تقرير
Wasserstein convergence in Bayesian and frequentist deconvolution models
| العنوان: | Wasserstein convergence in Bayesian and frequentist deconvolution models |
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| المؤلفون: | Rousseau, Judith, Scricciolo, Catia |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics Statistics |
| مصطلحات موضوعية: | Mathematics - Statistics Theory |
| الوصف: | We study the multivariate deconvolution problem of recovering the distribution of a signal from independent and identically distributed observations additively contaminated with random errors (noise) from a known distribution. For errors with independent coordinates having ordinary smooth densities, we derive an inversion inequality relating the $L^1$-Wasserstein distance between two distributions of the signal to the $L^1$-distance between the corresponding mixture densities of the observations. This smoothing inequality outperforms existing inversion inequalities. As an application of the inversion inequality to the Bayesian framework, we consider $1$-Wasserstein deconvolution with Laplace noise in dimension one using a Dirichlet process mixture of normal densities as a prior measure on the mixing distribution (or distribution of the signal). We construct an adaptive approximation of the sampling density by convolving the Laplace density with a well-chosen mixture of normal densities and show that the posterior measure concentrates around the sampling density at a nearly minimax rate, up to a log-factor, in the $L^1$-distance. The same posterior law is also shown to automatically adapt to the unknown Sobolev regularity of the mixing density, thus leading to a new Bayesian adaptive estimation procedure for mixing distributions with regular densities under the $L^1$-Wasserstein metric. We illustrate utility of the inversion inequality also in a frequentist setting by showing that an appropriate isotone approximation of the classical kernel deconvolution estimator attains the minimax rate of convergence for $1$-Wasserstein deconvolution in any dimension $d\geq 1$, when only a tail condition is required on the latent mixing density and we derive sharp lower bounds for these problems Comment: arXiv admin note: text overlap with arXiv:2111.06846 |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.48550/arXiv.2111.06846 |
| URL الوصول: | http://arxiv.org/abs/2309.15300 |
| رقم الأكسشن: | edsarx.2309.15300 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.48550/arXiv.2111.06846 |
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