Open Access
October 2001 Improper Regular Conditional Distributions
Joseph B. Kadane, Mark J. Schervish, Teddy Seidenfeild
Ann. Probab. 29(4): 1612-1624 (October 2001). DOI: 10.1214/aop/1015345764


Improper regular conditional distributions (rcd’s) given a $\sigma$-field $\mathscr{A}$ have the following anomalous property. For sets $A \in \mathscr{A}, \mathrm{Pr}(A|\mathscr{A})$ is not always equal to the indicator of $A$. Such a property makes the conditional probability puzzling as a representation of uncertainty. When rcd’s exist and the$\sigma$-field $\mathscr{A}$ is countably generated, then almost surely the rcd is proper. We give sufficient conditions for an rcd to be improper in a maximal sense, and show that these conditions apply to the tail $\sigma$-field and the $\sigma$-field of symmetric events.


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Joseph B. Kadane. Mark J. Schervish. Teddy Seidenfeild. "Improper Regular Conditional Distributions." Ann. Probab. 29 (4) 1612 - 1624, October 2001.


Published: October 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1017.60007
MathSciNet: MR1880234
Digital Object Identifier: 10.1214/aop/1015345764

Primary: 60A10

Keywords: Completion of $\sigma$-field , countably generated $\sigma$-field , nonmeasurable set , symmetric $\sigma$-field , tail $\sigma$-field

Rights: Copyright © 2001 Institute of Mathematical Statistics

Vol.29 • No. 4 • October 2001
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