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We study the Bayesian density estimation of data living in the offset of anunknown submanifold of the Euclidean space. In this perspective, we introduce anew notion of anisotropic H\"older for the underlying density and obtainposterior rates that are minimax optimal and adaptive to the regularity of thedensity, to the intrinsic dimension of the manifold, and to the size of theoffset, provided that the latter is not too small -- while still allowed to goto zero. Our Bayesian procedure, based on location-scale mixtures of Gaussians,appears to be convenient to implement and yields good practical results, evenfor quite singular data.

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