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.
Citation
Joseph B. Kadane. Mark J. Schervish. Teddy Seidenfeild. "Improper Regular Conditional Distributions." Ann. Probab. 29 (4) 1612 - 1624, October 2001. https://doi.org/10.1214/aop/1015345764
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