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$\pi ^{\nu}$, 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 ≢ 1/2, then frequentist consistency obtains: that is, for any ν with infinite support, the posterior of$\pi ^{\nu}$concentrates on any L¹ neighborhood of f₀. Solution of an associated large-deviations problem is central to the consistency proof.