Project Euclid

A noninvertible function of a first-order Markov process or of a nearest-neighbor Markov random field is called a hidden Markov model. Hidden Markov models are generally not Markovian. In fact, they may have complex and long range interactions, which is largely the reason for their utility. Applications include signal and image processing, speech recognition and biological modeling. We show that hidden Markov models are dense among essentially all finite-state discrete-time stationary processes and finite-state lattice-based stationary random fields. This leads to a nearly universal parameterization of stationary processes and stationary random fields, and to a consistent nonparametric estimator. We show the results of attempts to fit simple speech and texture patterns.