Existence of infinite Viterbi path for pairwise Markov models

Lember, Jüri; Sova, Joonas

doi:10.48550/arxiv.1708.03799

Cited by 1 publication

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“…In other words, for every n big enough, there exists at least one Viterbi path so that v(x 1:n ) 1:t = v 1:t . For the definition of infinite Viterbi path in the case Y is infinite, see [6,28].…”

Section: Infinite Viterbi Pathmentioning

confidence: 99%

“…As we shall see, for many PMM's the infinite Viterbi path exists for almost every realization of X. However, it is possible to construct a model where almost no realization has an infinite Viterbi path (see [28,Example 1.1]).…”

Section: Infinite Viterbi Pathmentioning

confidence: 99%

“…It has been recently proven [28] that under some conditions the infinite Viterbi path exists for almost every realization x 1:∞ of X, allowing to define an infinite Viterbi decoding of X, called the Viterbi process. This was done trough construction of barriers.…”

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

“…Having infinitely many barriers is not necessary for existence of infinite Viterbi path (see [28,Example 1.2]), but the barrier-construction has several advantages. One of them is that it allows to construct the infinite path piecewise, meaning that to determine the first k elements v 1:k of the infinite path it suffices to observe x 1:n for n big enough.…”

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

“…In Subsection 1.3, the segmentation problem, infinite Viterbi path, barriers and many other concepts are introduced and defined. In Subsection 1.4 we state the two barrier construction theorems from [28] (Theorem 1.1 concerning HMM and Theorem 1.2 concerning general PMM) as well as introduce some general Markov chain terminology. In Section 2 we prove our main regeneration-theorem.…”

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

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Regenerativity of Viterbi Process for Pairwise Markov Models

Lember¹,

Sova²

2020

J Theor Probab

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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.

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Section: Infinite Viterbi Pathmentioning

confidence: 99%

Section: Infinite Viterbi Pathmentioning

confidence: 99%

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

Section: Introduction and Preliminaries 1introductionmentioning

confidence: 99%

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