Ensemble updating of categorical state vectors

An ensemble updating method for categorical state vectors is proposed. The method is based on a Bayesian view of the ensemble Kalman filter (EnKF). In the EnKF, Gaussian approximations to the forecast and filtering distributions are introduced, and the forecast ensemble is updated with a linear shift. Given that the Gaussian approximation to the forecast distribution is correct, the EnKF linear update corresponds to conditional simulation from a Gaussian distribution with mean and covariance such that the posterior samples marginally are distributed according to the Gaussian approximation to the filtering distribution. In the proposed approach for categorical vectors, the Gaussian approximations are replaced with a (possibly higher order) Markov chain model, and the linear update is replaced with simulation based on a class of decomposable graphical models. To make the update robust against errors in the assumed forecast and filtering distributions, an optimality criterion is formulated, for which the resulting optimal updating procedure can be found by solving a linear program. We explore the properties of the proposed updating procedure in a simulation example where each state variable can take three values.