Minimal dimensional linear filters for discrete-time Markov processes with finite state space

We consider a filtering problem for a discrete-time Markov process with k states observed in white Gaussian noise. It is known that in this situation the best linear estimate is given by a k-dimensional Kalman filter, and in some cases the dimension of such a filter can be reduced. Here, using a backward semimartingale description of the process and results from stochastic realization theory, we provide an algorithm for the construction of the minimal dimensional linear filter.