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

We consider the problem of nonparametric mixing distribution estimation for discrete exponential family models. It has been recently shown that, under the Gaussian-smoothed optimal transport (GOT) distance, i.e., the 1-Wasserstein distance between the Gaussian convolved distributions, the accuracy of the nonparametric maximum likelihood estimator (NPMLE) is improved to a polynomial rate from the sub-polynomial rate with respect to the standard 1-Wasserstein distance. The focus of this work is on studying the problem taking a Bayesian nonparametric approach. We provide conditions under which the Bayes’ estimator of the true mixing distribution converges at least as fast as the NPMLE in the GOT distance.