Equivalence of state representations for hidden Markov models

In this paper we consider the following problem for hidden Markov models: given a minimal hidden Markov model, derive conditions for another hidden Markov model to be equivalent and give a description of the complete set of equivalent models. A distinction is made between quasi- and positive hidden Markov models and between Mealy and Moore hidden Markov models. We derive a condition for two positive Mealy models to be equivalent and give a description of the complete set of Mealy models that are equivalent to a given Mealy model. We show that under certain conditions minimal quasi-Moore models are unique up to a permutation of the states. We derive a condition for two positive Moore models to be equivalent and give a description of the complete set of Moore models equivalent to a given Moore model. Finally, we compare the results for hidden Markov models and linear Gaussian systems.