Model averaging in the space of probability distributions

In the modern era, the complexity and density of data structures have significantly increased, particularly with the advent of technologies such as cloud computing, sensor networks and manifold-based data representations. A notable case within this landscape is the class of measure-valued data, which encompasses data best represented through probability distributions rather than individual observations (Ranjan and Gneiting, 2010; Gneiting and Ranjan, 2013). This framework is prevalent across various fields, including actuarial science, economics and finance, environmental sciences, etc where uncertainty and heterogeneity are inherent and models must reflect the full distributional information. For instance, in economics integrating diverse models allows for the generation of numerous meaningfull probabilistic scenarios that can effectively inform future decision-making (Moral-Benito, 2015; Hong and Martin, 2017; Christensen et al., 2018; Steel, 2020; Koundouri et al., 2024). In environmental sciences, the prediction of future states through stochastic simulation models is crucial for evaluating the consequences of natural hazards (Muis et al., 2015; Hsiang et al., 2017; Fronzek et al., 2022) or improving climatic forecasts (Friederichs and Thorarinsdottir, 2012; Scheuerer and M oller, 2015; Papayiannis et al., 2018).