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

_t
,
Y

_t
)
_{t ¿ 1}
, observations
X

₁
,
X

₂
,... are assumed to be conditionally independent given a Markov process
Y

₁
,
Y

₂
,..., which itself is not observed; moreover, the conditional distribution of
X

_t
depends solely on
Y

_t
. Central to the theory and applications of HMM is the Viterbi algorithm to find a maximum
a posteriori
probability (MAP) estimate
v
(
x

_1:

T
)=(
v

₁
,
v

₂
,...,
vT
) of
Y

_1:

T
given observed data
x

_1:

T
. Maximum
a posteriori
paths are also known as the Viterbi paths, or alignments. Recently, attempts have been made to study behavior of the Viterbi alignments when
T
¿ ¿. Thus, it has been shown that in some 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.