Posterior concentration rates for mixtures of normals in random design regression

Naulet, Zacharie; Rousseau, Judith

doi:10.1214/17-ejs1344

Cited by 1 publication

(1 citation statement)

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“…Nonparametric location mixtures of Gaussians are known to be flexible models for densities, and adaptive minimax rates of convergence on Hölder types spaces have been obtained using Bayesian or frequentist estimation procedures based on location mixtures of normals, see Kruijer et al (2010), Shen et al (2013), and Ghosal and Van Der Vaart (2007) for Bayesian methods and Maugis-Rabusseau and Michel (2013) for a penalized likelihood approach. However, location mixtures are not versatile enough since the covariance matrix remains fixed across the components, so we instead take advantage of the flexibility of location-scale mixtures of Gaussians, as studied in Canale and De Blasi (2017), and more particularly of its hybrid version, for which optimal posterior concentration rates have been derived in the Euclidean case (Naulet and Rousseau, 2017).…”

Section: Introduction 1motivationmentioning

confidence: 99%

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

Berenfeld¹,

Rosa²,

Rousseau³

2022

Preprint

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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.

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