Square-root variational Gaussian filter with natural gradient descent

We consider nonlinear state filtering. For tractability, the state predictive distribution and posterior are approximated using two Gaussian densities found by solving two variational problems. The derivation of such a variational Gaussian filter (VGF) is re-visited with the state predictive distribution and posterior constrained to the same exponential family. We show that the state prediction is achieved via moment matching. Moreover, when the measurements have additive Gaussian noise and only one iteration of natural gradient descent (NGD) is performed to compute the state posterior, the measurement update becomes identical to that of a linear Kalman filter. The corresponding linear measurement model has multiple perturbed measurements with independent noise. Its specific form depends on the method employed to evaluate numerically the Gaussian integral required by the NGD update. Square-root VGF realizations are developed for this scenario. Their performance under a variety of numerical integration techniques is compared using simulations.