Theory of Continuous-State Hidden Markov Models and Hidden Gauss-Markov Models

Abstract

Abstract : A general theory of continuous-state hidden Markov models is developed, with continuous-state analogs of the Baum, Viterbi, and Baum-Welch algorithms formulated for this class of models. The algorithms are specialized to models with linear Gaussian densities, thereby unifying the theory of hidden Markov models and Kalman filters. The Baum and Viterbi algorithms for Gaussian models are shown to be implemented by two different formulations of the fixed-interval Kalman smoother. Moreover, the measurement likelihoods obtained from the forward pass of the Baum algorithm and from the Kalman-filter Innovation sequence are found to be equivalent A direct link between the Baum-Welch algorithm and an existing expectation-maximization algorithm for linear Gaussian models is demonstrated. The general continuous-state and Gaussian models are extended to incorporate mixture densities for the prior probability of the initial state. For the Gaussian models, a new expression for the cross covariance between time adjacent states is derived from the off-diagonal block of the conditional joint covariance matrix and a parameter invariance structure is observed when the system matrices are time invariant.