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14-18 August 2017
Uppsala University
Europe/Stockholm timezone

Viterbi process for pairwise Markov models

17 Aug 2017, 10:00
30m
Ångströmslaboratoriet (Uppsala University)

Ångströmslaboratoriet

Uppsala University

Speaker

Speaker

Mr Joonas Sova (University of Tartu)

Description

My talk is based on ongoing joint work with my supervisor Jüri Lember. We consider a Markov chain $Z = \{Z_k\}_{k \geq 1}$ with product state space $\mathcal{X}\times \mathcal{Y}$, where $\mathcal{Y}$ is a finite set (state space) and $\mathcal{X}$ is an arbitrary separable metric space (observation space). Thus, the process $Z$ decomposes as $Z=(X,Y)$, where $X=\{X_k \}_{k\geq 1}$ and $Y=\{Y_k \}_{k\geq 1}$ are random processes taking values in $\mathcal{X}$ and $\mathcal{Y}$, respectively. Following cite{pairwise,pairwise2,pairwise3}, we call the process $Z$ a \textit{pairwise Markov model}. The process $X$ is identified as an observation process and the process $Y$, sometimes called the \textit{regime}, models the observations-driving hidden state sequence. Therefore our general model contains many well-known stochastic models as a special case: hidden Markov models, Markov switching models, hidden Markov models with dependent noise and many more. The \textit{segmentation} or \textit{path estimation} problem consists of estimating the realization of $(Y_1,\ldots,Y_n)$ given a realization $x_{1:n}$ of $(X_1,\ldots,X_n)$. A standard estimate is any path $v_{1:n}\in \mathcal{Y}^n$ having maximum posterior probability: $$v_{1:n}=\mathop{\mathrm{argmax}}_{y_{1:n}}P(Y_{1:n}=y_{1:n}|X_{1:n}=x_{1:n}).$$ Any such path is called \textit{Viterbi path} and we are interested in the behaviour of $v_{1:n}$ as $n$ grows. The study of asymptotics of Viterbi path is complicated by the fact that adding one more observation, $x_{n+1}$ can change the whole path, and so it is not clear, whether there exists a limiting infinite Viterbi path. We show that under some conditions the infinite Viterbi path indeed exists for almost every realization $x_{1:\infty}$ of $X$, thereby defining an infinite Viterbi decoding of $X$, called the \textit{Viterbi process.} This is done trough construction of \textit{barriers}. A barrier is a fixed-sized block in the observations $x_{1:n}$ that fixes the Viterbi path up to itself: for every continuation of $x_{1:n}$, the Viterbi path up to the barrier remains unchanged. Therefore, if almost every realization of $X$-process contains infinitely many barriers, then the Viterbi process exists. Having infinitely many barriers is not necessary for existence of infinite Viterbi path, but the barrier-construction has several advantages. One of them is that it allows to construct the infinite path \textit{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. Barrier construction has another great advantage: namely, the process $(Z,V)=\{(Z_k,V_k)\}_{k \geq 1}$, where $V= \{V_k\}_{k \geq 1}$ denotes the Viterbi process, is under certain conditions regenerative. This is can be proven by, roughly speaking, applying the Markov splitting method to construct regeneration times for $Z$ which coincide with the occurrences of barriers. Regenerativity of $(Z,V)$ allows to easily prove limit theorems to understand the asymptotic behaviour of inferences based on Viterbi paths. In fact, in a special case of hidden Markov model this regenerative property has already been known to hold and has found several applications cite{AV,AVacta,Vsmoothing,Vrisk, iowa}.

Primary author

Mr Joonas Sova (University of Tartu)

Presentation Materials

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