  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.
