# Mixability of Integral Losses: a Key to Efficient Online Aggregation of Functional and Probabilistic Forecasts

arXiv: https://arxiv.org/abs/1912.07048 · HTML: https://mlsanity.com/paper/1912.07048/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2019.

## Built on by (most cited first, 7 of 7)

- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Wasserstein Iterative Networks for Barycenter Estimation](https://mlsanity.com/paper/2201.12245/) (2022)
- [CRPS Learning](https://mlsanity.com/paper/2102.00968/) (2021)
- [Wasserstein Geodesic Generator for Conditional Distributions](https://mlsanity.com/paper/2308.10145/) (2023)
- [Online Aggregation of Probability Forecasts with Confidence](https://mlsanity.com/paper/2109.14309/) (2021)
- [Multivariate Uncertainty Quantification with Tomographic Quantile Forests](https://mlsanity.com/paper/2512.16383/) (2025)
- [Online Algorithm for Aggregating Experts' Predictions with Unbounded Quadratic Loss](https://mlsanity.com/paper/2501.06505/) (2025)

## Abstract (arXiv)

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 and Sliced Wasserstein-2 distance, Beta-2 and Kullback-Leibler divergences, Characteristic function and Maximum Mean Discrepancies.