An Outlier-Robust Kalman Filter Based on the Generalized $t$ Distribution with a Mixed Gaussian Approximation

In this paper, a novel robust Kalman filter (KF) based on the generalized
$t$
distribution is proposed to deal with measurement outliers. By ultilizing the relationship between the generalized
$t$
distribution and the Gauss-Inverse-Wishart distribution, a latent random variable is introduced, which helps us derive the accurate analytical form of the posterior density. After that, a mixed Gaussian approximation method is proposed to obtain an accurate approximation of the posterior mean and covariance. We also establish the best Gauss distribution to approximate the posterior distribution under the Kullback-Leibler (KL) divergence. Compared with the current robust KFs, our method is more implementation-friendly, and can achieve a higher accuracy due to the adoption of the latent matrix random variable and the best Gaussian approximation for the posterior distribution under the KL divergence.