Bellman Filtering for State-Space Models

Abstract

This article presents a new filter for state-space models based on Bellman's dynamic programming principle applied to the posterior mode. The proposed Bellman filter generalises the Kalman filter including its extended and iterated versions, while remaining equally inexpensive computationally. The Bellman filter is also (unlike the Kalman filter) robust under heavy-tailed observation noise and applicable to a wider range of models. Simulation studies reveal that the mean absolute error of the Bellman-filtered states using estimated parameters typically falls within a few percent of that produced by the mode estimator evaluated at the true parameters, which is optimal but generally infeasible.