A constructive proof of the existence of Viterbi processes

Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM $(X_i,Y_i){i\ge 1}$, observations $X_1,X_2,…$ are assumed to be conditionally independent given an ``explanatory’’ 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} (MAP) estimate $q{1:n}=(q_1,q_2,…,q_n)$ of $Y_{1:n}$ given observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also known as Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments when $n\to \infty$. Thus, it has been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more constructive manner and for a very general class of HMMs.

💡 Research Summary

The paper addresses a fundamental yet under‑explored question in hidden Markov models (HMMs): can the Viterbi algorithm, which yields a maximum‑a‑posteriori (MAP) state sequence for a finite observation window, be extended to an infinite observation horizon in a mathematically rigorous way? Earlier works have shown the existence of infinite Viterbi alignments only under very restrictive conditions—typically two‑state models or transition matrices with strictly positive entries—using existential arguments such as “meeting times” and “meeting states”. Those results do not readily generalize to the more common situation where the state space has three or more elements and the transition matrix may contain zeros.

The authors introduce two central constructs: nodes and barriers. A node is a time point (u) (with an associated order (r)) at which a particular state (l) dominates all alternative continuation paths when the likelihood contributions of the next (r) observations are taken into account. Formally, the dynamic‑programming score (\delta_u(l)) multiplied by the maximal (r)-step transition likelihood (p^{(r)}_{l j}(u)) must be at least as large as the corresponding product for any other state pair ((i,j)). When the inequality is strict the node is called strong; a strong node guarantees that the Viterbi path must pass through state (l) at time (u) regardless of any future observations.

A barrier is a finite block of observations (z_{1:b}) that, when inserted into any longer observation sequence, forces a particular time point to become a node of a prescribed order. The authors prove that for a suitably chosen block length (M) the probability that a randomly generated block of length (M) is a barrier is strictly positive. Because the underlying Markov chain (Y) is ergodic, the Borel‑Cantelli lemma implies that almost every infinite observation realization contains infinitely many barriers (and therefore infinitely many nodes).

With an infinite supply of barriers, the Viterbi alignment can be constructed piecewise. Let the barrier‑induced nodes occur at times (u_1<u_2<\dots). The observation interval (