Estimating a density near an unknown manifold: a Bayesian nonparametric approach
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arXiv
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| Format: | Preprint |
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2022
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| author | Berenfeld, Clément Rosa, Paul Rousseau, Judith |
| author_facet | Berenfeld, Clément Rosa, Paul Rousseau, Judith |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_15717 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Estimating a density near an unknown manifold: a Bayesian nonparametric approach Berenfeld, Clément Rosa, Paul Rousseau, Judith Statistics Theory 62G07, 62G20, 62R30 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. |
| title | Estimating a density near an unknown manifold: a Bayesian nonparametric approach |
| topic | Statistics Theory 62G07, 62G20, 62R30 |
| url | https://arxiv.org/abs/2205.15717 |