distribution on f as ∑kw kπk/∑kw k, where πk is uniform 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 at all f if wk decreases rapidly as k increase. Otherwise, the estimates are inconsistent at $f \equiv 1/2$.