arXiv:2605.26881v2 Announce Type: replace Abstract: We address observation noise misspecification in Bayesian filtering of dynamical systems via generalised Bayesian inference. Mismatch in tail decay between the true data generating process and an assumed observation model, often indicated via outliers, can strongly impact Bayesian updates in Kalman filters and their ensemble approximations. Adjusting information processing in the analysis step by employing diffusion score matching achieves robustness while maintaining closed-form Gaussian updates and good uncertainty quantification. We provide theoretical properties of this novel Kalman filter in linear Gaussian state space systems covering conjugacy of the analysis step, robustness, covariance stability, and tuning. We extend the approach to non-linear signal dynamics by adapting ensemble Kalman filter variants. Simulation studies on 2D target tracking and the Lorenz-63 and 40D Lorenz-96 systems highlight a critical trade-off between robustness and stability in filtering.