Skip to main content
SpringerLink
Log in

Intrinsic credible regions: An objective Bayesian approach to interval estimation

TEST volume 14, pages 317–384 (2005)Cite this article

Abstract

This paper definesintrinsic credible regions, a method to produce objective Bayesian credible regions which only depends on the assumed model and the available data.Lowest posterior loss (LPL) regions are defined as Bayesian credible regions which contain values of minimum posterior expected loss: they depend both on the loss function and on the prior specification. An invariant, information-theory based loss function, theintrinsic discrepancy is argued to be appropriate for scientific communication. Intrinsic credible regions are the lowest posterior loss regions with respect to the intrinsic discrepancy loss and the appropriate reference prior. The proposed procedure is completely general, and it is invariant under both reparametrization and marginalization. The exact derivation of intrinsic credible regions often requires numerical integration, but good analytical approximations are provided. Special attention is given to one-dimensional intrinsic credible intervals; their coverage properties show that they are always approximate (and sometimes exact) frequentist confidence intervals. The method is illustrated with a number of examples.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  • Barnard, G. A. (1980). Pivotal inference and the Bayesian controversy (with discussion). In J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, eds.,Bayesian Statistics 1, pp. 295–318, University Press, Valencia.

    Google Scholar 

  • Berger, J. O. (2000). Bayesian analysis: A look at today and thoughts of tomorrow.Journal of the American Statististical Association, 95:1269–1276.

    Article  MATH  Google Scholar 

  • Berger, J. O. andBernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors.Journal of the American Statististical Association, 84:200–207.

    Article  MATH  MathSciNet  Google Scholar 

  • Berger, J. O. andBernardo, J. M. (1992a). On the development of reference priors (with discussion). In J. M. Bernardo J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds.,Bayesian Statistics 4, pp. 35–60. Oxford University Press, Oxford.

    Google Scholar 

  • Berger, J. O. andBernardo, J. M. (1992b). Ordered group reference priors with applications to a multinomial problem.Biometrika, 79:25–37.

    Article  MATH  MathSciNet  Google Scholar 

  • Berger, J. O. andBernardo, J. M. (1992c). Reference priors in a variance components problem In P. K. Goel and N. S. Iyengar, eds.,Bayesian Analysis in Statistics and Econometrics, pp. 323–340. Springer-Verlag, Berlin.

    Google Scholar 

  • Berger, J. O. andPericchi, L. R. (1996). The intrinsic Bayes factor for model selection and prediction.Journal of the American Statististical Association, 91:109–122.

    Article  MATH  MathSciNet  Google Scholar 

  • Bernardo, J. M. (1979a). Expected information as expected utility.The Annals of Statistics, 7:686–690.

    MATH  MathSciNet  Google Scholar 

  • Bernardo, J. M. (1979b). Reference posterior distributions for Bayesian inference (with discussion).Journal of the Royal Statistical Society, Series B, 41:113–147. Reprinted inBayesian Inference (N. G. Polson and G. C. Tiao, eds.), Edward Elgar, Brookfield, VT, 1995, pp. 229–263.

    MATH  MathSciNet  Google Scholar 

  • Bernardo, J. M. (1997). Non-informative priors do not exist (with discussion).Journal of Statistical Planning and Inference, 65:159–189.

    Article  MathSciNet  Google Scholar 

  • Bernardo, J. M. (2001). Un programa de síntesis para la enseñanza universitaria de la estadística matemática contemporánea.Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales (España), 95(1–2):87–105.

    Google Scholar 

  • Bernardo, J. M. (2005a). Intrinsic point estimation of the normal variance. In S. K. Upadhyay, U. Singh, and D. K. Dey, eds.,Bayesian Statistics and its Applications. Anamaya Pub, New Delhi. In press.

    Google Scholar 

  • Bernardo, J. M. (2005b). Reference analysis. In D. Dey and C. R. Rao, eds.,Bayesian Thinking, Modeling and Computation, vol. 25 ofHandbook of Statistics. North Holland, Amsterdam. In press

    Google Scholar 

  • Bernardo, J. M. andJuárez, M. (2003). Intrinsic estimation. In J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, and M. West, eds.,Bayesian Statistics 7, pp. 465–476. Oxford University Press, Oxford.

