The Annals of Statistics

Dominating countably many forecasts

M. J. Schervish, Teddy Seidenfeld, and J. B. Kadane

Full-text: Open access

Abstract

We investigate differences between a simple Dominance Principle applied to sums of fair prices for variables and dominance applied to sums of forecasts for variables scored by proper scoring rules. In particular, we consider differences when fair prices and forecasts correspond to finitely additive expectations and dominance is applied with infinitely many prices and/or forecasts.

Article information

Source
Ann. Statist., Volume 42, Number 2 (2014), 728-756.

Dates
First available in Project Euclid: 20 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1400592176

Digital Object Identifier
doi:10.1214/14-AOS1203

Mathematical Reviews number (MathSciNet)
MR3210985

Zentralblatt MATH identifier
1295.62006

Subjects
Primary: 62A01: Foundations and philosophical topics
Secondary: 62C05: General considerations

Keywords
Proper scoring rule coherence conglomerable probability dominance finitely additive probability sure-loss

Citation

Schervish, M. J.; Seidenfeld, Teddy; Kadane, J. B. Dominating countably many forecasts. Ann. Statist. 42 (2014), no. 2, 728--756. doi:10.1214/14-AOS1203. https://projecteuclid.org/euclid.aos/1400592176


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References

  • Berti, P., Regazzini, E. and Rigo, P. (2001). Strong previsions of random elements. Statistical Methods and Applications 10 11–28.
  • Berti, P. and Rigo, P. (1992). Weak disintegrability as a form of preservation of coherence. Journal of the Italian Statistical Society 1 161–181.
  • Berti, P. and Rigo, P. (2000). Integral representation of linear functionals on spaces of unbounded functions. Proc. Amer. Math. Soc. 128 3251–3258.
  • Berti, P. and Rigo, P. (2002). On coherent conditional probabilities and disintegrations. Ann. Math. Artif. Intell. 35 71–82.
  • Crisma, L. and Gigante, P. (2001). A notion of coherent conditional prevision for arbitrary random quantities. Stat. Methods Appl. 10 29–40.
  • Crisma, L., Gigante, P. and Millossovich, P. (1997). A notion of coherent prevision for arbitrary random quantities. Journal of the Italian Statistical Society 6 233–243.
  • de Finetti, B. (1972). Probability, Induction, and Statistics. The Art of Guessing. Wiley, New York.
  • de Finetti, B. (1974). Theory of Probability: A Critical Introductory Treatment, Vol. 1. Wiley, New York.
  • de Finetti, B. (1975). Theory of Probability: A Critical Introductory Treatment, Vol. 2. Wiley, New York.
  • de Finetti, B. (1981). The role of “Dutch Books” and of “proper scoring rules”. British J. Philos. Sci. 32 55–56.
  • Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Probab. 3 89–99.
  • Dunford, N. and Schwartz, J. (1958). Linear Operators. Wiley, New York.
  • Gneiting, T. (2011a). Making and evaluating point forecasts. J. Amer. Statist. Assoc. 106 746–762.
  • Gneiting, T. (2011b). Quantiles as optimal point forecasts. International Journal of Forecasting 27 197–207.
  • Heath, D. and Sudderth, W. (1978). On finitely additive priors, coherence, and extended admissibility. Ann. Statist. 6 333–345.
  • Levi, I. (1980). The Enterprise of Knowledge. MIT Press, Cambridge, MA.
  • Regazzini, E. (1987). de Finetti’s coherence and statistical inference. Ann. Statist. 15 845–864.
  • Royden, H. L. (1963). Real Analysis. Macmillan, New York.
  • Savage, L. J. (1971). Elicitation of personal probabilities and expectations. J. Amer. Statist. Assoc. 66 783–801.
  • Schervish, M. J., Seidenfeld, T. and Kadane, J. B. (1984). The extent of nonconglomerability of finitely additive probabilities. Z. Wahrsch. Verw. Gebiete 66 205–226.
  • Schervish, M., Seidenfeld, T. and Kadane, J. (2008a). On the equivalence of conglomerability and disintegrability for unbounded random variables. Technical Report 864, Carnegie Mellon Univ., Pittsburgh, PA.
  • Schervish, M. J., Seidenfeld, T. and Kadane, J. B. (2008b). The fundamental theorems of prevision and asset pricing. Internat. J. Approx. Reason. 49 148–158.
  • Schervish, M., Seidenfeld, T. and Kadane, J. (2009). Proper scoring rules, dominated forecasts, and coherence. Decis. Anal. 6 202–221.
  • Schervish, M., Seidenfeld, T. and Kadane, J. (2014). Supplement to “Dominating countably many forecasts.” DOI:10.1214/14-AOS1203SUPP.
  • Seidenfeld, T., Schervish, M. J. and Kadane, J. B. (2009). Preference for equivalent random variables: A price for unbounded utilities. J. Math. Econom. 45 329–340.

Supplemental materials

  • Supplementary material: Infinite previsions and finitely additive expectations. The expectation of a random variable $X$ defined on $\Omega$ is usually defined as the integral of $X$ over the set $\Omega$ with respect to the underlying probability measure defined on subsets of $\Omega$. In the countably additive setting, such integrals can be defined (except for certain cases involving $\infty-\infty$) uniquely from a probability measure on $\Omega$. Dunford and Schwartz [(1958), Chapter III] give a detailed analysis of integration with respect to finitely additive measures that attempts to replicate the uniqueness of integrals. Their analysis requires additional assumptions if one wishes to integrate unbounded random variables. We choose the alternative of defining integrals as special types of linear functionals. This is the approach used in the study of the Daniell integral. [See Royden (1968), Chapter 13.] Then the measure of a set becomes the integral of its indicator function. De Finetti’s concept of prevision turns out to be a finitely additive generalization of the Daniell integral. (See Definition 10 in Appendix A.2.) We provide details on the finitely additive Daniell integral along with details about the meaning of infinite previsions and how to extend coherence$_{1}$ and coherence$_{3}$ to deal with random variables having infinite previsions. Infinite previsions invariably arise when dealing with general sets of unbounded random variables.