Department of Statistics Unitmark
Dietrich College of Humanities and Social Sciences

Improper Regular Conditional Distributions

Publication Date

October, 2000

Publication Type

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.