  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.
