ROBUST BAYES-LIKE ESTIMATION: RHO-BAYES ESTIMATION

• 成果类型: Article
• 署名作者: Baraud, Yannick; Birge, Lucien
• 署名单位: University of Luxembourg; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1948
• 发表日期: 2020
• 页码: 3699-3720
• 关键词: model selection; posterior; CONVERGENCE

摘要

We observe n independent random variables with joint distribution P and pretend that they are i.i.d. with some common density s (with respect to a known measure mu) that we wish to estimate. We consider a density model (S) over bar for s that we endow with a prior distribution pi (with support in (S) over bar) and build a robust alternative to the classical Bayes posterior distribution which possesses similar concentration properties around s whenever the data are truly i.i.d. and their density s belongs to the model (S) over bar. Furthermore, in this case, the Hellinger distance between the classical and the robust posterior distributions tends to 0, as the number of observations tends to infinity, under suitable assumptions on the model and the prior. However, unlike what happens with the classical Bayes posterior distribution, we show that the concentration properties of this new posterior distribution are still preserved when the model is misspecified or when the data are not i.i.d. but the marginal densities of their joint distribution are close enough in Hellinger distance to the model (S) over bar.