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On Bayes’s theorem for improper mixtures. (English) Zbl 1227.62007

Summary: Although Bayes’ theorem demands a prior that is a probability distribution on the parameter space, the calculus associated with Bayes’ theorem sometimes generates sensible procedures from improper priors, the Pitman’s estimator being a good example. However, improper priors may also lead to Bayes procedures that are paradoxical or otherwise unsatisfactory, prompting some authors to insist that all priors be proper.
This paper begins with the observation that an improper measure on \(\Theta \) satisfying J. F. C. Kingman’s [Poisson processes. Oxford: Clarendon Press (1993; Zbl 0771.60001)] countability condition is in fact a probability distribution on the power set. We show how to extend a model in such a way that the extended parameter space is the power set. Under an additional finiteness condition, which is needed for the existence of a sampling region, the conditions for Bayes’ theorem are satisfied by the extension. Lack of interference ensures that the posterior distribution in the extended space is compatible with the original parameter space. Provided that the key finiteness condition is satisfied, this probabilistic analysis of the extended model may be interpreted as a vindication of improper Bayes procedures derived from the original model.

MSC:

62C10 Bayesian problems; characterization of Bayes procedures
62A01 Foundations and philosophical topics in statistics
62F15 Bayesian inference

Citations:

Zbl 0771.60001
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References:

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