Hierarchical Variational Kalman Filtering

2026-07-01

AI Summary

This work addresses the inconsistent covariance estimation and slow convergence of conventional variational Kalman filters when noise statistics are unknown. To overcome these limitations, the authors propose a hierarchical variational Kalman filtering framework that explicitly models noise statistics by introducing auxiliary variables devoid of process noise. The coordinate-ascent variational inference is reformulated as a marginal maximum a posteriori estimation problem, and a single-step sliding-window hyperparameter optimization scheme is integrated to eliminate nested inner iterations. This design decouples covariance tracking from filter synthesis, naturally enabling high-order covariance adaptation and zero-phase estimation when full historical data are available. Simulation results demonstrate that the proposed method significantly outperforms existing approaches in both convergence speed and estimation accuracy.

Abstract

Traditional variational Kalman filtering with unknown noise statistics suffers from inconsistent process covariance estimation and slow convergence speed, limiting its practical utility. To address these issues, we introduce a surrogate variable representing the process-noise-free state, which enables explicit modeling and inference of process noise statistics. In addition, we reformulate the conventional coordinate ascent variation inference (CAVI) as a marginalized maximum a posteriori problem, followed by a single-step hyperparameter fitting. This reformulation obviates the need for multiple inner iterations inherent to CAVI and decouples the design of the covariance tracking filters. Consequently, this architecture permits the deployment of higher-order filters for covariance tracking and enables sliding-window hyperparameter estimation. Notably, when this window encompasses all historical data, the covariance tracking estimator intrinsically operates as a zero-phase filter. Numerical simulations validate the theoretical framework, demonstrating the enhanced convergence speed and superior estimation accuracy compared with existing methods.

Problem

• variational Kalman filtering
• unknown noise statistics
• covariance estimation
• convergence speed

Innovation

• Hierarchical Variational Kalman Filtering
• Surrogate State Variable
• Marginalized MAP
• Covariance Tracking
• Zero-Phase Filter