4Approximate 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 p_Y(·) of Y the weak realization problem consists in finding a Markov chain X and a function f such that, at least distributionally, Y∼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. nonnegative (componentwise) matrices related to the processes. To construct the realization we apply recent results on the approximate factorization of nonnegative matrices.