Strong posterior contraction rates via Wasserstein dynamics

TL;DR

This paper introduces a novel approach to Bayesian posterior contraction rates using Wasserstein dynamics, linking statistical convergence with classical analysis and probability problems, and applies it to various models.

Contribution

It develops a new method combining Wasserstein distances and Lipschitz continuity to analyze PCRs in infinite-dimensional Bayesian models, with broad applications.

Findings

Optimal PCRs in finite-dimensional models

Explicit influence of priors on infinite-dimensional PCRs

Novel techniques for Laplace integrals and Poincaré-Wirtinger constants

Abstract

In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov's large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincar\'e-Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincar\'e-Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to 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 approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs.