Gaussian Mixture Particle Filter Step based on Method of Moments

We propose a novel update step of a Gaussian mixture particle filter for nonlinear state estimation. The update procedure works as follows: First, unweighted samples are drawn in an optimal deterministic sense from a prior Gaussian mixture. These samples are then assigned weights from the likelihood function, and we compute higher-order moments from this sample-based posterior. These moment approximations converge with $L^{-1}$ instead of $L^{-1/2}$ as our samples are optimal deterministic. Finally, the continuous posterior approximation is determined as the Gaussian mixture that has minimal Fisher information under the constraint of having the aforementioned moments. To achieve this, we employ a closed-form solution of the Fisher information that involves Gaussian root mixture densities.