  This paper presents a new approach to the classical problem of quantifying
posterior contraction rates (PCRs) in Bayesian statistics. Our approach relies
on Wasserstein distance, and it leads to two main contributions which improve
on the existing literature of PCRs. The first contribution exploits the dynamic
formulation of Wasserstein distance, for short referred to as Wasserstein
dynamics, in order to establish PCRs under dominated Bayesian statistical
models. As a novelty with respect to existing approaches to PCRs, Wasserstein
dynamics allows us to circumvent the use of sieves in both stating and proving
PCRs, and it sets forth a natural connection between PCRs and three well-known
classical problems in statistics and probability theory: the speed of mean
Glivenko-Cantelli convergence, the estimation of weighted Poincar\'e-Wirtinger
constants and Sanov large deviation principle for Wasserstein distance. The
second contribution combines the use of Wasserstein distance with a suitable
sieve construction to establish PCRs under full Bayesian nonparametric models.
As a novelty with respect to existing literature of PCRs, our second result
provides with the first treatment of PCRs under non-dominated Bayesian models.
Applications of our results are presented for some classical Bayesian
statistical models, e.g., regular parametric models, infinite-dimensional
exponential families, linear regression in infinite dimension and nonparametric
models under Dirichlet process priors.
