Learning Probabilistic Filters with Strictly Proper Scoring Rules

Thu, 25 Jun 2026 00:00:00 +0000

The proper scoring ensemble filter (PSEF) learns an ensemble analysis operator that targets the complete Bayesian filtering distribution rather than only its conditional mean. The operator takes a forecast ensemble and a new observation as input and returns an analysis ensemble. A permutation-invariant transformer ensures that the result respects the exchangeability of ensemble members and can be evaluated at different ensemble sizes.

The key training device is a strictly proper scoring rule, implemented with the energy score. Its expected value is uniquely minimized by the true filtering distribution, yet its empirical form requires only simulated state–observation trajectories—not direct access to the generally intractable filtering distribution. Under a realizability assumption, the population objective recovers the Bayesian filter, and a mean-field time-averaging argument connects the practical single-trajectory objective to this population target.

Experiments cover a linear–Gaussian diagnostic, the strongly non-Gaussian doubling-angle model, Lorenz-63, and Lorenz-96. Energy-score training captures multimodal distributions missed by Gaussian filters and mean-based learning objectives. The results also reveal a useful architectural distinction: an EnKF correction provides an effective inductive bias in close-to-Gaussian problems, while a flexible end-to-end update is better suited to strongly non-Gaussian posterior geometry.

The manuscript has been submitted to the Journal of Machine Learning Research.

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Learning Probabilistic Filters with Strictly Proper Scoring Rules