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.
翻译:我们引入并研究一种基于最佳交通的新的贝叶斯学习模式选择战略及其相关的预测后继法:后继法的瓦森斯坦人口中间点。我们首先展示了这个叫巴伊西亚瓦西斯坦温中间点(BWB)的顶点是如何在一般的无参数巴伊西亚模式选择框架中自然产生的,当时考虑的巴伊西亚风险是瓦瑟斯坦距离。我们列举了一些实例,说明BWB如何扩展一些典型的参数和非参数选择战略。此外,我们还提供了明确的条件,允许BWB的存在和统计一致性,并讨论其一些一般和具体特性,提供了与通常选择相比,如模型平均估计者(BWBB)的优势。最后,我们展示了如何用瓦瑟斯坦空间的随机梯度梯系的算法(SGD)进行计算,该算法是在一份配套文件ArXiv:2204.232v2[math.OC]中引入的,并提供了一个用于实验性验证拟议方法的数字示例。