On the variational Gaussian filtering with natural gradient descent

Variational Gaussian filter (VGF) approximates the intractable posterior of the state of a non-linear non-Gaussian system using a single Gaussian density normally found through Kullback-Leibler divergence minimization. This paper focuses on the VGFs whose measurement update is realized by employing the natural gradient descent (NGD). Under the assumption that the state predictive distribution is also Gaussian, we re-examine the iterative NGD-based measurement update under two different parameterizations of the Gaussian posterior. The first one consists of the mean and covariance, while the other comprises the mean and precision matrix (i.e., the inverse of the covariance). Their NGD-based update rules are derived in an alternative but unified way using matrix calculus. They are compared against each other and with the one developed using the natural parameterization of the Gaussian density. Important new insights are obtained. Modifications to the established update rules, which guarantee the positive definiteness of the covariance/precision matrix of the Gaussian posterior, are re-visited as well. Simulations are used to corroborate the theoretical results and evaluate the performance of the developed algorithms in range-bearing tracking.