Estimating a density near an unknown manifold: a Bayesian nonparametric approach

Abstract

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 Hölder 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.

73 pages, 41 figures. [v2] Major structural changes and corrections of minor errors and typos. [v3] Major changes in the statement and proof of Thm 3.1 which now includes the isotropic case. Thanks to the work of three anonymous referees, numerous remarks have been added throughout the manuscript, which was accepted for publication in Annals of Statistics