Strong posterior contraction rates via Wasserstein dynamics

03/21/2022
by   Emanuele Dolera, et al.
0

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: the Laplace method for approximating integrals, Sanov's large deviation principles in the Wasserstein distance, rates of convergence of the mean Glivenko-Cantelli theorem, and estimates of weighted Poincaré-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 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. 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 approaches in the classical (frequentist) and Bayesian literature deal with spaces of density functions endowed with L^p norms or the Hellinger distance.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

11/29/2020

A new approach to posterior contraction rates via Wasserstein dynamics

This paper presents a new approach to the classical problem of quantifyi...
01/28/2022

Wasserstein posterior contraction rates in non-dominated Bayesian nonparametric models

Posterior contractions rates (PCRs) strengthen the notion of Bayesian co...
12/02/2010

Conjugate Projective Limits

We characterize conjugate nonparametric Bayesian models as projective li...
12/08/2017

Posterior distribution existence and error control in Banach spaces

We generalize the results of Christen2017 on expected Bayes factors (BF)...
09/23/2019

On uniform continuity of posterior distributions

In the setting of dominated statistical models, we provide conditions yi...
08/02/2019

Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space

This work establishes fast rates of convergence for empirical barycenter...
02/24/2021

Semiparametric counterfactual density estimation

Causal effects are often characterized with averages, which can give an ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.