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Introduction to Imprecise Probabilities

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Optimization Under Uncertainty with Applications to Aerospace Engineering

Abstract

Since uncertainty is persistent in engineering analyses, this chapter aimed to introduce methods to describe and reason with under uncertainty in various scenarios. Probability theory is the most widely used methodology for uncertainty quantification for a long time and has proven to be a powerful tool for this task. Nevertheless, the construction of stochastic models relies on very fine information, such as large amount of observations, which is not always available. Without it, the constructed models are only very rough approximations of the real laws and may cause incorrect decisions. In this chapter, we introduce other types of models, based on the theory of imprecise probability, which we are able to construct and reason with under situations with limited available knowledge.

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References

  1. D.A. Alvarez, On the calculation of the bounds of probability of events using infinite random sets. Int. J. Approx. Reason. 43 241–267 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Augustin, F.P.A. Coolen, Nonparametric predictive inference and interval probability. J. Stat. Plan. Inference 124, 251–272 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Augustin et al. (eds.), Introduction to Imprecise Probabilities (Wiley, New York, 2014), p. 432

    Google Scholar 

  4. M.S. Balch, Methods for rigorous uncertainty quantification with application to a Mars atmosphere model. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, 2010

    Google Scholar 

  5. M. Beer, S. Ferson, V. Kreinovich, Imprecise probabilities in engineering analyses. Mech. Syst. Signal Process. 37, 4–29 (2013)

    Article  ADS  Google Scholar 

  6. J.O. Berger et al., An overview of robust bayesian analysis. Test 3, 5–124 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. K.P.S. Bhaskara Rao, M.B. Rao, Theory of Charges: A Study of Finitely Additive Measures. Pure and Applied Mathematics (Elsevier Science, Amsterdam, 1983)

    MATH  Google Scholar 

  8. G. Boole, An Investigation of the Laws of Thought: On Which Are Founded Mathematical Theories of Logic and Probabilities (Dover, New York, 1854)

    MATH  Google Scholar 

  9. G. Casella, R.L. Berger, Statistical Inference (Thomson Learning, Pacific Grove, 2002, 2010)

    Google Scholar 

  10. G. Choquet, Theory of capacities: research on modern potential theory and Dirichlet problem. Technical note, University of Kansas, Dept. of Mathematics, 1954

    Google Scholar 

  11. F.P.A. Coolen, Low structure imprecise predictive inference for Bayes’ problem. Stat. Probab. Lett. 36, 349–357 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. G. de Cooman, Possibility theory I: the measure- and integral-theoretic groundwork. Int. J. Gen. Syst. 25, 291–323 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. G. de Cooman, M.C.M. Troffaes, E. Miranda, A unifying approach to integration for bounded positive charges. J. Math. Anal. Appl. 340, 982–999 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. B. de Finetti, La Prévision: Ses Lois Logiques, Ses Sources Subjectives. Ann. Inst. Henri Poincaré 17, 1–68 (1937)

    MATH  Google Scholar 

  15. B. de Finetti, Theory of Probability: A Critical Introductory Treatment (Wiley, New York, 2017)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. A.P. Dempster, New methods for reasoning towards posterior distributions based on sample data. Ann. Math. Stat. 37, 355–374 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. A.P. Dempster, The Dempster–Shafer calculus for statisticians. Int. J. Approx. Reason. 48, 365–377 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Denneberg, Non-Additive Measure and Integral (Springer, Dordrecht, 1994)

    Book  MATH  Google Scholar 

  20. S. Ferson et al., Constructing probability boxes and Dempster-Shafer structures. Tech. rep., Sandia National Laboratories, 2003

    Google Scholar 

  21. S. Ferson et al., Dependence in probabilistic modeling, Dempster–Shafer theory, and probability bounds analysis. Tech. rep., Sandia National Laboratories, 2004

    Google Scholar 

  22. T. Fetz, M. Oberguggenberger, Propagation of uncertainty through multivariate functions in the framework of sets of probability measures. Reliab. Eng. Syst. Saf. 85, 73–87 (2004)

