Pairwise Markov models
Date
2021-07-12
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Abstract
Varjatud 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.
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.
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Keywords
mathematical statistics, Markov chains, sequences (math.)