On the likelihood function of HMMs for a long data sequence

Hidden Markov models (HMMs) are widely applied to the analysis of time-dependent data sequences, such as nonlinear signal processing, natural language processing, and bioinformatics. Training data in HMMs have two possible formats: a large set of time-dependent sequential data and an infinitely long sequence. The learning process is one of the main concerns in machine learning. For a large set of time-dependent sequential data, the generalization ability can be determined based on algebraic geometry. However, there has been no theoretical analysis for the case of an infinitely long sequence. Therefore, the present paper experimentally determines a number of unique properties of the likelihood function and explains these properties theoretically. The results indicate that the likelihood function implicitly includes a local maximum factor, which can make the learning process slow, and that this slow learning enables high performance in a stationary state evaluation.