Abstract
A betting game establishes a sense in which confidence measures, confidence distributions in the form of probability measures, are the only reliable inferential probability distributions. In addition, because confidence measures are Kolmogorov probability distributions, they are as coherent as Bayesian posterior distributions in their avoidance of sure loss under the usual Dutch-book betting game.
Although a confidence measure can be computed without any prior, previous knowledge can be incorporated into confidence-based reasoning by combining the confidence measure from the observed data with one or more independent confidence measures representing previous agent opinion. The representation of subjective knowledge in terms of confidence measures rather than more general priors preserves approximate frequentist validity and thus reliability in the first game.
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References
Armendt, B. (1992). Dutch strategies for diachronic rules: when believers see the sure loss coming. In PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992, pp. 217–229.
Barnard, G.A. (1987). R. A. Fisher: a true Bayesian? International Statistical Review, 55, 183–189.
Barndorff-nielsen, O.E. and Cox, D.R. (1994). Inference and Asymptotics. CRC Press, London.
Berger, J.O., Bernardo, J.M. and Sun, D. (2009). The formal definition of reference priors. Ann. Statist., 37, 905–938.
Berger, J.O. (2004). The case for objective Bayesian analysis. Bayesian Anal., 1, 1–17.
Bernardo, J.M. (1979). Reference posterior distributions for Bayesian inference. J. R. Stat. Soc. Ser. B, 41, 113–147.
Bernardo, J.M. (1997). Noninformative priors do not exist: a discussion. J. Statist. Plann. Inference, 65, 159–189.
Bickel, D.R. (2006). Incorporating expert knowledge into frequentist results by combining subjective prior and objective posterior distributions: a generalization of confidence distribution combination. Technical Report, Pioneer Hi-Bred International, arXiv:math.ST/0602377v2.
Bickel, D.R. (2011). Estimating the null distribution to adjust observed confidence levels for genome-scale screening. Biometrics, 67, 363–370.
Bickel, D.R. (2012a). Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing. Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/23124.
Bickel, D.R. (2012b). Coherent frequentism: a decision theory based on confidence sets. Comm. Statist. Theory Methods, 41, 1478–1496.
Bickel, D.R. (2012c). Controlling the degree of caution in statistical inference with the Bayesian and frequentist approaches as opposite extremes. Electron. J. Stat., 6, 686–709.
Bickel, D.R. (2012d). Empirical Bayes interval estimates that are conditionally equal to unadjusted confidence intervals or to default prior credibility intervals. Stat. Appl. Genet. Mol. Biol., 11, art. 3.
Bickel, D.R. (2012e). A prior-free framework of coherent inference and its derivation of simple shrinkage estimators. Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/23093.
Bickel, D.R. (2012f). The strength of statistical evidence for composite hypotheses: Inference to the best explanation. Statistica Sinica, 22, 1147–1198.
Brazzale, A.R., Davison, A.C. and Reid, N. (2007). Applied Asymptotics: Case Studies in Small-sample Statistics. Cambridge University Press, Cambridge.
Buehler, R.J. (1977). Conditional confidence statements and confidence estimators: Comment. J. Amer. Statist. Assoc., 72, 813–814.
Carnap, R. (1971). A basic system of inductive logic, part 1. Studies in Inductive Logic and Probability, Vol. 1. University of California Press, Berkeley, pp. 3–165.
Chaloner, K. (1996). The elicitation of prior distributions. Bayesian Biostatistics. Marcel Dekker, New York.
Clarke, B. (2007). Information optimality and Bayesian modelling. J. Econometrics, 138, 405–429.
Cornfield, J. (1969). The Bayesian outlook and its application. Biometrics, 25, 617–657.
Cox, D.R. (1958). Some problems connected with statistical inference. Annals of Mathematical Statistics, 29, 357–372.
Craig, P.S., Goldstein, M., Seheult, A.H. and Smith, J.A. (1998). Constructing partial prior specifications for models of complex physical systems. The Statistician, 47, 37–53.
Datta, G.S. and Mukerjee, R. (2004). Probability Matching Priors: Higher Order Asymptotics. Springer, New York.
De Finetti, B. (1970). Theory of Probability: a Critical Introductory Treatment, 1st Edition. John Wiley and Sons Ltd, New York.
Dempster, A.P. (2008). The Dempster-Shafer calculus for statisticians. Internat. J. Approx. Reason., 48, 365–377.
Edwards, A.W.F. (1992). Likelihood. Johns Hopkins Press, Baltimore.
Efron, B. (1993). Bayes and likelihood calculations from confidence intervals. Biometrika, 80, 3–26.
