mlsanityData to 14 Oct 2026

Proper scoring rules for estimation and forecast evaluation

arXiv 2504.01781 ↗First posted 2 Apr 2025math.ST stat.ML stat.THcited by 18

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Abstract (arXiv)

Proper scoring rules have been a subject of growing interest in recent years, not only as tools for evaluation of probabilistic forecasts but also as methods for estimating probability distributions. In this article, we review the mathematical foundations of proper scoring rules including general characterization results and important families of scoring rules. We discuss their role in statistics and machine learning for estimation and forecast evaluation. Furthermore, we comment on interesting developments of their usage in applications.

Key facts. Cited by 18 papers in the map (data through 14 Oct 2026). Builds on 42 of its 151 references that are in the map. First posted on arXiv 2 Apr 2025.

Builds on

42 of 151 references.

References · 42

  1. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,255
  2. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  3. 2015Generative Moment Matching Networkscited by 257
  4. 2015Training generative neural networks via Maximum Mean Discrepancy optimizationcited by 151
  5. 2016Learning in Implicit Generative Modelscited by 125
  6. 2022Score matching enables causal discovery of nonlinear additive noise modelscited by 111
  7. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 64
  8. 2023Engression: Extrapolation through the Lens of Distributional Regressioncited by 51
  9. 2020A Spectral Energy Distance for Parallel Speech Synthesiscited by 42
  10. 2021Survival Regression with Proper Scoring Rules and Monotonic Neural Networkscited by 42
  11. 2021Probabilistic Forecasting with Generative Networks via Scoring Rule Minimizationcited by 37
  12. 2023Non-adversarial training of Neural SDEs with signature kernel scorescited by 37
30 more
  1. 2018Countdown Regression: Sharp and Calibrated Survival Predictionscited by 31
  2. 2019Isotonic Distributional Regressioncited by 28
  3. 2022Generative machine learning methods for multivariate ensemble post-processingcited by 26
  4. 2025Distributional Diffusion Models with Scoring Rulescited by 25
  5. 2020Score Matched Neural Exponential Families for Likelihood-Free Inferencecited by 21
  6. 2015Estimation of High-Dimensional Graphical Models Using Regularized Score Matchingcited by 20
  7. 2023Variational Inference with Gaussian Score Matchingcited by 17
  8. 2016Variational Hamiltonian Monte Carlo via Score Matchingcited by 14
  9. 2017Strictly proper kernel scores and characteristic kernels on compact spacescited by 14
  10. 2017Regularizing with Bregman-Moreau envelopescited by 12
  11. 2021Generalized Bayesian Likelihood-Free Inferencecited by 11
  12. 2022Evaluating forecasts for high-impact events using transformed kernel scorescited by 11
  13. 2017Bayesian model comparison with the Hyv"arinen score: computation and consistencycited by 10
  14. 2023Proper Scoring Rules for Survival Analysiscited by 9
  15. 2023Decompositions of the mean continuous ranked probability scorecited by 9
  16. 2019Generative Adversarial Nets for Robust Scatter Estimation: A Proper Scoring Rule Perspectivecited by 8
  17. 2019Local scale invariance and robustness of proper scoring rulescited by 8
  18. 2021Comparing Sequential Forecasterscited by 8
  19. 2024Language Generation with Strictly Proper Scoring Rulescited by 8
  20. 2025Stein's unbiased risk estimate and Hyv"arinen's score matchingcited by 8
  21. 2021Valid sequential inference on probability forecast performancecited by 7
  22. 2021Pseudo-Spherical Contrastive Divergencecited by 7
  23. 2022The Geometry and Calculus of Lossescited by 7
  24. 2024Tail calibration of probabilistic forecastscited by 7
  25. 2024Is Score Matching Suitable for Estimating Point Processes?cited by 5
  26. 2021Proper Scoring Rules, Gradients, Divergences, and Entropies for Paths and Time Seriescited by 3
  27. 2022Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax riskcited by 3
  28. 2023Overview of AdaBoost : Reconciling its views to better understand its dynamicscited by 3
  29. 2025Fixing the Pitfalls of Probabilistic Time-Series Forecasting Evaluation by Kernel Quadraturecited by 2
  30. 2023Locally tail-scale invariant scoring rules for evaluation of extreme value forecastscited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 8; 2026: 10.

18 papers.

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  2. 2026Maximum Likelihood Reinforcement Learningcited by 36
  3. 2026Agentic Forecasting using Sequential Bayesian Updating of Linguistic Beliefscited by 10
  4. 2026Probabilistic Retrofitting of Learned Simulatorscited by 7
  5. 2026ScoringBench: A Benchmark for Evaluating Tabular Foundation Models with Proper Scoring Rulescited by 6
  6. 2025Enforcing tail calibration when training probabilistic forecast modelscited by 5
  7. 2025Uncertainty Quantification for Regression using Proper Scoring Rulescited by 5
  8. 2025Generative Machine Learning for Multivariate Angular Simulationcited by 3
  9. 2026Distributional Regression with Tabular Foundation Models: Evaluating Probabilistic Predictions via Proper Scoring Rulescited by 3
  10. 2025Uncertainty Quantification for Regression: A Unified Framework based on kernel scorescited by 2
  11. 2025Generalisation Bounds of Zero-Shot Economic Forecasting using Time Series Foundation Modelscited by 1
  12. 2025Multivariate Uncertainty Quantification with Tomographic Quantile Forestscited by 1
6 more
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  3. 2026Probabilistic bias adjustment of seasonal forecasts using generative machine learning: A case study of Arctic sea ice predictionscited by 0
  4. 2026Censoring-Aware In-Context Learning for Generalized Supplier Lead Time Estimation in Supply Chain Planningcited by 0
  5. 2026Expressivity In Multimodal Contrastive Learningcited by 0
  6. 2026Variational Streaming Flow: Probabilistic Forecasting in Physical Timecited by 0