The Score Kalman Filter

2026-05-15

AI Summary

This work addresses the challenge of efficiently representing posterior distributions in nonlinear Bayesian filtering under high-dimensional settings, where conventional moment-based methods suffer from computational intractability due to their reliance on the partition function. The authors propose a novel filtering framework that eliminates the need for the partition function by introducing score matching and Stein’s identity into Bayesian filtering for the first time. This approach enables density estimation through linear solves and integrates moment propagation with maximum entropy reconstruction to close both prediction and update steps. The method entirely avoids high-dimensional integration, scales to state spaces beyond 20 dimensions, and naturally reduces to the information-form Kalman filter in linear-Gaussian cases. Experiments on nonlinearly coupled oscillator networks demonstrate significantly lower RMSE compared to baseline methods including EKF, UKF, EnKF, and particle filters.

Abstract

A central obstacle in nonlinear Bayesian filtering is representing the belief distribution. Moment-based filters address this by propagating polynomial moments and reconstructing a density from them. Recent work completes the predict-update loop via the maximum-entropy (MaxEnt) principle, but each step requires the partition function and its gradient, both $n$-dimensional integrals whose cost scales exponentially, restricting the demonstrated MaxEnt moment filtering to $n \le 4$. We avoid the partition function entirely by combining score matching with Stein's identity. In our setting, score matching reduces the density fit to a single linear solve whose coefficients are assembled directly from the propagated moments. The same parameters then drive Stein's identity to close the moment hierarchy during prediction and to recover posterior moments after each Bayesian update, keeping the full predict-update loop free of partition function evaluation. The resulting Score Kalman Filter (SKF) reduces to the classical information-form Kalman filter as a special case and performs every step through linear algebra. On nonlinear coupled-oscillator networks, the SKF runs through $n=20$ and reports lower RMSE than the EKF, UKF, EnKF, and particle-filter baselines on the tested synthetic benchmarks.

Problem

• nonlinear Bayesian filtering
• belief distribution representation
• partition function
• high-dimensional inference
• moment-based filtering

Innovation

• Score Kalman Filter
• Score Matching
• Stein's Identity
• Moment-based Filtering
• Nonlinear Bayesian Filtering