Wasserstein posterior contraction rates in non-dominated Bayesian nonparametric models

Posterior contractions rates (PCRs) strengthen the notion of Bayesian consistency, quantifying the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of the “true model”, with probability tending to 1 or almost surely, as the sample size goes to infinity. In this paper, we study PCRs in non-dominated Bayesian nonparametric models for the observations, thus assuming that posterior distributions are available through a more general disintegration argument than Bayes formula. By means of a suitable sieve construction on the space of observations, in combination with an assumption of Lipschitz-continuity for the posterior distribution, our main result provides Wasserstein PCRs in non-dominated Bayesian nonparametric models. Besides the Lipschitz-continuity, our result relies on minimal modeling assumptions, and it is stated in a general form that allows for any nonparametric prior distributions. Refinements of our result are presented under additional assumptions on the prior distribution and the “true model”, which lead to PCRs that, up to a constant, are of the form n−1/(d+2), with n being the sample size and d being the dimension of the observations. Such a PCR is slightly slower than the optimal minimax rate for the estimation of smooth densities in Wasserstein distance, which, however, is obtained for dominated models for the observations, and under smoothness conditions for the (density of the) model. To date, we are not aware of any classical (frequentist) study on Wasserstein rates of consistency in the context of non-dominated nonparametric models, which would provide a fair term of comparison for our PCRs. Applications of our results are given with respect to the Dirichlet process prior, which is a conjugate prior, and the normalized extended Gamma process prior, which is a non-conjugate prior.