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.
Funding Statement
This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 834175).
Acknowledgments
We are very thankful to Marc Hoffmann and to Yann Chaubet for helpful discussions, as well as to Leo Zhang for spotting an initial mistake in the Gibbs sampling algorithm. We would also like to express our gratitude to the referees and the associate editor for the great care and attention shown to the manuscript, which helped us improve the results and their exposition.
Citation
Clément Berenfeld. Paul Rosa. Judith Rousseau. "Estimating a density near an unknown manifold: A Bayesian nonparametric approach." Ann. Statist. 52 (5) 2081 - 2111, October 2024. https://doi.org/10.1214/24-AOS2423
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