The complete realization problem for hidden Markov models: a survey and some new results

Suppose m is a positive integer, and let \({\mathcal{M} = \{1, \ldots ,m\}}\) . Suppose \({\{\mathcal{Y}_t \}}\) is a stationary stochastic process assuming values in \({\mathcal{M}}\) . In this paper we study the question: When does there exist a hidden Markov model (HMM) that reproduces the statistics of this process? This question is more than forty years old, and as yet no complete solution is available. In this paper, we begin by surveying several known results, and then we present some new results that provide ‘almost’ necessary and sufficient conditions for the existence of a HMM for a mixing and ultra-mixing process (where the notion of ultra-mixing is introduced here). In the survey part of the paper, consisting of Sects. 2 through 8, we rederive the following known results: (i) Associate an infinite matrix H with the process, and call it a ‘Hankel’ matrix (because of some superficial similarity to a Hankel matrix). Then the process has a HMM realization only if H has finite rank. (ii) However, the finite Hankel rank condition is not sufficient in general. There exist processes with finite Hankel rank that do not admit a HMM realization. (iii) An abstract necessary and sufficient condition states that a frequency distribution has a realization as an HMM if and only if it belongs to a ‘stable polyhedral’ convex set within the set of all frequency distributions on \({\mathcal{M}^{*}}\) , the set of all finite strings over \({\mathcal{M}}\) . While this condition may be ‘necessary and sufficient,’ it virtually amounts to a restatement of the problem rather than a solution of it, as observed by Anderson (Math Control Signals Syst 12(1):80–120, 1999). (iv) Suppose a process has finite Hankel rank, say r. Then there always exists a ‘regular quasi-realization’ of the process. That is, there exist a row vector, a column vector, and a set of matrices, each of dimension r or r × r as appropriate, such that the frequency of arbitrary strings is given by a formula that is similar to the corresponding formula for HMM’s. Moreover, all quasi-regular realizations of the process can be obtained from one of them via a similarity transformation. Hence, given a finite Hankel-rank process, it is a simple matter to determine whether or not it has a regular HMM in the conventional sense, by testing the feasibility of a linear programming problem. (v) If in addition the process is α-mixing, every regular quasi-realization has additional features. Specifically, a matrix associated with the quasi-realization (which plays the role of the state transition matrix in a HMM) is ‘quasi-row stochastic’ (in that its rows add up to one, even though the matrix may not be nonnegative), and it also satisfies the ‘quasi-strong Perron property’ (its spectral radius is one, the spectral radius is a simple eigenvalue, and there are no other eigenvalues on the unit circle). A corollary is that if a finite Hankel rank α-mixing process has a regular HMM in the conventional sense, then the associated Markov chain is irreducible and aperiodic. While this last result is not surprising, it does not seem to have been stated explicitly. While the above results are all ‘known,’ they are scattered over the literature; moreover, the presentation here is unified and occasionally consists of relatively simpler proofs than are found in the literature. Next we move on to present some new results. The key is the introduction of a property called ‘ultra-mixing.’ The following results are established: (a) Suppose a process has finite Hankel rank, is both α-mixing as well as ‘ultra-mixing,’ and in addition satisfies a technical condition. Then it has an irreducible HMM realization (and not just a quasi-realization). Moreover, the Markov process underlying the HMM is either aperiodic (and is thus α-mixing), or else satisfies a ‘consistency condition.’ (b) In the other direction, suppose a HMM satisfies the consistency condition plus another technical condition. Then the associated output process has finite Hankel rank, is α-mixing and is also ultra-mixing. Moreover, it is shown that under a natural topology on the set of HMMs, both ‘technical’ conditions are indeed satisfied by an open dense set of HMMs. Taken together, these two results show that, modulo two technical conditions, the finite Hankel rank condition, α-mixing, and ultra-mixing are ‘almost’ necessary and sufficient for a process to have an irreducible and aperiodic HMM.