  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.