Efron, B. (1998). R. A. Fisher in the 21st century, invited paper presented at the 1996 R. A. Fisher lecture. Statist. Sci., 13, 95–114.
Efron, B. and Tibshirani, R. (1998). The problem of regions. Ann. Statist. 26, 1687–1718.
Fisher, R.A. (1960). Scientific thought and the refinement of human reasoning. J. Oper. Res. Soc. Japan, 3, 1–10.
Fisher, R.A. (1973). Statistical Methods and Scientific Inference. Hafner Press, New York.
Fraser, D. (1978). Inference and Linear Models. McGraw-Hill, New York.
Fraser, D.A.S. (1968). The Structure of Inference. John Wiley, New York.
Fraser, D.A.S. (1977). Confidence, posterior probability, and the buehler example. Ann. Statist., 5, 892–898.
Fraser, D.A.S. (1991). Statistical inference: likelihood to significance. J. Amer. Statist. Assoc., 86, 258–265.
Fraser, D.A.S. (2004). Ancillaries and conditional inference. Statist. Sci., 19, 333–351.
Fraser, D.A.S. (2006). Did Lindley get the argument the wrong way around? Technical Report, Department of Statistics, University of Toronto.
Fraser, D.A.S. and Reid, N. (2002). Strong matching of frequentist and Bayesian parametric inference. J. Statist. Plann. Inference, 103, 263–285.
Freedman, D.A. and Purves, R.A. (1969). Bayes’ method for bookies. Annals of Mathematical Statistics, 40, 1177–1186.
Garthwaite, P.H., Kadane, J.B. and O’hagan, A. (2005). Statistical methods for eliciting probability distributions. J. Amer. Statist. Assoc., 100, 680–700.
Gleser, L.J. (2002). [setting confidence intervals for bounded parameters]: Comment. Statist. Sci., 17, 161–163.
Goldstein, M. (1997). Prior inferences for posterior judgements. In Structures and Norms in Science: Volume Two of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995 (M. L. D. Chiara, K. Doets, D. Mundici, & J. van Benthem Eds.). New York, Springer, pp. 55–71.
Goldstein, M. (2001). Avoiding foregone conclusions: Geometric and foundational analysis of paradoxes of finite additivity. J. Statist. Plann. Inference, 94, 73–87.
Goldstein, M. (2006). Subjective Bayesian analysis: principles and practice. Bayesian Anal., 1, 403–420.
Grundy, P.M. (1956). Fiducial distributions and prior distributions: an example in which the former cannot be associated with the latter. J. R. Stat. Soc. Ser. B, 18, 217–221.
Hacking, I. (1965). Logic of Statistical Inference. Cambridge University Press, Cambridge.
Hacking, I. (2001). An introduction to probability and inductive logic. Cambridge University Press, Cambridge.
Hannig, J. (2009). On generalized fiducial inference. Statist. Sinica, 19, 491–544.
Hannig, J. and Xie, M. (2009). A note on Dempster-Shafer recombination of confidence distributions. Electron. J. Stat., 6, 1943–1966.
Heath, D. and Sudderth, W. (1978). On finitely additive priors, coherence, and extended admissibility. Ann. Statist., 6, 333–345.
Heath, D. and Sudderth, W. (1989). Coherent inference from improper priors and from finitely additive priors. Ann. Statist., 17, 907–919.
Helland, I.S. (2004). Statistical inference under symmetry. International Statistical Review, 72, 409–422.
Hurwicz, L. (1951). The generalized Bayes-minimax principle: a criterion for decision-making under uncertainty. Cowles Commission Discussion Paper 355.
Hwang, J.T., Casella, G., Robert, C., Wells, M.T. and Farrell, R.H. (1992). Estimation of accuracy in testing. Ann. Statist., 20, 490–509.
Jaffray, J.-Y. (1989). Linear utility theory for belief functions. Oper. Res. Lett., 8, 107–112.
Jeffrey, R. (1986). Probabilism and induction. Topoi, 5, 51–58.
Kempthorne, O. (1976). Comment on E. T. Jaynes, ‘Confidence intervals vs Bayesian intervals’. Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. D. Reidel, Dordrecht-Holland, Ch. Confidence intervals vs Bayesian intervals, pp. 220–228.
Kiefer, J. (1977). Conditional confidence statements and confidence estimators: rejoinder. J. Amer. Statist. Assoc., 72, 822–827.
Kohlas, J. and Monney, P.-A. (2008). An algebraic theory for statistical information based on the theory of hints. Internat. J. Approx. Reason., 48, 378–398.
Kyburg, H.E. (2007). Probability and Inference. Texts in Philosophy 2. College Publications, London, Ch. Bayesian inference with evidential probability, pp. 281–296.
