  In recent years, several ensemble-based filtering methods have been proposed
and studied. The main challenge in such procedures is the updating of a prior
ensemble to a posterior ensemble at every step of the filtering recursions. In
the famous ensemble Kalman filter, the assumption of a linear-Gaussian state
space model is introduced in order to overcome this issue, and the prior
ensemble is updated with a linear shift closely related to the traditional
Kalman filter equations. In the current article, we consider how the ideas
underlying the ensemble Kalman filter can be applied when the components of the
state vectors are binary variables. While the ensemble Kalman filter relies on
Gaussian approximations of the forecast and filtering distributions, we instead
use first order Markov chains. To update the prior ensemble, we simulate
samples from a distribution constructed such that the expected number of equal
components in a prior and posterior state vector is maximised. We demonstrate
the performance of our approach in a simulation example inspired by the
movement of oil and water in a petroleum reservoir, where also a more na\"{i}ve
updating approach is applied for comparison. Here, we observe that the
Frobenius norm of the difference between the estimated and the true marginal
filtering probabilities is reduced to the half with our method compared to the
na\"{i}ve approach, indicating that our method is superior. Finally, we discuss
how our methodology can be generalised from the binary setting to more
complicated situations.
