Inserted: 1 dec 2020
Last Updated: 3 may 2022
We develop a novel approach to posterior contractions rates (PCRs), for both finite-dimensional (parametric) and infinite-dimensional (nonparametric) Bayesian models. Our approach combines an assumption of local Lipschitz-continuity for the posterior distribution with the dynamic formulation of the Wasserstein distance, and allows to set forth a connection between the problem of establishing PCRs and other classical problems: the Laplace method for approximating integrals, Sanov's large deviation principles, rates of convergence of the mean Glivenko-Cantelli theorem, and estimates of weighted Poincaré-Wirtinger constants. We present a theorem on PCRs for the regular infinite-dimensional exponential family of statistical models, and a theorem on PCRs for a general dominated statistical model. 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. In general, our results lead to optimal PCRs in finite dimension, whereas in infinite dimension it is shown how the prior distribution may affect PCRs.