Bayesian nonparametric mixing distribution estimation in the Gaussian-smoothed 1-Wasserstein distance

We study the problem of mixing distribution estimation for mixtures of discrete exponential family models, taking a Bayesian nonparametric approach. It has been recently shown that, under the Gaussian-smoothed optimal transport (GOT) distance, that is, the 1-Wasserstein distance between the Gaussian-convolved distributions, the accuracy of the nonparametric maximum likelihood estimator is improved to a nearly parametric rate from the sub-polynomial (logarithmic) rate relative to the standard 1-Wasserstein distance. We provide sufficient conditions under which the Bayes' estimator for the true mixing distribution also converges at a nearly parametric rate in the GOT distance, where $n^{-1/2}$ is shown to be a lower bound on the minimax GOT risk.