Bernoulli

  • Bernoulli
  • Volume 15, Number 3 (2009), 721-735.

Exchangeable lower previsions

Gert de Cooman, Erik Quaeghebeur, and Enrique Miranda

Full-text: Open access

Abstract

We extend de Finetti’s [Ann. Inst. H. Poincaré 7 (1937) 1–68] notion of exchangeability to finite and countable sequences of variables, when a subject’s beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We derive representation theorems in both the finite and countable cases, in terms of sampling without and with replacement, respectively.

Article information

Source
Bernoulli, Volume 15, Number 3 (2009), 721-735.

Dates
First available in Project Euclid: 28 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.bj/1251463278

Digital Object Identifier
doi:10.3150/09-BEJ182

Mathematical Reviews number (MathSciNet)
MR2555196

Zentralblatt MATH identifier
1202.60055

Keywords
Bernstein polynomials coherence convergence in distribution exchangeability imprecise probability lower prevision multinomial sampling representation theorem sampling without replacement

Citation

de Cooman, Gert; Quaeghebeur, Erik; Miranda, Enrique. Exchangeable lower previsions. Bernoulli 15 (2009), no. 3, 721--735. doi:10.3150/09-BEJ182. https://projecteuclid.org/euclid.bj/1251463278


Export citation

References

  • [1] Cifarelli, D.M. and Regazzini, E. (1996). De Finetti’s contributions to probability and statistics. Statist. Sci. 11 253–282.
  • [2] Dawid, A.P. (1985). Probability, symmetry, and frequency. British J. Philos. Sci. 36 107–128.
  • [3] de Cooman, G. and Miranda, E. (2008). Weak and strong laws of large numbers for coherent lower previsions. J. Statist. Plann. Inference 138 2409–2432.
  • [4] de Cooman, G. and Miranda, E. (2007). Symmetry of models versus models of symmetry. In Probability and Inference: Essays in Honor of Henry E. Kyburg, Jr. (W.L. Harper and G.R. Wheeler, eds.) 67–149. London: King’s College Publications.
  • [5] de Finetti, B. (1937). La prévision: Ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7 1–68. (English translation in [16].)
  • [6] de Finetti, B. (1970). Teoria delle Probabilità. Turin: Einaudi.
  • [7] de Finetti, B. (1974–1975). Theory of Probability: A Critical Introductory Treatment. Chichester: Wiley. (English translation of [6], two volumes.)
  • [8] Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences. Ann. Probab. 8 745–764.
  • [9] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, II. New York: Wiley.
  • [10] Heath, D.C. and Sudderth, W.D. (1976). De Finetti’s theorem on exchangeable variables. Amer. Statist. 30 188–189.
  • [11] Heitzinger, C., Hössinger, A. and Selberherr, S. (2003). On smoothing three-dimensional Monte Carlo ion implantation simulation results. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 22 879–883.
  • [12] Hewitt, E. and Savage, L.J. (1955). Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80 470–501.
  • [13] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. New York: Wiley.
  • [14] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. New York: Springer.
  • [15] Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. New York: Springer.
  • [16] Kyburg, H.E., Jr. and Smokler, H.E., eds. (1964). Studies in Subjective Probability, 2nd ed. New York: Wiley.
  • [17] Prautzsch, H., Boehm, W. and Paluszny, M. (2002). Bézier and B-Spline Techniques. Berlin: Springer.
  • [18] Trump, W. and Prautzsch, H. (1996). Arbitrary degree elevation of Bézier representations. Comput. Aided Geom. Design 13 387–398.
  • [19] Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall.