Integral Mixabilty: a Tool for Efficient Online Aggregation of Functional and Probabilistic Forecasts

In this paper we extend the setting of the online prediction with expert advice to function-valued forecasts. At each step of the online game several experts predict a function and the learner has to efficiently aggregate these functional forecasts into one a single forecast. We adapt basic mixable loss functions to compare functional predictions and prove that these "integral" expansions are also mixable. We call this phenomena integral mixability. As an application, we consider various loss functions for prediction of probability distributions and show that they are mixable by using our main result. The considered loss functions include Continuous ranking probability score (CRPS), Optimal transport costs (OT), Beta-2 and Kullback-Leibler (KL) divergences.