Skip to main content
SpringerLink
Log in

Markov and Time-Homogeneity Properties in Dempster-Shafer Random Walks

  • Conference paper
  • First Online:
Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2022)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1601))

  • 543 Accesses

Abstract

We generalize discrete-time finite-horizon random walks in order to deal with ambiguity or mis-specification inside the framework of belief functions. Referring to the product rule of conditioning for belief functions, we propose suitable definitions of Markov and time-homogeneity properties. Moreover, we investigate to what extent such properties can be constrained by means of one-step time-homogeneity.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Çinlar, E.: Introduction to Stochastic Processes. Prentice-Hall (1975)

    Google Scholar 

  2. Coletti, G., Petturiti, D., Vantaggi, B.: Conditional belief functions as lower envelopes of conditional probabilities in a finite setting. Inf. Sci. 339, 64–84 (2016)

    Article  MathSciNet  Google Scholar 

  3. Coletti, G., Petturiti, D., Vantaggi, B.: A Dutch book coherence condition for conditional completely alternating Choquet expectations. Bollet. dell’Unione Mate. Italiana 13(4), 585–593 (2020). https://doi.org/10.1007/s40574-020-00251-8

    Article  MathSciNet  MATH  Google Scholar 

  4. De Bock, J., de Cooman, G.: Imprecise Bernoulli processes. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012. CCIS, vol. 299, pp. 400–409. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31718-7_42

    Chapter  Google Scholar 

  5. de Cooman, G., De Bock, J., Lopatatzidis, S.: Imprecise stochastic processes in discrete time: global models, imprecise Markov chains, and ergodic theorems. Int. J. Approx. Reason. 76, 18–46 (2016)

    Article  MathSciNet  Google Scholar 

  6. de Cooman, G., Hermans, F., Quaeghebeur, E.: Imprecise Markov chains and their limit behaviour. Probab. Eng. Inf. Sci. 23(4), 597–635 (2009)

    Article  Google Scholar 

  7. Dempster, A.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38(2), 325–339 (1967)

    Article  MathSciNet  Google Scholar 

  8. Etner, J., Jeleva, M., Tallon, J.M.: Decision theory under ambiguity. J. Econ. Surv. 26(2), 234–270 (2012)

    Article  Google Scholar 

  9. Feng, D., Nguyen, H.: Choquet weak convergence of capacity functionals of random sets. Inf. Sci. 177(16), 3239–3250 (2007)

    Article  MathSciNet  Google Scholar 

  10. Gilboa, I., Marinacci, M.: Ambiguity and the Bayesian paradigm. In: Arló-Costa, H., Hendricks, V.F., van Benthem, J. (eds.) Readings in Formal Epistemology. SGTP, vol. 1, pp. 385–439. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-20451-2_21

  11. Set Functions, Games and Capacities in Decision Making. TDLC, vol. 46. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30690-2_7

  12. Hartfiel, D., Seneta, E.: On the theory of Markov set-chains. Adv. Appl. Probab. 26(4), 947–964 (1994)

    Article  MathSciNet  Google Scholar 

  13. Krak, T., T’Joens, N., de Bock, J.: Hitting times and probabilities for imprecise Markov chains. Proc. Mach. Learn. Res. 103, 265–275 (2019)

    MathSciNet  Google Scholar 

  14. Nendel, M.: On nonlinear expectations and Markov chains under model uncertainty. Int. J. Approx. Reason. 130, 226–245 (2021)

    Article  MathSciNet  Google Scholar 

  15. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)

    Book  Google Scholar 

  16. Škulj, D.: Discrete time Markov chains with interval probabilities. Int. J. Approx. Reason. 50(8), 1314–1329 (2009)

    Article  MathSciNet  Google Scholar 

  17. Škulj, D.: Random walks on graphs with interval weights and precise marginals. Int. J. Approx. Reason. 73, 76–86 (2016)

    Article  MathSciNet  Google Scholar 

  18. Suppes, P., Zanotti, M.: On using random relations to generate upper and lower probabilities. Synthese 36(4), 427–440 (1977)

    Article  MathSciNet  Google Scholar 

  19. T’Joens, N., De Bock, J., de Cooman, G.: A particular upper expectation as global belief model for discrete-time finite-state uncertain processes. Int. J. Approx. Reason. 131, 30–55 (2021)

    Article  MathSciNet  Google Scholar 

  20. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)

    Book  Google Scholar 

  21. Williams, P.: Notes on conditional previsions. Int. J. Approx. Reason. 44(3), 366–383 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The second author was partially supported by Fondazione Cassa di Risparmio di Perugia (grant n. 2018.0427).

Author information

Authors and Affiliations

  1. Department of MEMOTEF, “La Sapienza” University of Rome, Rome, Italy

    Andrea Cinfrignini & Barbara Vantaggi

  2. Department of Economics, University of Perugia, Perugia, Italy

    Davide Petturiti

Authors
  1. Andrea Cinfrignini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Davide Petturiti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Barbara Vantaggi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davide Petturiti .

Editor information

Editors and Affiliations

  1. University of Milano-Bicocca, Milan, Italy

    Davide Ciucci

  2. University of Oviedo, Oviedo, Spain

    Inés Couso

  3. University of Cádiz, Cádiz, Spain

    Jesús Medina

  4. University of Warsaw, Warsaw, Poland

    Dominik Ślęzak

  5. University of Perugia, Perugia, Italy

    Davide Petturiti

  6. Sorbonne Université, Paris, France

    Bernadette Bouchon-Meunier

  7. Iona College, New Rochelle, NY, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cinfrignini, A., Petturiti, D., Vantaggi, B. (2022). Markov and Time-Homogeneity Properties in Dempster-Shafer Random Walks. In: Ciucci, D., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. Communications in Computer and Information Science, vol 1601. Springer, Cham. https://doi.org/10.1007/978-3-031-08971-8_63

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-08971-8_63

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08970-1

  • Online ISBN: 978-3-031-08971-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation