Strong posterior contraction rates via Wasserstein dynamics

In this paper, we develop a novel approach to posterior contractions rates (PCRs), for both finite-dimensional (parametric) and infinite-dimensional (nonparametric) Bayesian models. Critical to our approach is the combination of an assumption of local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, here referred to as Wasserstein dynamics, which allows to set forth a connection between the problem of establishing PCRs and some classical problems in mathematical analysis, probability theory and mathematical statistics: Laplace methods for approximating integrals, Sanov's large deviation principle under the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincar\'e-Wirtinger constants. Under dominated Bayesian models, we present two main results: i) a theorem on PCRs for the regular infinite-dimensional exponential family of statistical models; ii) a theorem on PCRs for a general dominated statistical models. Some applications of our results are presented for the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. It turns out that our approach leads to optimal PCRs in finite dimension, whereas in infinite dimension it is shown explicitly how prior distributions affect the corresponding PCRs. In general, with regards to infinite-dimensional Bayesian models for density estimation, our approach to PCRs is the first to consider strong norm distances on parameter spaces of functions, such as Sobolev-like norms, as most of the literature deals with spaces of density functions endowed with $\mathrm{L}^p$ norms or the Hellinger distance.