# Evaluating the Discrimination Ability of Proper Multivariate Scoring Rules

arXiv: https://arxiv.org/abs/2101.12693 · HTML: https://mlsanity.com/paper/2101.12693/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2021.

## Built on by (most cited first, 8 of 8)

- [Generative machine learning methods for multivariate ensemble post-processing](https://mlsanity.com/paper/2211.01345/) (2022)
- [Regions of Reliability in the Evaluation of Multivariate Probabilistic Forecasts](https://mlsanity.com/paper/2304.09836/) (2023)
- [MVG-CRPS: A Robust Loss Function for Multivariate Probabilistic Forecasting](https://mlsanity.com/paper/2410.09133/) (2024)
- [Multivariate Latent Recalibration for Conditional Normalizing Flows](https://mlsanity.com/paper/2505.16636/) (2025)
- [Online Multivariate Regularized Distributional Regression for High-dimensional Probabilistic Electricity Price Forecasting](https://mlsanity.com/paper/2504.02518/) (2025)
- [Probabilistic intraday electricity price forecasting using generative machine learning](https://mlsanity.com/paper/2506.00044/) (2025)
- [Multivariate Uncertainty Quantification with Tomographic Quantile Forests](https://mlsanity.com/paper/2512.16383/) (2025)
- [Scoring rule nets: beyond mean target prediction in multivariate regression](https://mlsanity.com/paper/2409.14456/) (2024)

## Abstract (arXiv)

Proper scoring rules are commonly applied to quantify the accuracy of distribution forecasts. Given an observation they assign a scalar score to each distribution forecast, with the the lowest expected score attributed to the true distribution. The energy and variogram scores are two rules that have recently gained some popularity in multivariate settings because their computation does not require a forecast to have parametric density function and so they are broadly applicable. Here we conduct a simulation study to compare the discrimination ability between the energy score and three variogram scores. Compared with other studies, our simulation design is more realistic because it is supported by a historical data set containing commodity prices, currencies and interest rates, and our data generating processes include a diverse selection of models with different marginal distributions, dependence structure, and calibration windows. This facilitates a comprehensive comparison of the performance of proper scoring rules in different settings. To compare the scores we use three metrics: the mean relative score, error rate and a generalised discrimination heuristic. Overall, we find that the variogram score with parameter p=0.5 outperforms the energy score and the other two variogram scores.