Centered plug-in estimation of Wasserstein distances

TL;DR

This paper introduces centered plug-in estimators for Wasserstein distances that reduce bias, providing reliable bounds and applications in Bayesian inference and MCMC convergence assessment.

Contribution

The authors propose a simple centering technique to create unbiased, informative estimators of Wasserstein distances with practical applications in Bayesian computation.

Findings

Centered estimators decrease with true Wasserstein distance

Estimators serve as complementary bounds on the Wasserstein distance

Applications include bias estimation and convergence diagnostics in Bayesian methods

Abstract

The plug-in estimator of the squared Euclidean 2-Wasserstein distance is conservative, however due to its large positive bias it is often uninformative. We eliminate most of this bias using a simple centering procedure based on linear combinations. We construct a pair of centered plug-in estimators that decrease with the true Wasserstein distance, and are therefore guaranteed to be informative, for any finite sample size. Crucially, we demonstrate that these estimators can often be viewed as complementary upper and lower bounds on the squared Wasserstein distance. Finally, we apply the estimators to Bayesian computation, developing methods for estimating (i) the bias of approximate inference methods and (ii) the convergence of MCMC algorithms.