  This paper addresses state estimation of linear systems with special
attention on unknown process and measurement noise covariances, aiming to
enhance estimation accuracy while preserving the stability guarantee of the
Kalman filter. To this end, the full information estimation problem over a
finite interval is firstly addressed. Then, a novel adaptive variational
Bayesian (VB) moving horizon estimation (MHE) method is proposed, exploiting VB
inference, MHE and Monte Carlo integration with importance sampling for joint
estimation of the unknown process and measurement noise covariances, along with
the state trajectory over a moving window of fixed length. Further, it is
proved that the proposed adaptive VB MHE filter ensures mean-square boundedness
of the estimation error with any number of importance samples and VB
iterations, as well as for any window length. Finally, simulation results on a
target tracking example demonstrate the effectiveness of the VB MHE filter with
enhanced estimation accuracy and convergence properties compared to the
conventional non-adaptive Kalman filter and other existing adaptive filters.