    Google Scholar 

  • Bernardo, J. M. andRamón, J. M. (1998). An introduction to Bayesian reference analysis: Inference on the ration of multinomial parameters.Journal of the Royal Statistical Society, Series D. The Statistician, 47:1–35.

    Article  Google Scholar 

  • Bernardo, J. M. andRueda, R. (2002). Bayesian hypothesis testing: A reference approach.International Statistical Review, 70:351–372.

    Article  MATH  Google Scholar 

  • Bernardo, J. M. andSmith, A. F. M. (1994).Bayesian Theory, John Wiley & Sons, Chichester.

    MATH  Google Scholar 

  • Blyth, C. R. (1986). Approximate binomial confidence limits.Journal of the American Statististical Association, 81:843–855.

    Article  MATH  MathSciNet  Google Scholar 

  • Blyth, C. R. andStill, H. A. (1983). Binomial confidence intervals.Journal of the American Statististical Association, 78:108–116.

    Article  MATH  MathSciNet  Google Scholar 

  • Brown, L. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters.Annals of Mathematics and Statistics 39:29–48.

    MATH  Google Scholar 

  • Brown, L. D., Cai, T. T., andDasGupta, A. (2001). Interval estimation for a binomial proportion (with discussion).Statistical Science, 16:101–133.

    MATH  MathSciNet  Google Scholar 

  • Brown, L. D., Cai, T. T., andDasGupta, A. (2002). Confidence intervals for a binomial proportion and Edgeworth expansion.The Annals of Statistics, 30(1):160–201.

    Article  MATH  MathSciNet  Google Scholar 

  • Burdick, R. K. andGraybill, F. A. (1992).Confidence Intervals on Variance Components. Marcel Dekker, New York.

    MATH  Google Scholar 

  • Cano, J. A., Hernández, A., andMoreno, E. (1987). A note on maximized likelihood sets.European Journal of Operational Research, 32:291–293.

    Article  MATH  MathSciNet  Google Scholar 

  • Casella, G. (1986). Refining binomial confidence intervals.The Canadian Journal of Statistics, 14:113–129.

    MATH  MathSciNet  Google Scholar 

  • Casella, G., Hwang, J. T., andRobert, C. P. (1993). A paradox in decision-theoretic set estimation.Statistica Sinica, 3:141–155.

    MATH  MathSciNet  Google Scholar 

  • Casella, G., Hwang, J. T. andRobert, C. P. (1994). Loss function for set estimation. In J. O. Berger and S. S. Gupta, eds.,Statistical Decision Theory and Related Topics, V, pp. 237–252. Springer Verlag, New York.

    Google Scholar 

  • Datta, G. S. andMukerjee, R. (2004).Probability Matching Priors: Higher Order Asymptotics vol. 178 ofLecture Notes in Statistics. Springer Verlag, New York.

    Google Scholar 

  • Data, G. S. andSweeting, T. (2005). Probability matching priors. In D. Dey and C. R. Rao, eds.,Bayesian Thinking, Modeling and Computation, vol. 25 ofHandbook of Statistics, North Holland, Amsterdam. In press.

    Google Scholar 

  • de Finetti, B. (1970).Teoria delle Probabilità. Einaudi, Turin, English translation asTheory of Probability in 1974. John Wiley & Sons, Chichester.

    Google Scholar 

  • Eberly, L. E. andCasella, G. (2003). Estimating Bayesian credible intervals.Journal of Statistical Planning and Inference, 112:115–132.

    Article  MATH  MathSciNet  Google Scholar 

  • Efron, B. (1987). Better bootstrap confidence intervals (with discussion).Journal of the American Statististical Association, 82:171–200.

    Article  MATH  MathSciNet  Google Scholar 

  • George, E. I. andMcCulloch, R. E. (1993). On obtaining invariant prior distributions.Journal of Statistical Planning and Inference, 37:169–179.

    Article  MATH  MathSciNet  Google Scholar 

  • Geyer, C. J. andMeeden, G. D. (2005) Fuzzy and randomized confidence intervals andp-values (with discussion).Statistical Science. To appear.

  • Gleser, L. andHwang, J. T. (1987). The nonexistence of 100(1−α) confidence sets of finite expected diameter in errors-in-variable and related models.The Annals of Statistics, 15:1351–1362.

    MATH  MathSciNet  Google Scholar 

  • Granger, C. andLin, J.-L. (1994). Using the mutual information coefficient to identify lags in nonlinear models.The Journal of Time series Analysis, 15(4):371–384.

