Nonparametric Binary Regression: A Bayesian Approach

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 ∑ₖwₖπₖ/∑ₖwₖ, where πₖ is uniform on the set of f which only depend on the first k covariates and wₖ > 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 ≡ 1/2.