POSSIBILITY THEORY III: POSSIBILISTIC INDEPENDENCE

Abstract

The introduction of the notion of independence in possibility theory is a problem of long-standing interest. Many of the measure-theoretic definitions that have up to now been given in the literature face some difficulties as far as interpretation is concerned. Also, there are inconsistencies between the definition of independence of measurable sets and possibilistic variables. After a discussion of these definitions and their shortcomings, a new measure-theoretic definition is suggested, which is consistent in this respect, and which is a formal counterpart of the definition of stochastic independence in probability theory. In discussing the properties of possibilistic independence, I draw from the measure- and integral-theoretic treatment of possibility theory, discussed in Part I of this series of three papers. I also investigate the relationship between this definition of possibilistic independence and the definition of conditional possibility, discussed in detail in Part II of this series. Furthermore, I show that in the special case of classical, two-valued possibility the definition given here has a straightforward and natural interpretation.

INDEX TERMS:

• Possibility theory
• possibilistic independence
• logical independence
• conditional possibility