# Bayesian Learning with Wasserstein Barycenters

arXiv: https://arxiv.org/abs/1805.10833 · HTML: https://mlsanity.com/paper/1805.10833/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 May 2018.

## Built on by (most cited first, 12 of 12)

- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)
- [Learning to Generate Wasserstein Barycenters](https://mlsanity.com/paper/2102.12178/) (2021)
- [A novel notion of barycenter for probability distributions based on optimal weak mass transport](https://mlsanity.com/paper/2102.13380/) (2021)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [Learning where to learn: Training data distribution optimization for scientific machine learning](https://mlsanity.com/paper/2505.21626/) (2025)
- [Relax and Merge: A Simple Yet Effective Framework for Solving Fair k-Means and k-sparse Wasserstein Barycenter Problems](https://mlsanity.com/paper/2411.01115/) (2024)
- [Information-Geometric Barycenters for Bayesian Federated Learning](https://mlsanity.com/paper/2412.11646/) (2024)
- [Barycentric model aggregation in the Wasserstein space of distributions and a variational approach to consistency](https://mlsanity.com/paper/2507.11719/) (2025)
- [A Sub-4 Approximation for Fair k-Means](https://mlsanity.com/paper/2609.07974/) (2026)

## Abstract (arXiv)

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 arXiv:2201.04232v2 [math.OC], and provide a numerical example for experimental validation of the proposed method.