Chapter 36 - Qualitative Possibility Functions and Integrals

Abstract

This chapter elaborates various aspects of qualitative possibility functions and integrals. Measure theory relies on numerical set-functions. In this theory, the relations induced by set-functions on a set of events are considered. However, fields such as decision theory and artificial intelligence consider numerical set-functions as sophisticated notions for the representation of uncertainty viewed from a subjective point of view. The purpose of uncertainty modeling includes the representation of belief. The simplest confidence relations are comparative possibilities and necessities, because they are fully characterized by complete preordering of elementary states. Hence, the computational complexity of representations of such confidence relations is linear in the number of states, while it is exponential in general for confidence relations. A possibility distribution is supposed to represent the state of knowledge of an agent relative to the state of affairs, either the current one (as of now) or the usual one (generally). The chapter discusses the qualitative possibilistic counterpart of conditional probability. The notion of independence is provided a precise definition in the probabilistic framework. Logical independence is implied by stochastic independence.