    MATH  MathSciNet  Google Scholar 

  • Gutiérrez-Peña, E. (1992). Expected logarithmic divergence for exponential families. In J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds.,Bayesian Statistics 4, pp. 669–674. Oxford University Press, Oxford.

    Google Scholar 

  • Guttman, I. (1970).Statistical Tolerance Regions: Classical and Bayesian. Griffin, London.

    MATH  Google Scholar 

  • Hahn, G. J. andMeeker, W. Q. (1991).Statistical Intervals. John Wiley & Sons, Chichester.

    MATH  Google Scholar 

  • Hartigan, J. A. (1966). Note on the confidence prior of Welch and Peers.Journal of the Royal Statistical Society. Series B, 28:55–56.

    MATH  MathSciNet  Google Scholar 

  • Hartigan, J. A. (1983).Bayes Theory. Springer-Verlag. Berlin.

    MATH  Google Scholar 

  • Joe, H. (1989). Relative entropy measures of multivariate dependence.The Journal of American Statistical Association, 405:157–164.

    Article  MathSciNet  Google Scholar 

  • Juárez, M. (2004).Métodos Bayesianos Objetivos de Estimación y Contraste de Hipótesis. PhD Thesis, Universidad de Valencia, Spain.

    Google Scholar 

  • Kullback, S. (1968).Information Theory and Statistics. Dover, New York, 2nd ed. Reprinted in 1978, Peter Smith, Gloucester, MA.

    Google Scholar 

  • Kullback, S. andLeibler, R. A. (1951). On information and sufficiency.Annals of Mathematics and Statistics, 22:79–86.

    MathSciNet  MATH  Google Scholar 

  • Lindley, D. V. (1958). Fiducial distribution and Bayes' Theorem.Journal of the Royal Statistical Society. Series B, 20:102–107.

    MATH  MathSciNet  Google Scholar 

  • Lindley, D. V. (1970).Introduction to Probability and Statistics from a Bayesian Viewpoint. Part 2: Inference, Cambridge University Press, Cambridge.

    Google Scholar 

  • Peña, D. andvan der Linde, A. (2005). General measures of variability and dependence for multivariate continuous distribution. Manuscript submitted for publication.

  • Piccinato, L. (1984). A Bayesian property of the likelihood sets.Statistica, 3:367–372.

    MathSciNet  Google Scholar 

  • Robert, C. P. (1996). Intrinsic loss functionsTheory and Decision, 40:192–214.

    Article  Google Scholar 

  • Robert, C. P. (2001).The Bayesian Choice, Springer Verlag, New York, 2nd ed.

    MATH  Google Scholar 

  • Robert, C. P. andCasella, G. (1994). Distance penalized losses for testing and confidence set evaluation.Test, 3(1):163–182.

    Article  MATH  MathSciNet  Google Scholar 

  • Robinson, P. (1991). Consistent nonparametric entropy-based testing.Review of Economic Studies, 58:437–453.

    Article  MATH  MathSciNet  Google Scholar 

  • Rousseau, J. (1997). Expansions of penalized likelihood ratio statistics and consequences on matching priors for HPD regions. Tech. rep., CREST, INSEE, Paris.

  • Savage, L. J. (1954).The Foundations of Statistics. John Wiley & Sons, New York. Second edition in 1972, Dover, New York.

    MATH  Google Scholar 

  • Schervish, M. J. (1995).Theory of Statistics, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Wasserman, L. (1989). A robust Bayesian interpretation of likelihood regions.The Annals of Statistics, 17:1387–1393.

    MATH  MathSciNet  Google Scholar 

  • Welch, B. L. andPeers, H. W. (1963). On formulae for confidence points based on intervals of weighted likelihoods.Journal of the Royal Statistical Society. Series B, 25:318–329.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Estadística e I. O., Universidad de Valencia, Spain

    José M. Bernardo

Authors
  1. José M. Bernardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José M. Bernardo.

Additional information

Work partially supported by grant MTM2004-05956 of MEC, Madrid, Spain

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bernardo, J.M. Intrinsic credible regions: An objective Bayesian approach to interval estimation. TEST 14, 317–384 (2005). https://doi.org/10.1007/BF02595408

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02595408

Key Words

AMS subject classification

search

Navigation