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EnglishWe study the reknown deconvolution problem of recovering a distribution function from independent replicates (signal) additively contaminated with random errors (noise), whose distribution is known. We investigate whether a Bayesian nonparametric approach for modelling the latent distribution of the signal can yield inferences with asymptotic frequentist validity under the $L^1$-Wasserstein metric. When the error density is ordinary smooth, we develop two inversion inequalities relating either the $L^1$ or the $L^1$-Wasserstein distance between two mixture densities (of the observations) to the $L^1$-Wasserstein distance between the corresponding distributions of the signal. This smoothing inequality improves on those in the literature. We apply this general result to a Bayesian approach on a Dirichlet process mixture of normal distributions as a prior on the mixing distribution (or distribution of the signal), with a Laplace or Linnik noise. In particular, we construct an adaptive approximation of the density of the observations by the convolution of a Laplace (or Linnik) with a well chosen mixture of normal densities and show that the posterior concentrates at the minimax rate up to a logarithmic factor. The same prior law is shown to also adapt to the Sobolev regularity level of the mixing density, thus leading to a new Bayesian estimation method, relative to the Wasserstein distance, for distributions with smooth densities.
| Titolo: | Wasserstein convergence in Bayesian deconvolution models | |
| Autori: | ||
| Data di pubblicazione: | 2021 | |
| Abstract: | We study the reknown deconvolution problem of recovering a distribution function from independent replicates (signal) additively contaminated with random errors (noise), whose distribution is known. We investigate whether a Bayesian nonparametric approach for modelling the latent distribution of the signal can yield inferences with asymptotic frequentist validity under the $L^1$-Wasserstein metric. When the error density is ordinary smooth, we develop two inversion inequalities relating either the $L^1$ or the $L^1$-Wasserstein distance between two mixture densities (of the observations) to the $L^1$-Wasserstein distance between the corresponding distributions of the signal. This smoothing inequality improves on those in the literature. We apply this general result to a Bayesian approach on a Dirichlet process mixture of normal distributions as a prior on the mixing distribution (or distribution of the signal), with a Laplace or Linnik noise. In particular, we construct an adaptive approximation of the density of the observations by the convolution of a Laplace (or Linnik) with a well chosen mixture of normal densities and show that the posterior concentrates at the minimax rate up to a logarithmic factor. The same prior law is shown to also adapt to the Sobolev regularity level of the mixing density, thus leading to a new Bayesian estimation method, relative to the Wasserstein distance, for distributions with smooth densities. | |
| Handle: | http://hdl.handle.net/11562/1052925 | |
| Appare nelle tipologie: | 07.14 Rapporti di ricerca |
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| FILE ArXiv 2111.06846.pdf | ArXiv | Documento in Pre-print |  | Open Access Visualizza/Apri |
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