Existence of infinite Viterbi path for pairwise Markov models

TL;DR

This paper proves that under certain conditions, an infinite Viterbi path exists for pairwise Markov models, allowing for an infinite Viterbi decoding of observation sequences, extending the concept beyond traditional hidden Markov models.

Contribution

It establishes the existence of an infinite Viterbi path for pairwise Markov models, generalizing the decoding process for more complex Markovian systems.

Findings

Infinite Viterbi path exists under certain conditions.

Construction of barriers ensures Viterbi path passes through specific states.

Enables infinite Viterbi decoding for pairwise Markov models.

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, where the joint process consisting of finite-state hidden regime and observation process is assumed to be a Markov chain. We prove that under some conditions it is possible to extend the Viterbi path to infinity for almost every observation sequence which in turn enables to define an infinite Viterbi decoding of the observation process, called the Viterbi process. This is 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.