Abstract
The formalism of operational statistics, a generalized approach to probability and statistics, provides a setting within which inference strategies can be studied with great clarity. This paper is concerned with the asymptotic behavior of the Bayesian inference strategy in this setting. We consider a sequence of posterior distributions, obtained from a prior as a result of successive conditionings by the events of an admissible sequence. We identify certain statistical hypotheses whose limiting posterior probabilities converge to one. We describe these hypotheses, and show that when the prior is vague, they contain those probability models which represent the long-run relative frequencies of occurrence for the events in the sequence.
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References
D. J. Foulis and C. H. Randall, “Operational statistics, I, Basic concepts,”J. Math. Phys. 13, 1667 (1972).
C. H. Randall and D. J. Foulis, “Operational statistics, II, Manuals of operations and their logics,”J. Math. Phys. 14, 1472 (1973).
C. H. Randall and D. J. Foulis, “A mathematical setting for inductive reasoning,” inFoundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, C. Hooker, ed., Vol. III (D. Reidel, Dordrecht, Holland, 1976).
C. H. Randall and D. J. Foulis, “The operational approach to quantum mechanics,” inThe Logico-Algebraic Approach to Quantum Mechanics, C. Hooker, ed., Vol. III (D. Reidel, Dordrecht, Holland, 1978).
B. De Finetti,Probability, Induction, and Statistics: The Art of Guessing (Wiley, New York, 1972).
L. J. Savage,The Foundations of Statistics (Wiley, New York, 1954).
J. Aitchison and I. R. Dunsmore,Statistical Prediction Analysis (Cambridge University Press, Cambridge, 1975).
M. A. Gaudard, “The correspondence between credibilities and induced betting rate assignments,”Found. Phys. 14, 431 (1984).
M. E. Munroe,Measure and Integration (Addison-Wesley, Reading, Massachusetts 1971).
P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).
D. H. Hadwin and M. A. Gaudard, “Sigma-algebras in the space of probability measures,” in preparation.
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Gaudard, M. Convergence of posterior probabilities in the Bayesian inference strategy. Found Phys 15, 49–62 (1985). https://doi.org/10.1007/BF00738737
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DOI: https://doi.org/10.1007/BF00738737