Abstract—Methods for generating predictions online and in the form of probability distributions of future outcomes are considered. The difference between the probabilistic forecast (probability distribution) and the numerical outcome is measured using the loss function (scoring rule). In practical statistics, the continuous ranked probability score (CRPS) is often used to estimate the discrepancy between probabilistic forecasts and (quantitative) outcomes. The paper considers the case when several competing methods (experts) give their online predictions as distribution functions. An algorithm is proposed for online aggregation of these distribution functions. The performance bounds of the proposed algorithm are obtained in the form of a comparison of the cumulative loss of the algorithm and the loss of expert hypotheses. Unlike existing estimates, the proposed estimates do not depend on time. The results of numerical experiments illustrating the proposed methods are presented.

摘要-考虑了在线生成预测结果以及未来结果的概率分布形式的方法。概率预测（概率分布）与数值结果之间的差异是使用损失函数（评分规则）来衡量的。在实际统计中，连续排名概率评分（CRPS）通常用于估计概率预测与（定量）结果之间的差异。本文考虑了几种竞争方法（专家）将其在线预测作为分布函数的情况。提出了一种在线集成这些分布函数的算法。通过比较算法的累积损失和专家假设的损失来获得所提出算法的性能界限。与现有估算不同，拟议的估计数不取决于时间。数值实验结果说明了所提出的方法。