Wasserstein convergence in Bayesian deconvolution models

TL;DR

This paper investigates Bayesian nonparametric methods for deconvolution, establishing Wasserstein convergence rates and adaptive estimation of the latent distribution under known noise, with improved inequalities and practical priors.

Contribution

It develops new inversion inequalities relating mixture densities to signal distributions, and demonstrates adaptive Bayesian estimation with minimax optimal rates under Wasserstein metrics.

Findings

Wasserstein convergence rates are established for Bayesian deconvolution.

New smoothing inequalities improve existing bounds in the literature.

The proposed Bayesian method adapts to the regularity of the true distribution.

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 bayes 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 \textit{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.