Consistency of Bayes estimators of a binary regression function

• 成果类型: Article
• 署名作者: Coram, Marc; Lalley, Steven P.
• 署名单位: University of Chicago
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000236
• 发表日期: 2006
• 页码: 1233-1269
• 关键词: shannon-mcmillan theorem; posterior distributions; ergodic theorem; limit-theorem; CONVERGENCE; rates

摘要

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 v be any probability distribution on the nonnegative integers. To sample a function f from the prior pi(v), first sample m from v 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 f(0) not equivalent to 1/2, then frequentist consistency obtains: that is, for any V with infinite support, the posterior of pi(v) concentrates on any L-1 neighborhood of fo. Solution of an associated large-deviations problem is central to the consistency proof.