Regenerativity of Viterbi Process for Pairwise Markov Models

Jüri Lember () and Joonas Sova ()

Journal of Theoretical Probability, 2021, vol. 34, issue 1, 1-33

Abstract: Abstract For hidden Markov models, one of the most popular estimates of the hidden chain is the Viterbi path—the path maximising the posterior probability. We consider a more general setting, called the pairwise Markov model (PMM), where the joint process consisting of finite-state hidden process and observation process is assumed to be a Markov chain. It has been recently proven that under some conditions the Viterbi path of the PMM can almost surely be extended to infinity, thereby defining the infinite Viterbi decoding of the observation sequence, called the Viterbi process. This was done by constructing a block of observations, called a barrier, which ensures that the Viterbi path goes through a given state whenever this block occurs in the observation sequence. In this paper, we prove that the joint process consisting of Viterbi process and PMM is regenerative. The proof involves a delicate construction of regeneration times which coincide with the occurrences of barriers. As one possible application of our theory, some results on the asymptotics of the Viterbi training algorithm are derived.

Keywords: Viterbi path; MAP path; Viterbi training; Viterbi algorithm; Markov switching model; Hidden Markov model; 62M05; 60G35; 60G05 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-01022-z

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