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In this paper, state and noise covariance estimation problems for linearsystem with unknown multiplicative noise are considered. The measurementlikelihood is modelled as a mixture of two Gaussian distributions and aStudent's t distribution, respectively. The unknown covariance ofmultiplicative noise is modelled as an inverse Gamma/Wishart distribution andthe initial condition is formulated as the nominal covariance. By using robustdesign and choosing hierarchical priors, two variational Bayesian based robustKalman filters are proposed. Stability and covergence of the proposed filters,the covariance parameters, the VB inference, and the estimation error dynamicsare analyzed. The lower and upper bounds are also provided to guarantee theperformance of the proposed filters. A target tracking simulation is providedto validate the effectiveness of the proposed filters.

Robust Kalman filters with unknown covariance of multiplicative noise