Uncertainty and the Conditioning of Beliefs

Abstract

Uncertainty is part of the human condition. Whetherwe will or no, we must act, we must make decisions, in the face of uncertainty. Some authors have proposed that uncertainty be regarded as essentially a subjective matter. Our first goal is to draw the teeth of the classical subjectivistic argument that one must be prepared to meet all bets on the basis of one’s “degrees of belief.” The Dutch book theorem, which purports to have this as a consequence, is stated and criticized. Other criticisms of logical and subjective probability are considered. This leads to the consideration of alternative conceptions of how to represent epistemic uncertainty. A variety of alternatives have been offered, including, recently, Glenn Shafer’s theory of belief functions. An exposition of Shafer’s theory is offered. We then relate Shafer’s theory of belief functions to a theory that represents (and updates) uncertainty in terms of convex sets of classical probability functions. Finally, we discuss the question of the decision principles that can be employed in the case of both the convex set representation and the belief function representation of uncertainty.

Research on which this work is based was partially supported by the U.S. Army Signals Warfare Center.