Learning Probabilistic Filters with Strictly Proper Scoring Rules

Traditional data assimilation methods, such as the Ensemble Kalman Filter (EnKF), are computationally efficient but struggle to accurately represent the true Bayesian filtering distribution in highly nonlinear or non-Gaussian systems. Particle filters can capture these distributions but suffer from weight degeneracy in high dimensions. This paper addresses the challenge of learning a robust, amortized analysis operator that can approximate the full filtering distribution—the natural object for uncertainty quantification—using only synthetic data.

The authors introduce the Proper Scoring Ensemble Filter (PSEF), a machine learning framework that treats the analysis step as a learned operator. The model uses a permutation-invariant, transformer-based architecture to process forecast ensembles and observations, outputting an analysis ensemble. Crucially, the model is trained using strictly proper scoring rules (specifically the energy score). This objective function rewards probabilistic accuracy across the entire distribution rather than just minimizing point-wise errors, such as mean-squared error. The authors provide a theoretical foundation by proving that, under a realizability assumption, the population-level objective is minimized by the true Bayesian filtering distribution.

Numerical experiments demonstrate that the PSEF outperforms both classical methods and standard learning-based approaches in challenging, non-Gaussian, and multi-modal filtering tasks. The authors show that for systems close to Gaussian, the model effectively learns a correction to the EnKF. Conversely, for highly non-Gaussian systems, an end-to-end approach that discards traditional inductive biases proves superior. The use of a transformer architecture allows the filter to handle varying ensemble sizes, facilitating efficient training on small ensembles followed by fine-tuning on larger ones.