Updating Gaussian Mixture Weights Using Posterior Estimates

Gaussian mixture model (GMM) filters tackle the intricacies of nonlinear and multimodal systems by representing probability distributions as a weighted sum of Gaussian components. However, traditional GMM approaches often update component weights based on prior state estimates, which can lead to filter divergence and degeneracy. Therefore, in this work, weights based on posterior state estimates are used instead, which provide a more accurate and dynamic reflection of the system's state after receiving new measurement data. The posterior-based approach extends to GMM filters that update individual Gaussian components using linearization techniques, such as the extended Kalman filter and the Bayesian recursive update filter. In addition, a Jacobian-free version, using importance sampling, is proposed for sigma-point-based methods, such as the unscented Kalman filter and the cubature Kalman filter. Empirical results from a 2-D Avocado example and a cislunar orbit determination example show that updating weights using posterior estimates improves accuracy and consistency, while maintaining computational efficiency comparable to prior-based approaches.