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In this paper we extend the setting of the online prediction with expertadvice to function-valued forecasts. At each step of the online game severalexperts predict a function, and the learner has to efficiently aggregate thesefunctional forecasts into a single forecast. We adapt basic mixable (andexponentially concave) loss functions to compare functional predictions andprove that these adaptations are also mixable (exp-concave). We call thisphenomena integral mixability (exp-concavity). As an application of our mainresult, we prove that various loss functions used for probabilistic forecastingare mixable (exp-concave). The considered losses include Sliced ContinuousRanking Probability Score, Energy-Based Distance, Optimal Transport Costs &Sliced Wasserstein-2 distance, Beta-2 & Kullback-Leibler divergences,Characteristic function and Maximum Mean Discrepancies.

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