HEAVY-TAILED BAYESIAN NONPARAMETRIC ADAPTATION

• 成果类型: Article
• 署名作者: Agapiou, Sergios; Castillo, Ismael
• 署名单位: University of Cyprus; Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2397
• 发表日期: 2024
• 页码: 1433-1459
• 关键词: von mises theorems; Inverse problems; Adaptive estimation; convergence-rates; contraction; inference; bounds; priors

摘要

We propose a new Bayesian strategy for adaptation to smoothness in nonparametric models based on heavy-tailed series priors. We illustrate it in a variety of settings, showing in particular that the corresponding Bayesian posterior distributions achieve adaptive rates of contraction in the minimax sense (up to logarithmic factors) without the need to sample hyperparameters. Unlike many existing procedures, where a form of direct model (or estimator) selection is performed, the method can be seen as performing a soft selection through the prior tail. In Gaussian regression, such heavy-tailed priors are shown to lead to (near-)optimal simultaneous adaptation both in the L-2- and L-infinity-sense. Results are also derived for linear inverse problems, for anisotropic Besov classes, and for certain losses in more general models through the use of tempered posterior distributions. We present numerical simulations corroborating the theory.