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 a single forecast. We adapt basic mixable (and exponentially concave) loss functions to compare functional predictions and prove that these adaptations are also mixable (exp-concave). We call this phenomenon mixability (exp-concavity) of integral loss functions. As an application of our main result, we prove that various loss functions used for probabilistic forecasting are mixable (exp-concave). The considered losses include Sliced Continuous Ranked Probability Score, Energy-Based Distance, Optimal Transport Costs & Sliced Wasserstein-2 distance, Beta-2 & Kullback-Leibler divergences, Characteristic function and Maximum Mean Discrepancies.

在本文中，我们将具有专家建议的在线预测设置扩展到函数值预测。在网络游戏的每一步，都有几个专家预测一个函数，学习者必须有效地将这些函数预测聚合成一个单一的预测。我们采用基本的可混合（和指数凹）损失函数来比较功能预测，并证明这些适应也是可混合的（exp-concave）。我们称这种现象为积分损失函数的混合性（exp-concavity）。作为我们主要结果的应用，我们证明了用于概率预测的各种损失函数是可混合的（exp-concave）。考虑的损失包括切片连续排名概率分数、基于能量的距离、最佳运输成本和切片 Wasserstein-2 距离、Beta-2 和 Kullback-Leibler 分歧，