Skip to main content
SpringerLink
Log in

Prior assessments for bands of probability measures: Empirical bayes analysis

  • Published:
Test Aims and scope Submit manuscript

Summary

A useful non-parametric class of priors is formed as those probability measures which lie between an upper and a lower measure and it is called a Wand of probability measures Lavine (1991), Moreno and Pericchi (1991), Wasserman and Kadane (1992). This class allows considerable freedom in tail behaviour as long as the upper and lower measures have different tails. This property is inherited from the class of Interval of measures DeRobertis and Hartigan (1981), from which the Band of probability measures class is obtained by restricting attention to those measures that integrate to one, Although the analysis of this restricted class is somewhat more involved, the restriction is quite natural and may lead to robustness in situations in which the larger class is not robust. In fact, it is a generalization of the class implicitly considered by Edwards, Lindman and Savage (1963) in “Precise Measurement Theory”. An empirical analysis is performed with the two classes above. That is, we choose within a class of priors that measure which maximizes the probability of the actual data Good (1983). This “optimal” measure is used in this robust Bayesian context to assess the band of probability measures.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Berger, J. (1984). The robust Bayesian viewpoint.Robustness of Bayesian Analysis (J. Kadane, ed.). Amsterdam: North-Holland, 63–144, (with discussion).

    Google Scholar 

  • Berger, J. (1985).Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.

    MATH  Google Scholar 

  • Berger, J. (1989). Robust Bayesian Analysis: Sensitivity to the Prior.J. Statist. Planning and Inference 25, 303–328.

    Article  Google Scholar 

  • Berger, J. and Berliner, M. L. (1986). Robust Bayes and Empirical Bayes Analysis with ε-contaminated priors.Ann. Statist. 14, 461–486.

    MathSciNet  MATH  Google Scholar 

  • DeRobertis, L. and Hartigan, J. A. (1981). Bayesian Inference using intervals of measures.Ann. Statist. 9, 235–244.

    MathSciNet  Google Scholar 

  • Edwards, W., Lindman, H. and Savage, L. J. (1963). Bayesian Statistical Inference for Psychological Research.Psychological Review 10, 3, 450–501.

    Google Scholar 

  • Good, I. J. (1983).Good Thinking: The Foundations of Probability and its Applications. Minneapolis: University of Minnesota Press.

    MATH  Google Scholar 

  • Lavine, M. (1991). Sensitivity in Bayesian statistics: the prior and the likelihood.J. Amer. Statist. Assoc. 86, 400–403.

    Article  MathSciNet  MATH  Google Scholar 

  • Lavine, M., Wasserman, L. and Wolpert, R. (1994). Linearization of Bayesian Robustness problems.J. Statist. Planning and Inference (to appear).

  • Moreno, E. and Cano, J. A. (1991). Robust Bayesian Analysis with ε-contaminations partially known.J. Roy. Statist. Soc. B 53, 143–155.

    MathSciNet  MATH  Google Scholar 

  • Moreno, E. and González, A. (1990). Empirical Bayes Analysis for ε-contaminated priors with shape and quantile constraints.Brazilian Journal of Probability and Statistics 4, 177–200.

    MATH  Google Scholar 

  • Moreno, E. and Pericchi, L. R. (1990). Sensitivity of the Bayesian analysis to the priors: Structural contaminations with specified quantiles of parametric families.Actas III Congreso Latinoamericano en Probabilidad y Estadística Matemática, 143–158.

  • Moreno, E. and Pericchi, L. R. (1992). Bands of probability measures: A Robust Bayesian Analysis.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 707–715.

    Google Scholar 

  • Moreno, E., Pericchi, L. R. and Kadane, J. B. (1993). Precise bounds for precise measurement. In preparation.

  • Pericchi, L. R. and Walley, P. (1991). Robust Bayesian credible intervals and prior ignorance.Internat. Statist. Rev. 59, 1–23.

    Article  Google Scholar 

  • Sivaganesan, S. and Berger, J. (1989). Ranges of posterior measures for priors with unimodal contaminations.Ann. Statist. 17, 868–889.

    MathSciNet  MATH  Google Scholar 

  • Wasserman, L. and Kadane, J. B. (1992). Computing bounds on Expectations.J. Amer. Statist. Assoc. 87, 516–523.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. de Estadística, Universidad de Granada Facultad de Ciencias, 18071, Granada, Spain

    Elías Moreno

  2. Dept. de Matemáticas, Universidad Simón Bolivar, Apdo. 89000, 1080A, Caracas, Venezuela

    Luís Raúl Pericchi

Authors
  1. Elías Moreno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luís Raúl Pericchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moreno, E., Pericchi, L.R. Prior assessments for bands of probability measures: Empirical bayes analysis. Test 2, 101–110 (1993). https://doi.org/10.1007/BF02562670

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02562670

Keywords

Search

Navigation