  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.
Under the Bayesian nonparametric framework, a common assumption in the study of
PCRs is that the model is dominated for the observations; that is, it is
assumed that the posterior can be written through the Bayes formula. In this
paper, we consider the problem of establishing PCRs in Bayesian nonparametric
models where the posterior distribution is not available through the Bayes
formula, and hence models that are non-dominated for the observations. By means
of the Wasserstein distance and a suitable sieve construction, our main result
establishes PCRs in Bayesian nonparametric models where the posterior is
available through a more general disintegration than the Bayes formula. To the
best of our knowledge, this is the first general approach to provide PCRs in
non-dominated Bayesian nonparametric models, and it relies on minimal modeling
assumptions and on a suitable continuity assumption for the posterior
distribution. Some refinements of our result are presented under additional
assumptions on the prior distribution, and applications are given with respect
to the Dirichlet process prior and the normalized extended Gamma process prior.
