New Filter Learns Probabilistic Distributions with Proper Scoring Rules

Bayesian filtering aims to infer the evolving conditional distribution of a dynamical system's state given partial and noisy observations, providing crucial uncertainty quantification. However, this true filtering distribution is rarely available as a direct supervised learning target. To address this, the Proper Scoring Ensemble Filter (PSEF) is introduced. This ensemble data assimilation method trains an analysis map, represented as a permutation-invariant, transformer-based network, to approximate the filtering distribution. It achieves this by using only synthetic state-observation trajectories. The training is uniquely based on strictly proper scoring rules, specifically the energy score, which rewards probabilistic accuracy across the entire probability distribution. The paper proves that, under a realizability assumption, the population objective is minimized by the true Bayesian filtering distribution. Numerical experiments demonstrate PSEF's ability to accurately approximate challenging filtering distributions, including those that are nonlinear, non-Gaussian, and multi-modal. It outperforms classical methods and other learning-based methods that rely on mean-squared-error objectives, especially for highly non-Gaussian problems.