Laplace State Space Filter with Exact Inference and Moment Matching

We present a Bayesian filter for state space models with Laplace-distributed observation noise that is robust to heavy-tailed and outlier-ridden univariate time-series data. We analytically derive a closed-form expression of the exact posterior for a Laplace likelihood conditioned on a Gaussian prior. Posterior statistics are propagated forward in time by a proxy Gaussian density. The forward Kullback-Leibler divergence from the posterior to the Gaussian is minimized by matching their moments. The proposed method supports both linear and non-linear systems, and has a fast recursive structure analogous to the Kalman filter that enables online inference. Results show that the new method outperforms existing approximate inference methods, especially in challenging scenarios where the system's parameters are uncertain.