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Improper versus finitely additive distributions as limits of countably additive probabilities

Author

Listed:
  • Pierre Druilhet

    (Université Clermont Auvergne)

  • Erwan Saint Loubert Bié

    (Université Clermont Auvergne)

Abstract

The Bayesian paradigm with proper priors can be extended either to improper distributions or to finitely additive probabilities (FAPs). Improper distributions and diffuse FAPs can be seen as limits of proper distribution sequences for specific convergence modes. In this paper, we compare these two kinds of limits. We show that improper distributions and FAPs represent two distinct features of the limit behavior of a sequence of proper distribution. More specifically, an improper distribution characterizes the behavior of the sequence inside the domain, whereas diffuse FAPs characterizes how the mass concentrates on the boundary of the domain. Therefore, a diffuse FAP cannot be seen as the counterpart of an improper distribution. As an illustration, we consider several approach to define uniform FAP distributions on natural numbers as an equivalent of improper flat prior. We also show that expected logarithmic convergence may depend on the chosen sequence of compact sets.

Suggested Citation

  • Pierre Druilhet & Erwan Saint Loubert Bié, 2021. "Improper versus finitely additive distributions as limits of countably additive probabilities," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(6), pages 1187-1202, December.
    • Handle: RePEc:spr:aistmt:v:73:y:2021:i:6:d:10.1007_s10463-020-00779-8
    DOI: 10.1007/s10463-020-00779-8
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    References listed on IDEAS

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    1. Joseph B. Kadane & Jiashun Jin, 2014. "Uniform Distributions on the Integers: A connection to the Bernouilli Random Walk," Econometric Reviews, Taylor & Francis Journals, vol. 33(1-4), pages 372-378, June.
    2. Taraldsen, Gunnar & Lindqvist, Bo Henry, 2010. "Improper Priors Are Not Improper," The American Statistician, American Statistical Association, vol. 64(2), pages 154-158.
    3. Gunnar Taraldsen & Bo Henry Lindqvist, 2016. "Conditional probability and improper priors," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(17), pages 5007-5016, September.
    4. Leendert Huisman, 2016. "Infinitesimal Distributions, Improper Priors and Bayesian Inference," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(2), pages 324-346, August.
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