Black box variational inference to adaptive kalman filter with unknown process noise covariance matrix

Abstract

Adaptive Kalman filter (AKF) is concerned with jointly estimating the system state and the unknown parameters of the state-space models. In this paper, we treat the model uncertainty of the process noise covariance matrix (PNCM) from black box variational inference (BBVI) perspective. In order to lay the foundation for research, we prove that the probabilistic model for online Bayesian inference of the system state and PNCM is non-conjugate, so the traditional coordinate-ascent variational inference (CAVI) cannot deal with this problem. To fill this gap, we propose an AKF in the presence of unknown PNCM based on the BBVI method (which is recently introduced to conduct the approximate Bayesian inference for the non-conjugate probabilistic model). Firstly, we introduce a structured posterior model of the system state and PNCM, by which the posterior distributions of the system state and the PNCM can be calculated efficiently. Then, the BBVI online inference for the posterior distribution of the PNCM is derived. In what follows, we use the intrinsically Bayesian robust KF (IBR-KF) to calculate the state posterior distribution. In addition, a special case, when the structure of the PNCM is known, is explored. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed filters.