Convergence of the maximum a posteriori path estimator in hidden Markov models

In a hidden Markov model (HMM) the underlying finite-state Markov chain cannot be observed directly but only by an additional process. We are interested in estimating the unknown path of the Markov chain. The most widely used estimator is the maximum a posteriori path estimator (MAP path estimator). It can be calculated effectively by the Viterbi (1967) algorithm as is, e.g., frequently done in the field of coding theory, correction of intersymbol interference, and speech recognition. We investigate (component-wise) convergence of the MAP path estimator. Convergence is shown under the condition of unbounded likelihood ratios. This condition is satisfied in the important case of HMMs with additive white Gaussian noise. We also prove convergence, if the Markov chain has two states. The so-called Viterbi paths are an important tool for obtaining these results.