ESTIMATING A DENSITY NEAR AN UNKNOWN MANIFOLD: A BAYESIAN NONPARAMETRIC APPROACH

• 成果类型: Article
• 署名作者: Berenfeld, Clement; Rosa, Paul; Rousseau, Judith
• 署名单位: University of Potsdam; University of Oxford
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2423
• 发表日期: 2024
• 页码: 2081-2111
• 关键词: CONVERGENCE-RATES; mixtures; SPACES

摘要

We study the Bayesian density estimation of data living in the offset of an unknown submanifold of the Euclidean space. In this perspective, we introduce a new notion of anisotropic H & ouml;lder for the underlying density and obtain posterior rates that are minimax optimal and adaptive to the regularity of the density, to the intrinsic dimension of the manifold, and to the size of the offset, provided that the latter is not too small-while still allowed to go to zero. Our Bayesian procedure, based on location-scale mixtures of Gaussians, appears to be convenient to implement and yields good practical results, even for quite singular data.