Identification of Hidden Markov Models Using Spectral Learning with Likelihood Maximization

In this paper, we consider identifying a hidden Markov model (HMM) with the purpose of computing estimates of joint and conditional (posterior) probabilities over observation sequences. The classical maximum likelihood estimation algorithm (via the Baum-Welch/expectation-maximization algorithm), has recently been challenged by methods of moments. Such methods employ low-order moments to provide parameter estimates and have several benefits, including consistency and low computational cost. This paper aims to reduce the gap in statistical efficiency that results from restricting to only low-order moments in the training data. In particular, we propose a two-step procedure that combines spectral learning with a single Newton-like iteration for maximum likelihood estimation. We demonstrate an improved statistical performance using the proposed algorithm in numerical simulations.