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August 2011 On Bayes’s theorem for improper mixtures
Peter McCullagh, Han Han
Ann. Statist. 39(4): 2007-2020 (August 2011). DOI: 10.1214/11-AOS892


Although Bayes’s theorem demands a prior that is a probability distribution on the parameter space, the calculus associated with Bayes’s theorem sometimes generates sensible procedures from improper priors, 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 Θ satisfying Kingman’s 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’s 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.


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Peter McCullagh. Han Han. "On Bayes’s theorem for improper mixtures." Ann. Statist. 39 (4) 2007 - 2020, August 2011.


Published: August 2011
First available in Project Euclid: 24 August 2011

zbMATH: 1227.62007
MathSciNet: MR2893859
Digital Object Identifier: 10.1214/11-AOS892

Primary: 62F15
Secondary: 62C10

Keywords: Countable measure , lack of interference , marginalization paradox

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 4 • August 2011
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