    Article  Google Scholar 

  23. R.A. Fisher, Inverse probability. Math. Proc. Camb. Philos. Soc. 26, 528–535 (1930)

    Article  ADS  MATH  Google Scholar 

  24. D.A.S. Fraser, The Structure of Inference (Wiley, New York, 1968)

    MATH  Google Scholar 

  25. P.J. Huber, E.M. Ronchetti, Robust Statistics (Wiley, New York, 2009)

    Book  MATH  Google Scholar 

  26. E.T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, Cambridge, 2003)

    Book  MATH  Google Scholar 

  27. A.N. Kolmogorov, Foundations of the Theory of Probability (AMS Chelsea Publication, New York, 1956)

    MATH  Google Scholar 

  28. D.P. Kroese, T. Taimre, Z.I. Botev, Handbook of Monte Carlo Methods (Wiley, New York, 2011)

    Book  MATH  Google Scholar 

  29. R. Martin, C. Liu, Inferential Models: Reasoning with Uncertainty (Chapman and Hall/CRC, Boca Raton, 2015)

    Book  MATH  Google Scholar 

  30. G. Matheron, Random Sets and Integral Geometry (Wiley, New York, 1975)

    MATH  Google Scholar 

  31. I. Molchanov, Theory of Random Sets (Springer, London, 2005)

    MATH  Google Scholar 

  32. R.F. Nau, De Finetti was right: probability does not exist. Theory Decis. 51, 89–124 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  34. M. Oberguggenberger, W. Fellin, Reliability bounds through random sets: non-parametric methods and geotechnical applications. Comput. Struct. 86, 1093–1101 (2008)

    Article  Google Scholar 

  35. L.J. Savage, The Foundations of Statistics (Dover, New York, 2012)

    Google Scholar 

  36. J.G. Saw, M.C.K. Yang, T.C. Mo, Chebyshev inequality with estimated mean and variance. Am. Stat. 38, 130–132 (1984)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  38. M. Troffaes, Optimality, uncertainty, and dynamic programming with lower previsions, Jan 2005

    Google Scholar 

  39. M.C.M. Troffaes, G. de Cooman, Lower Previsions (Wiley, New York, 2014), pp. 37–75

    MATH  Google Scholar 

  40. P. Walley, Statistical Reasoning with Imprecise Probabilities. Chapman & Hall/CRC Monographs on Statistics & Applied Probability (Taylor & Francis, London, 1991)

    Google Scholar 

  41. P. Walley, Towards a unified theory of imprecise probability. Int. J. Approx. Reason. 24, 125–148 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. P. Walley, T.L. Fine, Towards a frequentist theory of upper and lower probability. Ann. Stat. 10, 741–761 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Weichselberger, The theory of interval-probability as a unifying concept for uncertainty. Int. J. Approx. Reason. 24, 149–170 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  44. P. Whittle, Probability via Expectation (Springer, New York, 1992)

    Book  MATH  Google Scholar 

  45. P.M. Williams, Indeterminate probabilities, in Formal Methods in the Methodology of Empirical Sciences: Proceedings of the Conference for Formal Methods in the Methodology of Empirical Sciences, Warsaw, 17–21 June 1974, ed. by M. Przełecki et al. (Springer, Dordrecht, 1976), pp. 229–246

    Chapter  Google Scholar 

  46. P.M. Williams, Notes on conditional previsions. Int. J. Approx. Reason. 44, 366–383 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. M.g. Xie, K. Singh, Confidence distribution, the frequentist distribution estimator of a parameter: a review. Int. Stat. Rev. 81, 3–39 (2013)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Durham University, Durham, UK

    Daniel Krpelík & Tathagata Basu

  2. Department of Applied Mathematics, VŠB - Technical University of Ostrava, Ostrava, Czechia

    Daniel Krpelík

Authors
  1. Daniel Krpelík
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  2. Tathagata Basu
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Corresponding author

Correspondence to Daniel Krpelík .

Editor information

Editors and Affiliations

  1. University of Strathclyde, Glasgow, UK

    Massimiliano Vasile

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Krpelík, D., Basu, T. (2021). Introduction to Imprecise Probabilities. In: Vasile, M. (eds) Optimization Under Uncertainty with Applications to Aerospace Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-60166-9_2

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