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Improper Regular Conditional Distributions

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

Abstract:

Improper regular conditional distributions (rcd's) given a $\sigma$-field $\cal{A}$ have the following anomalous property. For sets $A \in \cal{A}$, $\Pr(A \mid \cal{A})$ 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 $\cal{A}$ 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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