Adaptive Bayesian inference on the mean of an infinite-dimensional normal distribution

• 成果类型: Article
• 署名作者: Belitser, E; Ghosal, S
• 署名单位: Utrecht University; North Carolina State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2003
• 页码: 536-559
• 关键词: Nonparametric regression; posterior distributions; asymptotic equivalence; white-noise; CONVERGENCE; rates

摘要

We consider the problem of estimating the mean of an infinite-dimensional normal distribution from the Bayesian perspective. Under the assumption that the unknown true mean satisfies a smoothness condition, we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators. Although the posterior distribution can achieve the optimal rate of convergence, the required prior depends on a smoothness parameter q. When this parameter q is unknown, besides the estimation of the mean, we encounter the problem of selecting a model. In a Bayesian approach, this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model. We show that if q takes values only in a discrete set, the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model. A slightly weaker result is presented when q is unrestricted. An adaptive point estimator based on the posterior distribution is also constructed.