Kyburg, H.E. and Teng, C.M. (2001). Uncertain Inference. Cambridge University Press, Cambridge.
Lele, S.R. (2004). Elicit data, not prior: On using expert opinion in ecological studies. The Nature of Scientific Evidence: Statistical, Philosophical, and Empirical Considerations. University of Chicago Press, Chicago, pp. 410–436.
Lindley, D.V. (1958). Fiducial distributions and Bayes’ theorem. J. R. Stat. Soc. Ser. B, 20, 102–107.
Liu, R. and Singh, K. (1997). Notions of limiting P values based on data depth and bootstrap. J. Amer. Statist. Assoc., 92, 266–277.
Maher, P. (1992). Diachronic rationality. Philos. Sci., 59, 120–141.
Mccullagh, P. (2002). What is a statistical model? Ann. Statist., 30, 1225–1267.
Paris, J.B. (1994). The Uncertain Reasoner’s Companion: A Mathematical Perspective. Cambridge University Press, New York.
Polansky, A.M. (2007). Observed Confidence Levels: Theory and Application. Chapman and Hall, New York.
Robins, J. and Wasserman, L. (2000). Conditioning, likelihood, and coherence: A review of some foundational concepts. J. Amer. Statist. Assoc., 95, 1340–1346.
Royall, R. (1997). Statistical Evidence: A Likelihood Paradigm. CRC Press, New York.
Royall, R. (2000). On the probability of observing misleading statistical evidence. J. Amer. Statist. Assoc., 95, 760–768.
Savage, L.J. (1954). The Foundations of Statistics. John Wiley and Sons, New York.
Scheffe, H. (1977). A note on a reformulation of the s-method of multiple comparison. J. Amer. Statist. Assoc., 72, 143–146.
Schervish, M.J. (1995). Theory of Statistics. Springer-Verlag, New York.
Schweder, T., Hjort, N.L. (2002). Confidence and likelihood. Scand. J. Stat., 29, 309–332.
Seidenfeld, T. (2007). Probability and Inference. Texts in Philosophy 2. College Publications, London, Ch. Forbidden fruit: When epistemological probability may not take a bite of the Bayesian apple, pp. 267–279.
Shafer, G. (2011). A betting interpretation for probabilities and Dempster-Shafer degrees of belief. Internat. J. Approx. Reason., 52, 127–136.
Sharma, S.S. (1980). On hacking’s fiducial theory of inference. Canad. J. Statist., 8, 227–233.
Singh, K., Xie, M. and Strawderman, W.E. (2005). Combining information from independent sources through confidence distributions. Ann. Statist., 33, 159–183.
Sprott, D.A. (2000). Statistical Inference in Science. Springer, New York.
Troffaes, M.C.M. (2007). Decision making under uncertainty using imprecise probabilities. Internat. J. Approx. Reason., 45, 17–29.
Ville, J. (1939). Gauthier-Villars, Paris.
Vos, P. (2008). Boyles, R.A. (2008), “The role of likelihood in interval estimation,” The American Statistician, 62, 22–26: Comment by Vos and reply. Amer. Statist., 62, 274–275.
Wilkinson, G.N. (1977). On resolving the controversy in statistical inference (with discussion). J. R. Stat. Soc. Ser. B, 39, 119–171.
Williamson, J. (2009). Objective Bayesianism, Bayesian conditionalisation and voluntarism. Synthese, 178, 1–19.
Zabell, S.L. (1992). R. A. Fisher and the fiducial argument. Statist. Sci., 7, 369–387.
Acknowledgement.
I am grateful to the anonymous referee for comments that lead to several improvements in clarity and completeness, most notably the inclusion of Sections 3.2.2 and 5.2. Matthias Kohl kindly provided R code (R Development Core Team, 2004) used to compute the convolution of the double exponential distribution (“R-help” list message posted on 12/23/05). I thank Mark Cooper for helpful feedback and Jean Peccoud, Mark Whitsitt, Chris Martin, and Bob Merrill for their support of the seed of this paper (Bickel, 2006) at Pioneer Hi-Bred, International. Subsequent developments were partially supported by the Canada Foundation for Innovation and by the Ministry of Research and Innovation of Ontario.
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Bickel, D.R. A frequentist framework of inductive reasoning. Sankhya A 74, 141–169 (2012). https://doi.org/10.1007/s13171-012-0020-x
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DOI: https://doi.org/10.1007/s13171-012-0020-x
Keywords and phrases.
- Artificial intelligence
- betting
- coherence
- confidence distribution
- confidence posterior
- expert system
- foundations of statistics
- inductive reasoning
- interpretation of probability
- machine learning
- personal probability
- prior elicitation
- subjective probability