Volume 26, 2022
|Page(s)||436 - 472|
|Published online||08 December 2022|
Bayesian learning with Wasserstein barycenters*
Faculty of Mathematics, University of Vienna, Vienna, Austria
2 Center for Mathematical Modeling and Department of Mathematical Engineering, Universidad de Chile, Santiago, Chile
3 NoiseGrasp SpA, Santiago, Chile
4 Initiative for Data & Artificial Intelligence and Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile
** Corresponding author: email@example.com
Accepted: 6 November 2022
We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper, and provide a numerical example for experimental validation of the proposed method.
Mathematics Subject Classification: 62G05 / 62FL5 / 62H10 / 62G20
Key words: Bayesian learning / non-parametric estimation / Wasserstein distance and barycenter / consistency / MCMC / stochastic gradient descent in Wasserstein space
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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