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.Comment: Submitted to the IEEE Transactions on Information Theory, focuses on the proofs of the results presented in arXiv:0709.2317, and arXiv:0803.239

On the accuracy of the Viterbi alignment

In a hidden Markov model, the underlying Markov chain is usually hidden. Often, the maximum likelihood alignment (Viterbi alignment) is used as its estimate. Although having the biggest likelihood, the Viterbi alignment can behave very untypically by passing states that are at most unexpected. To avoid such situations, the Viterbi alignment can be modified by forcing it not to pass these states. In this article, an iterative procedure for improving the Viterbi alignment is proposed and studied. The iterative approach is compared with a simple bunch approach where a number of states with low probability are all replaced at the same time. It can be seen that the iterative way of adjusting the Viterbi alignment is more efficient and it has several advantages over the bunch approach. The same iterative algorithm for improving the Viterbi alignment can be used in the case of peeping, that is when it is possible to reveal hidden states. In addition, lower bounds for classification probabilities of the Viterbi alignment under different conditions on the model parameters are studied

A generalized risk approach to path inference based on hidden Markov models

Motivated by the unceasing interest in hidden Markov models (HMMs), this paper re-examines hidden path inference in these models, using primarily a risk-based framework. While the most common maximum a posteriori (MAP), or Viterbi, path estimator and the minimum error, or Posterior Decoder (PD), have long been around, other path estimators, or decoders, have been either only hinted at or applied more recently and in dedicated applications generally unfamiliar to the statistical learning community. Over a decade ago, however, a family of algorithmically defined decoders aiming to hybridize the two standard ones was proposed (Brushe et al., 1998). The present paper gives a careful analysis of this hybridization approach, identifies several problems and issues with it and other previously proposed approaches, and proposes practical resolutions of those. Furthermore, simple modifications of the classical criteria for hidden path recognition are shown to lead to a new class of decoders. Dynamic programming algorithms to compute these decoders in the usual forward-backward manner are presented. A particularly interesting subclass of such estimators can be also viewed as hybrids of the MAP and PD estimators. Similar to previously proposed MAP-PD hybrids, the new class is parameterized by a small number of tunable parameters. Unlike their algorithmic predecessors, the new risk-based decoders are more clearly interpretable, and, most importantly, work "out of the box" in practice, which is demonstrated on some real bioinformatics tasks and data. Some further generalizations and applications are discussed in conclusion.Comment: Section 5: corrected denominators of the scaled beta variables (pp. 27-30), => corrections in claims 1, 3, Prop. 12, bottom of Table 1. Decoder (49), Corol. 14 are generalized to handle 0 probabilities. Notation is more closely aligned with (Bishop, 2006). Details are inserted in eqn-s (43); the positivity assumption in Prop. 11 is explicit. Fixed typing errors in equation (41), Example

On the Viterbi process with continuous state space

This paper deals with convergence of the maximum a posterior probability path estimator in hidden Markov models. We show that when the state space of the hidden process is continuous, the optimal path may stabilize in a way which is essentially different from the previously considered finite-state setting.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ294 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Paariviisi Markovi ahelad

Väitekirja elektrooniline versioon ei sisalda publikatsiooneVarjatud muutujatega Markovi mudelid on kaasaegse statistika suur edulugu. Tänapäeval on üha suurem vajadus analüüsida keerulise struktuuriga andmeid, mis ei järgi klassikalise statistika eeldusi, nagu valimi liikmete sõltumatus ja sama jaotus. Teisalt varjatud muutujatega Markovi mudelid võimaldavad rakendada erinevaid kergesti kohandatavaid meetodeid analüüsimaks keerulisi omavahel sõltuvaid andmeid. Käesolev doktoritöö uurib laia selliste mudelite klassi nimega „paariviisi Markovi mudelid“ (PMM). PMM on lihtsalt mistahes varjatud muutujatega mudel, mille puhul varjatud ehk latentne kiht ning vaatluste kiht koos moodustavad Markovi ahela. PMM hõlmab väga laia mudelite hulka, kuid enimtuntud ja praktikas kõige rohkem rakendatud on kindlasti varjatud Markovi mudel. Viimase näol on tegemist PMM-i erijuhuga, mille puhul vaatlused sõltuvad üksteisest ainult läbi mudeli varjatud kihi. Käesolev doktoritöö annab ülevaate kolmest artiklist, mis kõik käsitlevad PMM-ide teatud aspekte. Esimeses artiklis uurime mudeli varjatud kihi suurima tõepära hinnangu – nn Viterbi joonduse – asümptootilist käitumist. Täpsemalt näitame, et teatud tingimustel on võimalik Viterbi joonduse laiendamine lõpmatusse. Teises artiklis uurime täpsemalt lõpmatu Viterbi joonduse tõenäosuslikku käitumist ning muuhulgas näitame, et üldistel tingimustel see järgib suurte arvude seadust. Kolmanda artikli temaatika erineb mõnevõrra kahest eelmisest: uurime PMM-ide silumistõenäosusi ning tõestame, et üldistel tingimustel kehtivad eksponentsiaalsed unustusomadused.Latent variable Markovian models are a great success story of modern statistics. Nowadays there is increasing prevalence of data where the classic assumptions of statistics, such as independence and equal distribution of observations, cannot be assumed. In contrast, the latent variable Markovian models offer a wide range of highly adaptable methodologies for analyzing complex inter-dependent data. This thesis investigates a wide class of such models, namely the “pairwise Markov model” (PMM). PMM is simply any model for which the hidden or latent layer and the observed layer both together constitute a Markov chain. As such, the PMM encompasses a wide range of models, but among them the most notable and most frequently applied is certainly the hidden Markov model. This latter model is a special case of the PMM in which case the observations depend on each other only through the hidden layer. This thesis is an overview of three papers, all of which deal with some aspects of the PMM. In the first paper we study the maximal likelihood estimate of the hidden layer, known as the Viterbi estimate. We show that under certain conditions it is possible to extend this estimate to infinity. In the second paper we study the probabilistic behavior of this infinite Viterbi estimate – among other things we show that the law of large numbers applies. The theme of the third paper differs from the previous two and is centered on the smoothing probabilities of the PMM. In particular, we prove that under general conditions those smoothing probabilities follow exponential forgetting properties.https://www.ester.ee/record=b545039