## Improper Regular Conditional Distributions

October, 2000

Tech Report

### Author(s)

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.