# Scoring rule nets: beyond mean target prediction in multivariate regression

arXiv: https://arxiv.org/abs/2409.14456 · HTML: https://mlsanity.com/paper/2409.14456/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 22 Sep 2024.

## Builds on (most cited first, 9 of 9)

- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression](https://mlsanity.com/paper/2005.14458/) (2020)
- [Countdown Regression: Sharp and Calibrated Survival Predictions](https://mlsanity.com/paper/1806.08324/) (2018)
- [Deep Gaussian Mixture Models](https://mlsanity.com/paper/1711.06929/) (2017)
- [Log-normal distribution based EMOS models for probabilistic wind speed forecasting](https://mlsanity.com/paper/1407.3252/) (2014)
- [Evaluating the Discrimination Ability of Proper Multivariate Scoring Rules](https://mlsanity.com/paper/2101.12693/) (2021)
- [Cholesky-based multivariate Gaussian regression](https://mlsanity.com/paper/2102.13518/) (2021)
- [Probabilistic Deep Learning to Quantify Uncertainty in Air Quality Forecasting](https://mlsanity.com/paper/2112.02622/) (2021)
- [Sample-based Uncertainty Quantification with a Single Deterministic Neural Network](https://mlsanity.com/paper/2209.08418/) (2022)

## Abstract (arXiv)

Probabilistic regression models trained with maximum likelihood estimation (MLE), can sometimes overestimate variance to an unacceptable degree. This is mostly problematic in the multivariate domain. While univariate models often optimize the popular Continuous Ranked Probability Score (CRPS), in the multivariate domain, no such alternative to MLE has yet been widely accepted. The Energy Score - the most investigated alternative - notoriously lacks closed-form expressions and sensitivity to the correlation between target variables. In this paper, we propose Conditional CRPS: a multivariate strictly proper scoring rule that extends CRPS. We show that closed-form expressions exist for popular distributions and illustrate their sensitivity to correlation. We then show in a variety of experiments on both synthetic and real data, that Conditional CRPS often outperforms MLE, and produces results comparable to state-of-the-art non-parametric models, such as Distributional Random Forest (DRF).