  In this paper, state and noise covariance estimation problems for linear
system with unknown multiplicative noise are considered. The measurement
likelihood is modelled as a mixture of two Gaussian distributions and a
Student's t distribution, respectively. The unknown covariance of
multiplicative noise is modelled as an inverse Gamma/Wishart distribution and
the initial condition is formulated as the nominal covariance. By using robust
design and choosing hierarchical priors, two variational Bayesian based robust
Kalman filters are proposed. Stability and covergence of the proposed filters,
the covariance parameters, the VB inference, and the estimation error dynamics
are analyzed. The lower and upper bounds are also provided to guarantee the
performance of the proposed filters. A target tracking simulation is provided
to validate the effectiveness of the proposed filters.
