Regenerativity of Viterbi process for pairwise Markov models

TL;DR

This paper proves the regenerativity of the Viterbi process in pairwise Markov models, extending the understanding of infinite Viterbi decoding and its implications for training algorithms.

Contribution

It establishes the regenerative property of the joint process of the Viterbi path and PMM, providing a foundation for analyzing asymptotics of Viterbi training.

Findings

Viterbi process can be extended infinitely under certain conditions

Joint process of Viterbi path and PMM is regenerative

Results on asymptotics of Viterbi training algorithm derived

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 trough 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.