Maximum likelihood hidden Markov modeling using a dominant sequence of states

Approximate maximum likelihood (ML) hidden Markov modeling using the most likely state sequence (MLSS) is examined and compared with the exact ML approach that considers all possible state sequences. It is shown that for any hidden Markov model (HMM), the difference between the approximate and the exact normalized likelihood functions cannot exceed the logarithm of the number of states divided by the dimension of the output vectors (frame length). Furthermore, for Gaussian HMMs and a given observation sequence, the MLSS is typically the sequence of nearest neighbor states in the Itakura-Saito sense, and the posterior probability of any state sequence which departs from the MLSS in a single time instant, decays exponentially with the frame length. Hence, for a sufficiently large frame length the exact and approximate ML approach provide similar model estimates and likelihood values