# Proper Scoring Rules for Multivariate Probabilistic Forecasts based on Aggregation and Transformation

arXiv: https://arxiv.org/abs/2407.00650 · HTML: https://mlsanity.com/paper/2407.00650/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2024.

## Built on by (most cited first, 9 of 9)

- [Generative Machine Learning for Multivariate Angular Simulation](https://mlsanity.com/paper/2504.21505/) (2025)
- [Distributional Regression with Tabular Foundation Models: Evaluating Probabilistic Predictions via Proper Scoring Rules](https://mlsanity.com/paper/2603.08206/) (2026)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Multivariate Uncertainty Quantification with Tomographic Quantile Forests](https://mlsanity.com/paper/2512.16383/) (2025)
- [Decision-Aligned Evaluation of Uncertainty Quantification](https://mlsanity.com/paper/2606.26990/) (2026)
- [On the sensitivity of machine-learned probabilistic weather forecast models to scale-aware scoring rules](https://mlsanity.com/paper/2607.19161/) (2026)
- [Variable transformations in consistent loss functions](https://mlsanity.com/paper/2502.16542/) (2025)
- [Reliability of Probabilistic Emulation of Physical Systems](https://mlsanity.com/paper/2606.12997/) (2026)
- [From Jumps to Signatures: a Generative Method for Temporal Point Processes](https://mlsanity.com/paper/2607.06652/) (2026)

## Abstract (arXiv)

Proper scoring rules are an essential tool to assess the predictive performance of probabilistic forecasts. However, propriety alone does not ensure an informative characterization of predictive performance and it is recommended to compare forecasts using multiple scoring rules. With that in mind, interpretable scoring rules providing complementary information are necessary. We formalize a framework based on aggregation and transformation to build interpretable multivariate proper scoring rules. Aggregation-and-transformation-based scoring rules are able to target specific features of the probabilistic forecasts; which improves the characterization of the predictive performance. This framework is illustrated through examples taken from the literature and studied using numerical experiments showcasing its benefits. In particular, it is shown that it can help bridge the gap between proper scoring rules and spatial verification tools.