Learning Probabilistic Filters with Strictly Proper Scoring Rules

arXiv cs.LG Papers
probabilistic-filters strictly-proper-scoring-rules bayesian-filtering ensemble-data-assimilation transformer energy-score machine-learning

Summary

This paper introduces the Proper Scoring Ensemble Filter (PSEF), a transformer-based method for Bayesian filtering that trains an analysis map using strictly proper scoring rules on synthetic state-observation trajectories. It demonstrates superior performance in nonlinear, non-Gaussian filtering tasks compared to classical and learning-based methods.

arXiv:2606.26497v1 Announce Type: new 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 the natural object for uncertainty quantification, but it is rarely available as a supervised learning target. However, one can often use the forecast model to generate synthetic system trajectories, along with synthetic observations. We introduce the proper scoring ensemble filter (PSEF), an ensemble data assimilation method based on training an analysis map to approximate the filtering distribution using only synthetic state--observation trajectories. The analysis step is represented as a permutation-invariant, transformer-based map that takes as input a forecast ensemble and observations, producing an analysis ensemble. 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. We prove that, under a realizability assumption, the population objective is minimized by the true Bayesian filtering distribution. We also derive the finite-ensemble empirical objective used in training and relate its single state--observation trajectory form to the population objective, using a mean-field consistency argument. Numerical experiments show that the learned filter accurately approximates challenging filtering distributions, including nonlinear, 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. For close-to-Gaussian problems, learning a correction to the EnKF is the best approach, while for highly non-Gaussian problems an end-to-end approach that discards this inductive bias is superior.
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# Learning Probabilistic Filters with Strictly Proper Scoring Rules
Source: [https://arxiv.org/abs/2606.26497](https://arxiv.org/abs/2606.26497)
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> 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 the natural object for uncertainty quantification, but it is rarely available as a supervised learning target\. However, one can often use the forecast model to generate synthetic system trajectories, along with synthetic observations\. We introduce the proper scoring ensemble filter \(PSEF\), an ensemble data assimilation method based on training an analysis map to approximate the filtering distribution using only synthetic state\-\-observation trajectories\. The analysis step is represented as a permutation\-invariant, transformer\-based map that takes as input a forecast ensemble and observations, producing an analysis ensemble\. 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\. We prove that, under a realizability assumption, the population objective is minimized by the true Bayesian filtering distribution\. We also derive the finite\-ensemble empirical objective used in training and relate its single state\-\-observation trajectory form to the population objective, using a mean\-field consistency argument\. Numerical experiments show that the learned filter accurately approximates challenging filtering distributions, including nonlinear, 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\. For close\-to\-Gaussian problems, learning a correction to the EnKF is the best approach, while for highly non\-Gaussian problems an end\-to\-end approach that discards this inductive bias is superior\.

## Submission history

From: Bohan Chen \[[view email](https://arxiv.org/show-email/a97b8888/2606.26497)\] **\[v1\]**Thu, 25 Jun 2026 01:04:21 UTC \(5,398 KB\)

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