The Kalman Laplace filter: A new deterministic algorithm for nonlinear Bayesian filtering

We propose a new recursive algorithm for nonlinear Bayesian filtering, where the prediction step is performed like in the extended Kalman filter, and the update step is done thanks to the Laplace method for integral approximation. This algorithm is called the Kalman Laplace filter (KLF). The KLF provides a closed-form non-Gaussian approximation of the posterior density. The hidden state is estimated by the maximum a posteriori, using a dimension reduction method to alleviate the computation cost of the maximization. The KLF is tested on three simulated nonlinear filtering problems: target tracking with angle measurements, population dynamics monitoring, motion reconstruction by neural decoding. It exhibits a good performance, especially when the observation noise is small.