A Comparative Study of the Particle Filter and the Ensemble Kalman Filter

Abstract

Non-linear Bayesian estimation, or estimation of the state of a non-linear stochastic system from a set of indirect noisy measurements is a problem encountered in several fields of science. The particle filter and the ensemble Kalman filter are both used to get sub-optimal solutions of Bayesian inference problems, particularly for high-dimensional non-Gaussian and non-linear models. Both are essentially Monte Carlo techniques that compute their results using a set of estimated trajectories of the variable to be monitored. It has been shown that in a linear and Gaussian environment, solutions obtained from both these filters converge to the optimal solution obtained by the Kalman Filter. However, it is of interest to explore how the two filters compare to each other in basic methodology and construction, especially due to the similarity between them. In this work, we take up a specific problem of Bayesian inference in a restricted framework and compare analytically the results obtained from the particle filter and the ensemble Kalman filter. We show that for the chosen model, under certain assumptions, the two filters become methodologically analogous as the sample size goes to infinity.