Posterior asymptotics in Wasserstein metrics on the real line

TL;DR

This paper investigates the asymptotic behavior of Bayesian posterior distributions using Wasserstein metrics, establishing conditions for consistency and convergence rates, with applications to density estimation and empirical simulations.

Contribution

It provides new sufficient conditions for posterior consistency and convergence rates in Wasserstein metrics, extending existing theory with sharp tail and moment conditions.

Findings

Posterior consistency under Wasserstein metrics is achieved with specific moment conditions.

Convergence rates depend on stronger tail conditions, which are shown to be sharp.

Application to density estimation demonstrates practical relevance and effectiveness.

Abstract

In this paper, we use the class of Wasserstein metrics to study asymptotic properties of posterior distributions. Our first goal is to provide sufficient conditions for posterior consistency. In addition to the well-known Schwartz's Kullback--Leibler condition on the prior, the true distribution and most probability measures in the support of the prior are required to possess moments up to an order which is determined by the order of the Wasserstein metric. We further investigate convergence rates of the posterior distributions for which we need stronger moment conditions. The required tail conditions are sharp in the sense that the posterior distribution may be inconsistent or contract slowly to the true distribution without these conditions. Our study involves techniques that build on recent advances on Wasserstein convergence of empirical measures. We apply the results to density estimation with a Dirichlet process mixture prior and conduct a simulation study for further illustration.