m at h . ST ] 1 4 O ct 2 00 5 CONSISTENCY OF BAYES ESTIMATORS OF A BINARY REGRESSION FUNCTION

When do nonparametric Bayesian procedures “overfit?” To shed light on this question, we consider a binary regression problem in detail and establish frequentist consistency for a certain class of Bayes procedures based on hierarchical priors, called uniform mixture priors. These are defined as follows: let ν be any probability distribution on the nonnegative integers. To sample a function f from the prior π , first sample m from ν and then sample f uniformly from the set of step functions from [0, 1] into [0, 1] that have exactly m jumps (i.e. sample all m jump locations and m + 1 function values independently and uniformly). The main result states that if a data-stream is generated according to any fixed, measurable binary-regression function f0 6≡ 1/2 then frequentist consistency obtains: i.e. for any ν with infinite support, the posterior of π concentrates on any L neighborhood of f0. Solution of an associated largedeviations problem is central to the consistency proof.