Hierarchical Variational Kalman Filtering
For practitioners of Kalman filtering with unknown noise statistics, this work provides a more efficient and accurate method, though it is an incremental improvement over existing variational approaches.
This paper addresses inconsistent process covariance estimation and slow convergence in variational Kalman filtering by introducing a surrogate variable for process-noise-free state and reformulating CAVI as a marginalized MAP problem with single-step hyperparameter fitting, achieving faster convergence and superior estimation accuracy.
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.