Imprecise Markov chains with absorption
type: Item
instance of: scholarly article
title: Imprecise Markov chains with absorption
zbMATH Open document ID: 1211.60029
published in: International Journal of Approximate Reasoning
publication date: 2011-01-31T00:00:00Z
review text: The authors consider convergence of Markov chains with uncertain parameters, known as imprecise Markov chains, which contain an absorbing state. The paper is structured in 6 sections. After Introduction, in Section 2 are given the most necessary conditions of the model of imprecise Markov chains used. For a Markov chain \(\chi= \{X(n), n= 0,\dots\}\), where \(X(n)\) represents the value (equivalently, the state) of the process at time \(n\), the set of states is denoted as follows: \[ S=:\{-1,0,\dots,s\}:= \{-1\}\cup C, \] where \(-1\) is an absorbing state, and \(C\) is a finite set of transient states. Let \(P\) be a transition matrix. All potential transition matrices for a given time step belong, by definition, to the set \[ {\mathcal P}:= \{P|{\mathbf p}^{(i)}\in{\mathcal P}^{(i)},\,\forall i\in S\}= \{1,0,\dots, 0\}\times \prod_{i\in C}{\mathcal P}^{(i)}, \] where \({\mathcal P}^{(i)}\) are closed sets of probabilities. Among other conditions it is supposed that for all \(P\in{\mathcal P}\), \(C\) is a single aperiodic communicating class. Any set of initial distributions, denoted in general by \({\mathcal D}_0\), will have the property that \({\mathcal D}_0\subseteq{\mathcal M}_0\), where \({\mathcal M}_0\) is the notation for the set of all possible probability mass functions. Then, methods for calculating sets of distributions at general steps are given in Section 3. In the next section is presented, for the imprecise case, a method for conditioning on non-absorption. Section 5 deals with convergence to equilibrium. The authors prove that as time \(n\) approaches infinity the set \({\mathcal D}^C_n\) will tend to \({\mathcal M}^C_\infty\) independently of the set \({\mathcal D}^C_0\subseteq{\mathcal M}_0^C\) chosen as the set of initial distributions; it is not necessary to begin with \({\mathcal M}^C_0\). They emphasize that this process requires a distance function between probability distributions, and based on this a distance between sets of probabiliti…
reviewed by: Gabriel V. Orman
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 37A50
zbMATH DE Number: 5843185
zbMATH Keywords: interval probabilities
zbMATH Keywords: Markov chains
zbMATH Keywords: quasi-stationary distributions
author: Richard J. Crossman
author: Damjan Škulj
describes a project that uses: MCQueue
MaRDI profile type: Publication
full work available at URL: j.ijar.2010.08.008
OpenAlex ID: W2010231692
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DOI: 10.1016/J.IJAR.2010.08.008
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description: scientific article; zbMATH DE number 5843185