Lower bounds for moments of global scores of pairwise Markov chains

Lember, Jüri; Matzinger, Heinrich; Sova, Joonas; Zucca, Fabio

doi:10.1016/j.spa.2017.08.009

Cited by 5 publications

(4 citation statements)

References 33 publications

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“…Within that framework, easy modifications of the tools in our approach also lead to a central limit theorem. Even more general linear lower bounds are obtained in various settings in [13] and [20] and the many references therein.…”

Section: Remark 11 (I)mentioning

confidence: 99%

On the Order of the Central Moments of the Length of the Longest Common Subsequences in Random Words

Houdré

2016

High Dimensional Probability VII

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Abstract. We investigate the order of the r-th, 1 ≤ r < +∞, central moment of the length of the longest common subsequence of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, a lower bound is shown to be of order n r/2 . This result complements a generic upper bound also of order n r/2 .

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Section: Remark 11 (I)mentioning

confidence: 99%

On the Order of the Central Moments of the Length of the Longest Common Subsequences in Random Words

Houdré

2016

High Dimensional Probability VII

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“…The PMMs with transition matrix (8) were introduced in [17] to model dependencies between two-state Markov chains, see also [18]. The model can be generalized to the case with bigger alphabets, for such a generalization with P X = P Y , see [21].…”

Section: The Related Markov Chains Modelmentioning

confidence: 99%

Hybrid classifiers of pairwise Markov models

Kuljus¹,

Lember²

2022

Preprint

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The article studies segmentation problem (also known as classification problem) with pairwise Markov models (PMMs). A PMM is a process where the observation process and underlying state sequence form a two-dimensional Markov chain, it is a natural generalization of a hidden Markov model. To demonstrate the richness of the class of PMMs, we examine closer a few examples of rather different types of PMMs: a model for two related Markov chains, a model that allows to model an inhomogeneous Markov chain as a homogeneous one and a semi-Markov model. The segmentation problem assumes that one of the marginal processes is observed and the other one is not, the problem is to estimate the unobserved state path given the observations. The standard state path estimators often used are the so-called Viterbi path (a sequence with maximum state path probability given the observations) or the pointwise maximum a posteriori (PMAP) path (a sequence that maximizes the conditional state probability for given observations pointwise). Both these estimators have their limitations, therefore we derive formulas for calculating the so-called hybrid path estimators which interpolate between the PMAP and Viterbi path. We apply the introduced algorithms to the studied models in order to demonstrate the properties of different segmentation methods, and to illustrate large variation in behaviour of different segmentation methods in different PMMs. The studied examples show that a segmentation method should always be chosen with care by taking into account the particular model of interest.

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“…In fact, depending on , n and N , the set (21) might well be empty. But in many instances (21) does have a positive µ 2N ·(n−1)+1 -measure, regardless of the choice of N . The following example demonstrates this.…”

Section: Barrier Set Construction Theoremmentioning

confidence: 99%

“…Let X = Y = {1, 2}, and assume that X and Y are Markov chains both having the transition matrix p 1 − p q 1 − q , where p, q ∈ (0, 1). Then, as is shown in[21], the transition matrix of Z…”

mentioning

confidence: 99%

Existence of infinite Viterbi path for pairwise Markov models

Lember

Sova

2020

Stochastic Processes and their Applications

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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, 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.

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Lower bounds for moments of global scores of pairwise Markov chains

Cited by 5 publications

References 33 publications

On the Order of the Central Moments of the Length of the Longest Common Subsequences in Random Words

On the Order of the Central Moments of the Length of the Longest Common Subsequences in Random Words

Hybrid classifiers of pairwise Markov models

Existence of infinite Viterbi path for pairwise Markov models

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