The Annals of Probability

Improper Regular Conditional Distributions

Joseph B. Kadane, Mark J. Schervish, and Teddy Seidenfeild

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Abstract

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.

Article information

Source
Ann. Probab., Volume 29, Number 4 (2001), 1612-1624.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1015345764

Digital Object Identifier
doi:10.1214/aop/1015345764

Mathematical Reviews number (MathSciNet)
MR1880234

Zentralblatt MATH identifier
1017.60007

Subjects
Primary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}

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

Citation

Seidenfeild, Teddy; Schervish, Mark J.; Kadane, Joseph B. Improper Regular Conditional Distributions. Ann. Probab. 29 (2001), no. 4, 1612--1624. doi:10.1214/aop/1015345764. https://projecteuclid.org/euclid.aop/1015345764


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