Abstract
This paper defends the fiducial argument. In particular, an interpretation of the fiducial argument is defended in which fiducial probability is treated as being subjective and the role taken by pivots in a more standard interpretation is taken by what are called primary random variables, which in fact form a special class of pivots. The resulting methodology, which is referred to as subjective fiducial inference, is outlined in the first part of the paper. This is followed by a defence of this methodology arranged in a series of criticisms and responses. These criticisms reflect objections that are often raised against standard fiducial inference and incorporate more specific concerns that are likely to exist with respect to subjective fiducial inference. It is hoped that the responses to these criticisms clarify the contribution that a system of fiducial reasoning can make to statistical inference.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barnard, G.A.: R. A. Fisher: a true Bayesian? Int. Stat. Rev. 55, 183–189 (1987)
Barnard, G.A.: Pivotal models and the fiducial argument. Int. Stat. Rev. 63, 309–323 (1995)
Bernardo, J.M.: Reference posterior distributions for Bayesian inference (with discussion). J. Royal Stat. Soc. B 41, 113–147 (1979)
Brown, L.: The conditional level of Student’s t test. Ann. Math. Stat. 38, 1068–1071 (1967)
Buehler, R.J., Feddersen, A.P.: Note on a conditional property of Student’s t. Ann. Math. Stat. 34, 1098–1100 (1963)
Cox, D.R.: Comments on ‘The Philosophy of Statistics’ by D. V. Lindley. J. Royal Stat. Soc. D 49, 321–324 (2000)
Dawid, A.P., Stone, M., Zidek, J.V.: Marginalization paradoxes in Bayesian and structural inference (with discussion). J. Royal Stat. Soc. B 35, 189–233 (1973)
Dawid, A.P., Stone, M.: The functional-model basis of fiducial inference. Ann. Stat. 10, 1054–1067 (1982)
Dempster, A.P.: A generalization of Bayesian inference (with discussion). J. Royal Stat. Soc. B 30, 205–247 (1968)
Edwards, A.W.F.: Fiducial probability. Statistician 25, 15–35 (1976)
Efron, B.: R. A. Fisher in the 21st Century (with discussion). Stat. Sci. 13, 95–122 (1998)
de Finetti, B.: Theory of probability, vol. 1. Wiley, Chichester (1974)
de Finetti, B.: Theory of probability, vol. 2. Wiley, Chichester (1975)
Fine, T.L.: Theories of probability: an examination of foundations. Academic Press, New York (1973)
Fisher, R.A.: Inverse probability. Math. Proceed. Cambridge Philosop. Soc. 26, 528–535 (1930)
Fisher, R.A.: Statistical methods and scientific inference, 1st edn. Hafner Press, New York (1956) [2nd edn., 1959; 3rd edn., 1973]
Fraser, D.A.S.: On fiducial inference. Ann. Math. Stat. 32, 661–676 (1961)
Fraser, D.A.S.: Structural probability and a generalization. Biometrika 53, 1–9 (1966)
Fraser, D.A.S.: Events, information processing, and the structured model. In: Godambe, V.P., Sprott, D.A. (eds.) Foundations of Statistical Inference, pp. 32–55. Holt, Renehart and Winston, Toronto (1972)
Ghosh, M.: Objective priors: an introduction for frequentists (with discussion). Stat. Sci. 26, 187–211 (2011)
Good, I.J.: Kinds of probability. Science 129, 443–447 (1959)
Good, I.J.: The interface between statistics and the philosophy of science (with discussion). Stat. Sci. 3, 386–412 (1988)
Goutis, C., Casella, G.: Frequentist post-data inference. Int. Stat. Rev. 63, 325–344 (1995)
Hannig, J.: On generalized fiducial inference. Stat. Sinica 19, 491–544 (2009)
Hannig, J., Iyer, H., Patterson, P.: Fiducial generalized confidence intervals. J. Am. Stat. Assoc. 101, 254–269 (2006)
Jeffreys, H.: Theory of probability, 3rd edn. Oxford University Press, Oxford (1961)
Kass, R.E., Wasserman, L.: The selection of prior distributions by formal rules. J. Am. Stat. Assoc. 91, 1343–1370 (1996)
Lindley, D.: Some comments on ‘Non-informative priors do not exist’. J. Stat. Plan. Infer. 65, 182–184 (1997)
Pedersen, J.G.: Fiducial inference. Int. Stat. Rev. 46, 147–170 (1978)
Robinson, G.K.: Conditional properties of statistical procedures. Ann. Stat. 7, 742–755 (1979a)
Robinson, G.K.: Conditional properties of statistical procedures for location and scale families. Ann. Stat. 7, 756–771 (1979b)
Savage, L.J.: The foundations of statistics. Wiley, New York (1954)
Savage, L.J.: The foundations of statistics reconsidered. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1., pp. 575–586, University of California Press, Berkeley (1961)
Seidenfeld, T.: Why I am not an objective Bayesian; some reflections prompted by Rosenkrantz. Theory Decision 11, 413–440 (1979)
Shafer, G.: The unity and diversity of probability (with discussion). Stat Sci 5, 435–462 (1990)
Spetzler, C.S., Stael von Holstein, C.A.S.: Probability encoding in decision analysis. Manag. Sci. 22, 340–358 (1975)
Stone, M.: Strong inconsistency from uniform priors (with discussion). J. Am. Stat. Assoc. 71, 114–125 (1976)
Taraldsen, G., Lindqvist, B.H.: Fiducial theory and optimal inference. Ann. Stat. 41, 323–341 (2013)
Wang, C.M., Hannig, J., Iyer, H.K.: Fiducial prediction intervals. J. Stat. Plan. Infer. 142, 1980–1990 (2012)
Welch, B.L.: The generalization of ‘Student’s’ problem when several different population variances are involved. Biometrika 34, 28–35 (1947)
Wilkinson, G.N.: On resolving the controversy in statistical inference (with discussion). J. Royal Stat. Soc. B 39, 119–171 (1977)
Yates, F.: Fiducial probability, recognisable sub-sets and Behrens’ test. Biometrics 20, 343–360 (1964)
Zabell, S.L.: R. A. Fisher and the fiducial argument. Stat. Sci. 7, 369–387 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bowater, R.J. A defence of subjective fiducial inference. AStA Adv Stat Anal 101, 177–197 (2017). https://doi.org/10.1007/s10182-016-0285-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10182-016-0285-9