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METRON

, Volume 76, Issue 2, pp 251–268 | Cite as

Confidence, credibility and prediction

  • Murray AitkinEmail author
  • Charles Liu
Article
  • 70 Downloads

Abstract

This paper examines, from a historical and model-based Bayesian perspective, two inferential issues: (1) the relation between confidence coverage and credibility of interval statements about model parameters; (2) the prediction of new values of a random variable. Confidence and credible intervals have different properties. This worries some statisticians, who want them to have the same properties, in frequentist repeated sampling coverage and Bayesian credibility content. Research continues on the conditions under which intervals have the same repeated sampling coverage and credibility content. We conclude that these two inferential approaches generally have incommensurable properties, which converge only asymptotically in sample size, and in a restricted class of samples. The repeated confusion, by new students of statistics, of the coverage of a confidence interval with the credibility of the observed interval, suggests that in general the credibility of the interval is the more important inferential aim. We show how the credibility of a confidence interval can be assessed generally. Bayesian prediction of new values is well established using the posterior predictive distribution. This has some curious features, known since Laplace but not well understood. The prediction of new Bernoullis highlights these features. We suggest that the credibility of predictive intervals needs to be reassessed.

Keywords

Likelihood Confidence Credible Prediction intervals 

References

  1. 1.
    Agresti, A., Caffo, B.: Simple and effective confidence intervals for proportions and differences of proportions. Am. Stat. 54, 280–288 (2000)zbMATHGoogle Scholar
  2. 2.
    Anscombe, F.J.: Normal likelihood functions. Ann. Inst. Stat. Math. 16, 1–19 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aitchison, J., Dunsmore, I.R.: Statistical Prediction Analysis. Cambridge University Press, Cambridge (1975)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bayes, T.: An essay towards solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53, 370–418 (1764)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berger, J.O., Bernardo, J.M., Sun, D.: The formal definition of reference priors. Ann. Stat. 37, 905–938 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bjornstad, J.F.: Predictive likelihood: a review. Stat. Sci. 5, 242–265 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cox, D.R.: Principles of Statistical Inference. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Datta, G.S., Sweeting, T.J.: Probability matching priors. In: Dey, D.K., Rao, C.R. (eds.) Handbook of Statistics, 25: Bayesian Thinking: Modeling and Computation, pp. 91–114. Elsevier, Amsterdam (2005)CrossRefGoogle Scholar
  9. 9.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39, 1–38 (1977)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Diaz-Francés, E.: Simple estimation intervals for Poisson, exponential and inverse Gaussian means obtained by symmetrizing the likelihood function. Am. Stat. 70, 171–180 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fisher, R.A.: On an absolute criterion for fitting frequency curves. Messenger Math. 41, 155–160 (1912)Google Scholar
  12. 12.
    Fisher, R.A.: On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. A 222, 309–368 (1922)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fisher, R.A.: Theory of statistical estimation. Proc. Camb. Philos. Soc. 22, 700–725 (1925)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fraser, D.A.S.: Is Bayes just quick and dirty confidence? (with discussion). Stat. Sci. 26, 299–316 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Gelman, A., Meng, X.L., Stern, H.S.: Posterior predictive assessment of model fitness via realised discrepancies (with discussion). Stat. Sin. 6, 733–807 (1996)zbMATHGoogle Scholar
  16. 16.
    Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian date analysis, 3rd edn. CRC Press, Boca Raton (2014)zbMATHGoogle Scholar
  17. 17.
    Jeffreys, H.: Theory of Probability (3rd edn. 1961, reissued 1998). Oxford University Press, Oxford (1939)Google Scholar
  18. 18.
    Laplace, P.-S.: Essai philosophique sur les probabilités (1814). English translation A Philosophical Essay on Probabilities. Wiley (1902) and Dover, New York (1950)Google Scholar
  19. 19.
    Little, R.J.A.: Calibrated Bayes: a Bayes/frequentist roadmap. Am. Stat. 60, 213–223 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Little, R.J.A.: Calibrated Bayes, for statistics in general, and missing data in particular. Stat. Sci. 26, 162–174 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Little, R.J.A.: Calibrated Bayes, an inferential paradigm for official statistics in the era of big data. Stat. J. IAOS 31, 555–563 (2015)CrossRefGoogle Scholar
  22. 22.
    Müller, U.K., Noretz, A.: Coverage inducing priors in nonstandard inference problems. J. Am. Stat. Assoc. 111, 1233–1241 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pearson, K.: Contribution to the mathematical theory of evolution. Philos. Trans. R. Soc. Lond. A 185, 71–110 (1894)CrossRefzbMATHGoogle Scholar
  24. 24.
    Pearson, K.: On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos. Mag. 50, 157–175 (1900)CrossRefzbMATHGoogle Scholar
  25. 25.
    Pearson, K.: On the systematic fitting of curves to observations and measurements, Parts I. Biometrika I, 265–303 (1902)CrossRefGoogle Scholar
  26. 26.
    Pearson, K.: On the systematic fitting of curves to observations and measurements, Parts II. Biometrika II, 1–23 (1902)Google Scholar
  27. 27.
    Raiffa, H., Schlaifer, R.: Applied statistical decision theory. Harward Business School, Boston (1961)zbMATHGoogle Scholar
  28. 28.
    Rubin, D.B.: Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Stat. 12, 1151–1172 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tanner, M., Wong, W.: The calculation of posterior distributions by data augmentation. J. Am. Stat. Assoc. 82, 528–550 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sapienza Università di Roma 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  2. 2.Cytel Inc.BostonUSA

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