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IMPRECISE MARKOV CHAINS AND THEIR LIMIT BEHAVIOR

Published online by Cambridge University Press:  04 August 2009

Gert de Cooman ,
Filip Hermans  and
Erik Quaeghebeur
Gert de Cooman
Affiliation:
SYSTeMS Research Group, Ghent University, Technologiepark–Zwijnaarde 914, 9052 Zwijnaarde, Belgium E-mail: gert.decooman@ugent.be; filip.hermans@ugent.be; erik.quaeghebeur@ugent.be
Filip Hermans
Affiliation:
SYSTeMS Research Group, Ghent University, Technologiepark–Zwijnaarde 914, 9052 Zwijnaarde, Belgium E-mail: gert.decooman@ugent.be; filip.hermans@ugent.be; erik.quaeghebeur@ugent.be
Erik Quaeghebeur
Affiliation:
SYSTeMS Research Group, Ghent University, Technologiepark–Zwijnaarde 914, 9052 Zwijnaarde, Belgium E-mail: gert.decooman@ugent.be; filip.hermans@ugent.be; erik.quaeghebeur@ugent.be
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Abstract

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n→∞: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalization of the classical Perron–Frobenius theorem to imprecise Markov chains.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 23 , Issue 4 , October 2009 , pp. 597 - 635
Copyright
Copyright © Cambridge University Press 2009

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