A generalised and fully Bayesian framework for ensemble updating

We propose a generalised framework for the updating of a prior ensemble to a posterior ensemble, an essential yet challenging part in ensemble-based filtering methods. The proposed framework is based on a generalised and fully Bayesian view on the traditional ensemble Kalman filter (EnKF). In the EnKF, the updating of the ensemble is based on Gaussian assumptions, whereas in our general setup the updating may be based on another parametric family. In addition, we propose to formulate an optimality criterion and to find the optimal update with respect to this criterion. The framework is fully Bayesian in the sense that the parameters of the assumed forecast model are treated as random variables. As a consequence, a parameter vector is simulated, for each ensemble member, prior to the updating. In contrast to existing fully Bayesian approaches, where the parameters are simulated conditionally on all the forecast samples, the parameters are in our framework simulated conditionally on both the data and all the forecast samples, except the forecast sample which is to be updated. The proposed framework is studied in detail for two parametric families: the linear-Gaussian model and the finite state-space hidden Markov model. For both cases, we present simulation examples and compare the results with existing ensemble-based filtering methods. The results of the proposed approach indicate a promising performance. In particular, the filter based on the linear-Gaussian model gives a more realistic representation of the uncertainty than the traditional EnKF, and the effect of not conditioning on the forecast sample which is to be updated when simulating the parameters is remarkable.