Identifiability of hidden markov processes and their minimum degrees of freedom

There exists a class of stochastic processes based on a finite-state discrete-time Markov chain, where the states are not directly observable and only the output symbols generated by the states are observable. If different states produce the same output symbol, the process is called a hidden Markov process. It is very important both in theory and applications. It is known that two Markov processes with different transition matrices are equivalent hidden Markov processes. The problem has been considered interesting concerning by what structures such a situation can arise. This paper gives a complete solution to the identification problem of the hidden Markov process, which has long been proposed and is still unsolved. At the same time, a structure of this kind of process, which has been unknown, is revealed. An n algebraic technique is used in the discussion, where new concepts called Δ-subspace and Δ-cy-clic subspace are introduced into the framework of subspace and cyclic subspace in the usual linear algebra. In this paper, a Δ-subspace is actually constructed which gives the effective minimum degree of the hidden Markov process. It is shown also that two stochastic processes are equivalent if and only if there exists an isomorphism between the Δ-sub-spaces.