The realization problem for hidden Markov models

If {X(t)} is a finite-state Markov process, and {Y(t)} is a finite-valued output process with Y(t+1) depending (possibly probabilistically) on X(t), then the process pair is said to constitute a hidden Markov model. This paper considers the realization question: given the probabilities of all finite-length output strings, under what circumstances and how can one construct a finite-state Markov process and a state-to-output mapping which generates an output process whose finite-length strings have the given probabilities? After reviewing known results dealing with this problem involving Hankel matrices and polyhedral cones, we develop new theory on the existence and construction of the cones in question, which effectively provides a solution to the realization problem. This theory is an extension of recent theoretical developments on the positive realization problem of linear system theory.