Learning Probabilistic Filters with Strictly Proper Scoring Rules
Abstract
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.
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BibTeX
@misc{bach2026learning,
title = {{Learning Probabilistic Filters with Strictly Proper Scoring Rules}},
author = {Eviatar Bach and Ricardo Baptista and Jochen Bröcker and Bohan Chen and Andrew Stuart},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.26497},
eprint = {2606.26497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26497},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning Probabilistic Filters with Strictly Proper Scoring Rules
AU - Eviatar Bach
AU - Ricardo Baptista
AU - Jochen Bröcker
AU - Bohan Chen
AU - Andrew Stuart
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2606.26497
UR - https://arxiv.org/abs/2606.26497
AB - 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.
ER -