Advanced search
2 files | 1.07 MB

Imprecise Markov chains and their limit behavior

Gert De Cooman (UGent) , Filip Hermans (UGent) and Erik Quaeghebeur (UGent)
Author
Organization
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 goes to infinity: 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 generalisation of the classical Perron-Frobenius Theorem to imprecise Markov chains.
Keywords
probability tree, credal set, event tree, imprecise Markov chain, Markov chain, non-linear Perron-Frobenius Theorem, sensitivity analysis, stationarity, lower expectation, upper expectation, regularity, DECISION-PROCESSES, TRANSITION-PROBABILITIES, SET-CHAINS

Downloads

  • impmarkov.pdf
    • full text
    • |
    • open access
    • |
    • PDF
    • |
    • 412.97 KB
  • impmarkov-PEIS.pdf
    • full text
    • |
    • open access
    • |
    • PDF
    • |
    • 656.74 KB

Citation

Please use this url to cite or link to this publication:

Chicago
De Cooman, Gert, Filip Hermans, and Erik Quaeghebeur. 2009. “Imprecise Markov Chains and Their Limit Behavior.” Probability in the Engineering and Informational Sciences 23 (4): 597–635.
APA
De Cooman, Gert, Hermans, F., & Quaeghebeur, E. (2009). Imprecise Markov chains and their limit behavior. PROBABILITY IN THE ENGINEERING AND INFORMATIONAL SCIENCES, 23(4), 597–635.
Vancouver
1.
De Cooman G, Hermans F, Quaeghebeur E. Imprecise Markov chains and their limit behavior. PROBABILITY IN THE ENGINEERING AND INFORMATIONAL SCIENCES. 2009;23(4):597–635.
MLA
De Cooman, Gert, Filip Hermans, and Erik Quaeghebeur. “Imprecise Markov Chains and Their Limit Behavior.” PROBABILITY IN THE ENGINEERING AND INFORMATIONAL SCIENCES 23.4 (2009): 597–635. Print.
@article{498502,
  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 goes to infinity: 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 generalisation of the classical Perron-Frobenius Theorem to imprecise Markov chains.},
  author       = {De Cooman, Gert and Hermans, Filip and Quaeghebeur, Erik},
  issn         = {1469-8951},
  journal      = {PROBABILITY IN THE ENGINEERING AND INFORMATIONAL SCIENCES},
  keyword      = {probability tree,credal set,event tree,imprecise Markov chain,Markov chain,non-linear Perron-Frobenius Theorem,sensitivity analysis,stationarity,lower expectation,upper expectation,regularity,DECISION-PROCESSES,TRANSITION-PROBABILITIES,SET-CHAINS},
  language     = {eng},
  number       = {4},
  pages        = {597--635},
  title        = {Imprecise Markov chains and their limit behavior},
  url          = {http://dx.doi.org/10.1017/S0269964809990039},
  volume       = {23},
  year         = {2009},
}

Altmetric
View in Altmetric
Web of Science
Times cited: