# Probabilistic Forecasting with Generative Networks via Scoring Rule Minimization

arXiv: https://arxiv.org/abs/2112.08217 · HTML: https://mlsanity.com/paper/2112.08217/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 54 references that are in the map.
- First posted on arXiv 15 Dec 2021.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Skillful Precipitation Nowcasting using Deep Generative Models of Radar](https://mlsanity.com/paper/2104.00954/) (2021)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [The Cramer Distance as a Solution to Biased Wasserstein Gradients](https://mlsanity.com/paper/1705.10743/) (2017)
- [On GANs and GMMs](https://mlsanity.com/paper/1805.12462/) (2018)
- [Machine Learning for Stochastic Parameterization: Generative Adversarial Networks in the Lorenz '96 Model](https://mlsanity.com/paper/1909.04711/) (2019)
- [A Spectral Energy Distance for Parallel Speech Synthesis](https://mlsanity.com/paper/2008.01160/) (2020)
- [Estimating and Evaluating Regression Predictive Uncertainty in Deep Object Detectors](https://mlsanity.com/paper/2101.05036/) (2021)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [DISCO Nets: DISsimilarity COefficient Networks](https://mlsanity.com/paper/1606.02556/) (2016)
- [A generative adversarial network approach to (ensemble) weather prediction](https://mlsanity.com/paper/2006.07718/) (2020)
- [Combining distribution-based neural networks to predict weather forecast probabilities](https://mlsanity.com/paper/2103.14430/) (2021)
- [If You Like It, GAN It. Probabilistic Multivariate Times Series Forecast With GAN](https://mlsanity.com/paper/2005.01181/) (2020)
- [Approximate Bayesian computation via the energy statistic](https://mlsanity.com/paper/1905.05884/) (2019)

## Built on by (most cited first, 30 of 37)

- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [Generative machine learning methods for multivariate ensemble post-processing](https://mlsanity.com/paper/2211.01345/) (2022)
- [Distributional Diffusion Models with Scoring Rules](https://mlsanity.com/paper/2502.02483/) (2025)
- [Probabilistic neural operators for functional uncertainty quantification](https://mlsanity.com/paper/2502.12902/) (2025)
- [Proper scoring rules for estimation and forecast evaluation](https://mlsanity.com/paper/2504.01781/) (2025)
- [EnScale: Temporally-consistent multivariate generative downscaling via proper scoring rules](https://mlsanity.com/paper/2509.26258/) (2025)
- [Continuous Autoregressive Language Models](https://mlsanity.com/paper/2510.27688/) (2025)
- [Continuous Visual Autoregressive Generation via Score Maximization](https://mlsanity.com/paper/2505.07812/) (2025)
- [Stochastic Latent Transformer: Efficient Modelling of Stochastically Forced Zonal Jets](https://mlsanity.com/paper/2310.16741/) (2023)
- [Language Generation with Strictly Proper Scoring Rules](https://mlsanity.com/paper/2405.18906/) (2024)
- [A multi-scale loss formulation for learning a probabilistic model with proper score optimisation](https://mlsanity.com/paper/2506.10868/) (2025)
- [Quasi-Bayes meets Vines](https://mlsanity.com/paper/2406.12764/) (2024)
- [Enforcing tail calibration when training probabilistic forecast models](https://mlsanity.com/paper/2506.13687/) (2025)
- [Probabilistic Self-supervised Learning via Scoring Rules Minimization](https://mlsanity.com/paper/2309.02048/) (2023)
- [Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions](https://mlsanity.com/paper/2410.03973/) (2024)
- [A Gentle Introduction and Tutorial on Deep Generative Models in Transportation Research](https://mlsanity.com/paper/2410.07066/) (2024)
- [Sample-based Uncertainty Quantification with a Single Deterministic Neural Network](https://mlsanity.com/paper/2209.08418/) (2022)
- [Generative Machine Learning for Multivariate Angular Simulation](https://mlsanity.com/paper/2504.21505/) (2025)
- [Generalized Bayesian deep reinforcement learning](https://mlsanity.com/paper/2412.11743/) (2024)
- [Distributional Instrumental Variable Method](https://mlsanity.com/paper/2502.07641/) (2025)
- [Probabilistic intraday electricity price forecasting using generative machine learning](https://mlsanity.com/paper/2506.00044/) (2025)
- [Improved probabilistic regression using diffusion models](https://mlsanity.com/paper/2510.04583/) (2025)
- [Signature Kernel Scoring Rule: A Spatio-Temporal Diagnostic for Probabilistic Weather Forecasting](https://mlsanity.com/paper/2510.19110/) (2025)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Graph-based Neural Space Weather Forecasting](https://mlsanity.com/paper/2509.19605/) (2025)
- [Nonparametric estimation of conditional probability distributions using a generative approach based on conditional push-forward neural networks](https://mlsanity.com/paper/2511.14455/) (2025)
- [On the sensitivity of machine-learned probabilistic weather forecast models to scale-aware scoring rules](https://mlsanity.com/paper/2607.19161/) (2026)
- [Modeling Spatial Extremal Dependence of Precipitation Using Distributional Neural Networks](https://mlsanity.com/paper/2407.08668/) (2024)
- [Likelihood-Free Variational Autoencoders](https://mlsanity.com/paper/2504.17622/) (2025)
- [Moment Expansions of the Energy Distance](https://mlsanity.com/paper/2505.20647/) (2025)

## Abstract (arXiv)

Probabilistic forecasting relies on past observations to provide a probability distribution for a future outcome, which is often evaluated against the realization using a scoring rule. Here, we perform probabilistic forecasting with generative neural networks, which parametrize distributions on high-dimensional spaces by transforming draws from a latent variable. Generative networks are typically trained in an adversarial framework. In contrast, we propose to train generative networks to minimize a predictive-sequential (or prequential) scoring rule on a recorded temporal sequence of the phenomenon of interest, which is appealing as it corresponds to the way forecasting systems are routinely evaluated. Adversarial-free minimization is possible for some scoring rules; hence, our framework avoids the cumbersome hyperparameter tuning and uncertainty underestimation due to unstable adversarial training, thus unlocking reliable use of generative networks in probabilistic forecasting. Further, we prove consistency of the minimizer of our objective with dependent data, while adversarial training assumes independence. We perform simulation studies on two chaotic dynamical models and a benchmark data set of global weather observations; for this last example, we define scoring rules for spatial data by drawing from the relevant literature. Our method outperforms state-of-the-art adversarial approaches, especially in probabilistic calibration, while requiring less hyperparameter tuning.