Abstract
In this article, I develop a generalization of probability distributions that is rich enough to include distributions that can act as suitable replacements for locally integrable improper distributions. These replacements have unit weight but share with the improper distributions from which they originate the essential behavioral properties for which the latter were constructed. Their theory will be developed and associated posterior (generalized) distributions will be obtained. These posterior probability distributions are close to those obtained from the improper ones by a formal application of Bayes’s rule; they differ, in fact, by only an infinitesimal correction. This correction is small enough that Bayesian parametric inference can be performed using the formal posteriors without having to consider the possibility of strong inconsistency or marginalization paradoxes. For more general tasks involving the posteriors, the correction may have to be taken into account.
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References
Berger J.O. and Dongchu S 2008. Objective Priors for the Bivariate Normal Model, The Annals of Statistics, 36, 963–982.
Berger J.O., Bernardo J. M. and Dongchu S 2009. The Formal Definition of Reference Priors”, The Annals of Statistics, 37, 905–938.
Dawid A.P., Stone M. and Zidek J.V. 1973. Marginalization Paradoxes in Bayesian and Structural Inference, Journal of the Royal Statistical Society, Series B (Methodological), 35, 189–223.
Dawid A.P., Stone M. and Zidek J.V. 1996. Critique of E. T. Jaynes’s ‘Paradoxes of Probability Theory’, Research Report 172, Department of Statistical Science, University College London, available as http://www.ucl.ac.uk/stats/research/psfiles/172.zip..
Eaton M.L. and Sudderth W.D. 1995. The formal posterior of a standard flat prior in Manova is incoherent, Journal of the Italian Statistical Society, 2, 251–270.
Heath D and Sudderth W 1978. On Finitely Additive Priors, Coherence, and Extended Admissibility, The Annals of Statistics, 6, 333–345.
Heath D and Sudderth W 1989. Coherent Inference from Improper Priors and from Finitely Additive Priors, The Annals of Statistics, 17, 907–919.
Howson C and Urbach P 1993. Scientific Reasoning: The Bayesian Approach, Chicago: Open Court, Second edition.
Jaynes E.T. 1980. Marginalization and prior probabilities, Bayesian Analysis in Econometrics and Statistics, ed. A. Zellner, Amsterdam, North Holland.
Jeffreys H 1961. Theory of Probability, Oxford University Press.
Kass R.E. and Wasserman L. 1996. The Selection of Prior Probability Distributions by Formal Rules, Journal of the American Statistical Association, 91, 1343–1369.
Kolmogorov A.N. and Fomin S.V. 1970. Introductory Real Analysis, New York, Dover Publications.
Lindley D.V. 1973. Discussion on the Paper by Dr. Dawid, Professor Stone and Dr. Zidek, Journal of the Royal Statistical Society, Series B (Methodological), 35, 218–219.
Stone M 1970. Necessary and Sufficient Conditions for Convergence in Probability to Invariant Posterior Distributions”, The Annals of Mathematical Statistics, 41, 1349–1353.
Stone M and Dawid A.P. 1972. Un-Bayesian Implications of Improper Bayes Inference in Routine Statistical Problems, Biometrika, 59, 369–375.
Stone M 1976. Strong Inconsistency from Uniform Priors, Journal of the American Statistical Association, 71, 114–116.
Wallace D 1959. Conditional confidence level properties, Annals of Mathematical Statistics, 30, 864–876.
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Huisman, L. Infinitesimal Distributions, Improper Priors and Bayesian Inference. Sankhya A 78, 324–346 (2016). https://doi.org/10.1007/s13171-016-0092-0
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DOI: https://doi.org/10.1007/s13171-016-0092-0