Approximate realization of hidden Markov chains

In this paper we consider the approximate realization problem for finite valued hidden Markov models i.e. stochastic processes Y = f (X) where X is a finite state Markov chain and f a many-to-one function. Given the laws py(.) of Y the weak realization problem consists in finding a Markov chain X and a function f such that, at least. distributionally, Y similar to f (X). The approximate realization problem consists in finding X and f such that Y and f (X) are close. The approximation criterion we use is the informational divergence between properly defined non-negative (componentwise) matrices related to the processes. To construct the realization we apply recent results on the approximate factorization of nonnegative matrices.