# Learning Probabilistic Filters with Strictly Proper Scoring Rules

arXiv: https://arxiv.org/abs/2606.26497 · HTML: https://mlsanity.com/paper/2606.26497/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 110 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 13 of 13)

- [Set Transformer: A Framework for Attention-based Permutation-Invariant Neural Networks](https://mlsanity.com/paper/1810.00825/) (2018)
- [Validating Bayesian Inference Algorithms with Simulation-Based Calibration](https://mlsanity.com/paper/1804.06788/) (2018)
- [Skillful joint probabilistic weather forecasting from marginals](https://mlsanity.com/paper/2506.10772/) (2025)
- [FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale](https://mlsanity.com/paper/2507.12144/) (2025)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Likelihood-Free Inference with Generative Neural Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2205.15784/) (2022)
- [Nonlinear Filtering with Brenier Optimal Transport Maps](https://mlsanity.com/paper/2310.13886/) (2023)
- [Operator learning meets inverse problems: A probabilistic perspective](https://mlsanity.com/paper/2508.20207/) (2025)
- [In-Context Operator Learning on the Space of Probability Measures](https://mlsanity.com/paper/2601.09979/) (2026)
- [Learning Optimal Filters Using Variational Inference](https://mlsanity.com/paper/2406.18066/) (2024)
- [Machine Learning for Inverse Problems and Data Assimilation](https://mlsanity.com/paper/2410.10523/) (2024)
- [BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant Neural Operator Framework](https://mlsanity.com/paper/2410.16420/) (2024)
- [One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators](https://mlsanity.com/paper/2605.06873/) (2026)

## Built on by (most cited first, 6 of 6)

- [On the sensitivity of machine-learned probabilistic weather forecast models to scale-aware scoring rules](https://mlsanity.com/paper/2607.19161/) (2026)
- [Universal Approximation of Measure-to-Measure Operators by Pushforwards](https://mlsanity.com/paper/2609.35483/) (2026)
- [Attention Kernels for Learning Maps Between Heavy-Tailed Measures](https://mlsanity.com/paper/2610.00564/) (2026)
- [Improving Ensemble Filters with Flow Matching](https://mlsanity.com/paper/2609.28015/) (2026)
- [The interface of data assimilation and machine learning](https://mlsanity.com/paper/2610.07496/) (2026)
- [Stability of Measure-to-Measure Transformers on Sub-Gaussian Data](https://mlsanity.com/paper/2610.07717/) (2026)

## Abstract (arXiv)

Bayesian filtering of partially and noisily observed dynamical systems seeks to infer the evolving conditional distribution of the state of a dynamical system given observations, in an online fashion. This Bayesian filtering distribution is rarely available as a supervised learning target. However, one can often use the forecast model to generate synthetic trajectories, with corresponding synthetic observations. We introduce the proper scoring ensemble filter (PSEF), an ensemble data assimilation method trained using only synthetic trajectories. The analysis step is represented as a permutation-equivariant, transformer-based map. Training is based on strictly proper scoring rules---with the energy score used in our implementation---so that probabilistic accuracy is rewarded over the whole probability distribution. Under a realizability assumption, the population mean-field objective is minimized by the true Bayesian filtering distribution. Our methodology allows the same learned parameters to be shared between different ensemble sizes, subject to an ensemble-dependent fine-tuning. Numerical experiments show that the learned filter accurately approximates challenging filtering distributions, including highly non-Gaussian and multi-modal posteriors, and achieves stronger performance in data assimilation tasks than classical methods or learning-based methods with mean-squared-error objectives.