Infinite Viterbi alignments in the two state hidden Markov models

TL;DR

This paper proves that for nearly any two-state hidden Markov model, an infinite Viterbi alignment exists, extending previous results that required strong assumptions and providing a more general understanding of Viterbi paths.

Contribution

It establishes the existence of infinite Viterbi alignments in two-state HMMs under very general conditions, broadening the theoretical foundation of Viterbi path analysis.

Findings

Infinite Viterbi alignments exist for almost all two-state HMMs.

Previous strong assumptions are relaxed in this proof.

The result applies broadly to models used in various fields.

Abstract

Since the early days of digital communication, Hidden Markov Models (HMMs) have now been routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. An HMM $(X_i,Y_i)_{i\ge 1}$ assumes observations $X_1,X_2,...$ to be conditionally independent given an "explanotary" Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\em a maximum a posteriori} estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given the observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also called Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments of HMMs with two hidden states when $n$ tends to infinity. It has indeed been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions. This work proves the existence of infinite Viterbi alignments for virtually any HMM with two hidden states.