Noise Adaptive Kalman Filtering With Stochastic Natural Gradient Variational Inference

This article considers the adaptive Kalman filtering problem with unknown process noise and measurement noise covariances for linear dynamical systems. By formulating the joint estimation of system state and noise parameters as variational optimization problems, the joint posterior probability density function of latent variables (i.e., the noise covariance matrices and system state) is approximated by variational Bayesian (VB) inference. Different from the existing VB-based adaptive Kalman filter (VBAKF) methods, which update the variational hyperparameters analytically by constructing conjugate priors, this article presents a stochastic natural gradient-based VBAKF, referred to as NG AKF with unknown $Q$ and $R$ (AKF-QR), to directly optimize the intractable nonconjugate objectives. By splitting the optimization objective into conjugate and nonconjugate parts, the proposed NGAKF-QR updates the conjugate models of system state and measurement noise covariances with conjugate computations, and the nonconjugate models of process noise covariances with stochastic natural gradient, enabling effective and flexible Bayesian inference. Remarkably, the reparameterization trick for the inverse Wishart distribution is presented to decrease the stochastic gradient variance. Due to the direct estimation of state and noise covariance, the proposed NGAKF-QR has better filtering accuracy than the existing state-of-the-art VBAKF. The effectiveness of NGAKF-QR is validated through maneuvering target tracking scenarios in both simulated and real-world data.