Skip to main content
SpringerLink
Log in

Quantifying Predictive Uncertainty Using Belief Functions: Different Approaches and Practical Construction

  • Conference paper
  • First Online:
Predictive Econometrics and Big Data (TES 2018)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 753))

Included in the following conference series:

  • 2143 Accesses

Abstract

We consider the problem of quantifying prediction uncertainty using the formalism of belief functions. Three requirements for predictive belief functions are reviewed, each one of them inducing a distinct interpretation: compatibility with Bayesian inference, approximation of the true distribution, and frequency calibration. Construction procedures allowing us to build belief functions meeting each of these three requirements are described and illustrated using simple examples.

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 259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover 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. Aregui, A., Denœux, T.: Constructing predictive belief functions from continuous sample data using confidence bands. In: G. De Cooman, J. Vejnarová, M. Zaffalon (eds.) Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications (ISIPTA 2007), pp. 11–20. Prague, Czech Republic (2007)

    Google Scholar 

  2. Barndorff-Nielsen, O.E., Cox, D.R.: Prediction and asymptotics. Bernoulli 2(4), 319–340 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berger, J.O., Wolpert, R.L.: The likelihood principle: a review, generalizations, and statistical implications. Lecture Notes-Monograph Series, vol. 6, 2nd edn. Institute of Mathematical Statistics, Hayward (1988)

    Google Scholar 

  4. Birnbaum, A.: On the foundations of statistical inference. J. Am. Stat. Assoc. 57(298), 269–306 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  5. Coles, S.G.: An Introduction to Statistical Modelling of Extreme Values. Springer, London (2001)

    Book  MATH  Google Scholar 

  6. Dempster, A.P.: New methods for reasoning towards posterior distributions based on sample data. Ann. Math. Stat. 37, 355–374 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Dempster, A.P.: A generalization of Bayesian inference (with discussion). J. Roy. Stat. Soc. B 30, 205–247 (1968)

    MathSciNet  MATH  Google Scholar 

  9. Dempster, A.P.: Upper and lower probabilities generated by a random closed interval. Ann. Math. Stat. 39(3), 957–966 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  10. Denœux, T.: Constructing belief functions from sample data using multinomial confidence regions. Int. J. Approx. Reason. 42(3), 228–252 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Denœux, T.: Likelihood-based belief function: justification and some extensions to low-quality data. Int. J. Approx. Reason. 55(7), 1535–1547 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dubois, D., Foulloy, L., Mauris, G., Prade, H.: Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities. Reliab. Comput. 10(4), 273–297 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Edwards, A.W.F.: Likelihood, expanded edn. The John Hopkins University Press, Baltimore (1992)

    MATH  Google Scholar 

  14. Kanjanatarakul, O., Sriboonchitta, S., Denœux, T.: Forecasting using belief functions: an application to marketing econometrics. Int. J. Approx. Reason. 55(5), 1113–1128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kanjanatarakul, O., Sriboonchitta, S., Denœux, T.: Statistical estimation and prediction using belief functions: principles and application to some econometric models. Int. J. Approx. Reason. 72, 71–94 (2016)

    Article  MATH  Google Scholar 

  16. Lawless, J.F., Fredette, M.: Frequentist prediction intervals and predictive distribution. Biometrika 92(3), 529–542 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Martin, R., Lingham, R.T.: Prior-free probabilistic prediction of future observations. Technometrics 58(2), 225–235 (2016)

    Article  MathSciNet  Google Scholar 

  18. Martin, R., Liu, C.: Inferential Models: Reasoning with Uncertainty. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  19. Nguyen, H.T.: On random sets and belief functions. J. Math. Anal. Appl. 65, 531–542 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nguyen, H.T.: An Introduction to Random Sets. Chapman and Hall/CRC Press, Boca Raton (2006)

    Book  MATH  Google Scholar 

  21. Olkin, I., Gleser, L., Derman, C.: Probability Models and Applications. Macmillan, New York (1994)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  23. Wasserman, L.A.: Belief functions and statistical evidence. Can. J. Stat. 18(3), 183–196 (1990)

    Article  MATH  Google Scholar 

  24. Wilks, S.S.: The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Stat. 9(1), 60–62 (1938)

    Article  MATH  Google Scholar 

  25. Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhu, K., Thianpaen, N., Kreinovich, V.: How to make plausibility-based forecasting more accurate. In: Kreinovich, V., Sriboonchitta, S., Huynh, V.N. (eds.) Robustness in Econometrics, pp. 99–110. Springer, Berlin, Cham, Switzerland (2017)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sorbonne Universités, Université de Technologie de Compiègne, CNRS, UMR 7253 Heudiasyc, Compiègne, France

    Thierry Denœux

Authors
  1. Thierry Denœux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thierry Denœux .

Editor information

Editors and Affiliations

  1. Computer Science Department, University of Texas at El Paso, El Paso, Texas, USA

    Vladik Kreinovich

  2. International College, Chiang Mai University, Chiang Mai, Thailand

    Songsak Sriboonchitta

  3. International College, Chiang Mai University, Chiang Mai, Thailand

    Nopasit Chakpitak

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Denœux, T. (2018). Quantifying Predictive Uncertainty Using Belief Functions: Different Approaches and Practical Construction. In: Kreinovich, V., Sriboonchitta, S., Chakpitak, N. (eds) Predictive Econometrics and Big Data. TES 2018. Studies in Computational Intelligence, vol 753. Springer, Cham. https://doi.org/10.1007/978-3-319-70942-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-70942-0_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-70941-3

  • Online ISBN: 978-3-319-70942-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation