We study the problem of model aggregation within the Wasserstein space for probability measures on the real line. Given a fixed finite collection of candidate probability models, we consider the associated class of Wasserstein barycenters and develop a data-driven calibration framework in which the aggregation weights are statistically learned from empirical information associated with a target distribution. From a variational perspective based on $Γ$-convergence, we establish consistency of the resulting aggregation scheme, showing that empirical minimizers converge to the minimizers of the actual problem, along with the associated barycentric estimators, under mild conditions. The performance of the proposed method is evaluated through synthetic experiments and illustrated on a real dataset from a temperature monitoring network of sensors.

翻译：我们研究实线上概率测度的Wasserstein空间中的模型聚合问题。给定一个固定的有限候选概率模型集合，我们考虑相应的Wasserstein重心类，并发展一种数据驱动的校准框架，其中聚合权重根据与目标分布相关的经验信息进行统计学习。基于Γ-收敛的变分视角，我们建立了所得聚合方案的一致性，表明在温和条件下，经验最小化器收敛到实际问题的极小化器，同时相关的重心估计量也收敛。通过合成实验评估了所提方法的性能，并在来自温度监测传感器网络的真实数据集上进行了验证。