On some counter-counter-examples about classical inference


  • Benito Vittorio Frosini Università Cattolica del Sacro Cuore, Milano




This paper deals with theoretical concepts and practical examples, aimed at showing that non-Bayesian inference is liable to result in mistakes or unacceptable conclusions, and proves that they are not justified. Section 2 comments on examples when an objective prior distribution exists, and shows how widely one can be mistaken in using a prior quite distant from the real one. Section 3 comments on two results by Godambe, stressing that – in sampling from finite populations – no flat likelihood exists, while an unbiased linear “estimator” with zero variance does not exist, unless we reach a complete knowledge of the population. Section 4 stresses the fundamental difference between a “probability interval” for a parameter, and a “confidence interval” aimed at making inference on the parameter, thus summarizing all certain facts and constraints able to shrink such an inferential interval. Section 5 explains why we are justified in attaching an inductive meaning to a realized confidence interval. Finally, Section 6 counters some well known counter-examples spread in the Bayesian literature, showing that they are unacceptable from a sound inductive basis.


M.J. BAJARRI, , AND J.O. BERGER, (2004), The Interplay of Bayesian and Frequentist Analysis, “Statistical Science”, 19, pp. 58-80.

J.O. BERGER, (1980), Statistical Decision Theory, Springer, New York.

J.O. BERGER, (1985), Statistical Decision Theory and Bayesian Analysis, Springer, New York.

J.O. BERGER, (2006), The Case for Objective Bayesian Analysis, “Bayesian Analysis”, 1, pp. 385-402.

J.O. BERGER, AND R.L. WOLPERT (1988), The Likelihood Principle (Second Edition), Institute of Mathematical Statistics, Hayward.

J.M. BERNARDO, (1979), Reference Posterior Distributions for Bayesian Inference, “Journal of the Royal Statistical Society B”, 41, pp. 113-147.

J.M. BERNARDO, (2005), Reference Analysis, “Handbook of Statistics”, Vol. 25, pp. 17-90, Elsevier, Amsterdam.

J.M. BERNARDO, AND A.F.M. SMITH, (1994), Bayesian Theory, Wiley, Chichester.

R. CARNAP, (1962), Logical Foundations of Probability (Second Edition), The University of Chicago Press, Chicago.

D.R. COX, (1958), Some Problems Connected with Statistical Inference, “The Annals of Mathematical Statistics”, 29, pp. 357-372.

D.R. COX, (1986), Some General Aspects of the Theory of Statistics , “International Statistical Review”, 54, pp. 117-126.

D.R. COX, D.V. HINKLEY, (1974), Theoretical Statistics, Chapman and Hall, London.

B. EFRON, (1986), Why Isn’t Everyone a Bayesian?, “The American Statistician”, 40, pp. 1-5.

B. EFRON, AND C. MORRIS, (1971), Limiting the Risk of Bayes and Empirical Bayes Estimators – Part I: The Bayes Case, “Journal of the American Statistical Association”, 66, pp. 807-815.

R.A. FISHER, (1934), Two New Properties of Mathematical Likelihood, “Proceedings of the Royal Society”, A, 144, 285-307.

R.A. FISHER, (1935), The Logic of Inductive Inference (with discussion), “Journal of the Royal Statistical Society”, 98, Pt. 1, pp. 39-82.

B.V. FROSINI, (1989), La Statistica metodologica nei convegni della SIS, Società Italiana di Statistica, Atti del Convegno “Statistica e Società”, pp. 197-228, Pacini. Pisa.

B.V. FROSINI, (1991), On Some Applications of the Conditionality Principle, “Statistica Applicata”, 3, pp. 555-568.

B.V. FROSINI, (1992), Sui racconti con la morale, ovvero come rifiutare l’uso di informazioni certe, “Statistica Applicata”, 4, pp. 33-43.

B.V. FROSINI, (1993a), Likelihood Versus Probability, International Statistical Institute, Proceedings of the ISI 49-th Session, Vol. 1, pp. 359-376, Firenze.

B.V. FROSINI, (1993b), Global and conditional tests, “Metron”, 51, pp. 27-58.

B.V. FROSINI, (1996a), Likelihood, Superpopulation and Other Problems in Sampling from Finite Populations, Società Italiana di Statistica, 100 Anni di Indagini Campionarie, pp. 217-239, CISU, Roma.

B.V. FROSINI, (1996b), Some Reasons for Reconciling Confidence Intervals and Bayesian Intervals, “Statistica”, 56, pp. 301-311.

B.V. FROSINI, (1999), Conditioning, Information and Frequentist Properties, “Statistica Applicata”, 11, pp. 165-184.

B.V. FROSINI, (2001), Metodi Statistici, Carocci, Roma.

B.V. FROSINI, (2004), On Neyman-Pearson Theory: Information Content of an Experiment and a Fancy Paradox, “Statistica”, 64, pp. 271-286.

B.V. FROSINI, (2005), Objective Bayesian Intervals: Some Remarks on Gini’s Approach, “Metron”, 63, pp. 435-450.

V.P. GODAMBE, (1955), A Unified Theory of Sampling from Finite Populations, “Journal of the Royal Statistical Society B”, 17, pp. 269-278.

V.P. GODAMBE, (1965), A Review of the Contributions Towards a Unified Theory of Sampling from Finite Populations, “Review of the International Statistical Institute”, 33, pp. 242-258.

V.P. GODAMBE, (1969), A Fiducial Argument with Applications to Survey Sampling, “Journal of the Royal Statistical Society B”, 31, pp. 246-260.

C. HOWSON, AND P. URBACH, (1993), Scientific Reasoning: The Bayesian Approach (Second Edition), Open Court, Chicago.

E.T. JAYNES, (1968), Prior Probabilities, “IEEE Transactions on System, Science and Cybernetics”, SCC-4 (September 1968), pp. 227-241.

E.T. JAYNES, (1976), Confidence Intervals vs Bayesian Intervals, Harper and Hooker (eds), “Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science”, Vol. II, pp. 175-257.

H. JEFFREYS, (1939/1948), Theory of Probability, Oxford University Press, Oxford.

B.W. LINDGREN, (1962), Statistical Theory, Macmillan, New York.

D.V. LINDLEY, (1985), Making Decisions (Second Edition), Wiley, London.

M. MANDELKERN, (2002), Setting Confidence Intervals for Bounded Parameters (with discussion), “Statistical Science”, 17, pp. 149-172.

R. ROYALL, (1968), An Old Approach to Finite Population Sampling Theory, “Journal of the American Statistical Association”, 63, pp. 1269-1279.

B.L. WELCH, (1939), On confidence limits and sufficiency, with particular reference to parameters of location, “The Annals of Mathematical Statistics”, 10, pp. 58-69.




How to Cite

Frosini, B. V. (2008). On some counter-counter-examples about classical inference. Statistica, 68(2), 135–152. https://doi.org/10.6092/issn.1973-2201/3526