Bayes and maximum likelihood for $L^1$-Wasserstein deconvolution of Laplace mixtures

TL;DR

This paper investigates nonparametric deconvolution of Laplace mixtures, establishing convergence rates for Bayesian and MLE estimators in various metrics, and linking density estimation to Wasserstein distances for mixing distributions.

Contribution

It introduces new convergence rate results for Bayesian and MLE estimators in Laplace deconvolution, using an inversion inequality to connect density and Wasserstein metrics.

Findings

Bayes' density estimator achieves near-optimal convergence rates.

Rates of convergence in Wasserstein distance are derived for mixing distributions.

The discrepancy between Bayesian and MLE procedures is quantitatively assessed.

Abstract

We consider the problem of recovering a distribution function on the real line from observations additively contaminated with errors following the standard Laplace distribution. Assuming that the latent distribution is completely unknown leads to a nonparametric deconvolution problem. We begin by studying the rates of convergence relative to the $L^2$-norm and the Hellinger metric for the direct problem of estimating the sampling density, which is a mixture of Laplace densities with a possibly unbounded set of locations: the rate of convergence for the Bayes' density estimator corresponding to a Dirichlet process prior over the space of all mixing distributions on the real line matches, up to a logarithmic factor, with the $n^{-3/8}\log^{1/8}n$ rate for the maximum likelihood estimator. Then, appealing to an inversion inequality translating the $L^2$-norm and the Hellinger distance between general kernel mixtures, with a kernel density having polynomially decaying Fourier transform, into any $L^p$-Wasserstein distance, $p\geq1$, between the corresponding mixing distributions, provided their Laplace transforms are finite in some neighborhood of zero, we derive the rates of convergence in the $L^1$-Wasserstein metric for the Bayes' and maximum likelihood estimators of the mixing distribution. Merging in the $L^1$-Wasserstein distance between Bayes and maximum likelihood follows as a by-product, along with an assessment on the stochastic order of the discrepancy between the two estimation procedures.