A decision theory for non-probabilistic uncertainty and its applications

• Authors: Phan Hong Giang;Prakash P. Shenoy
• Venue: A decision theory for non-probabilistic uncertainty and its applications
• Year: 2003

In this dissertation, we study decision making under uncertainty, which is described by a family of non-probabilistic formalisms. This family includes Zadeh's fuzzy possibility theory, Spohn's theory of epistemic belief, extended likelihood functions and Walley's partially consonant belief functions. Possibility theory is apt for expressing the fuzziness of knowledge and the granularity of belief. Spohn's theory of epistemic belief, which can be interpreted as order-of-magnitude approximation of probability, is apt for describing the notion of plain belief. An extended likelihood function conveys relevant information obtainable from a statistical experiment when no prior information available. A variant of Dempster-Shafer's theory of belief functions, the theory of partially consonant belief functions, which include probability and possibility functions as extreme cases, offers a generalization of Bayesian statistics. We employ an axiomatic approach to the decision problem. Axiom systems for preference relations on non-probabilistic lotteries are proposed. The axioms are structurally similar to those of von Neumann-Morgenstern's linear utility theory. Our main results are theorems on the representation of the preference relations by utility functions. Unlike the linear utility of a probabilistic lottery, the utility of a non-probabilistic lottery is a binary construct. Each binary utility level is a pair of numbers that account for a positive and a negative sides of a lottery which are not strictly complementary. We compare our approach with proposals in literature. We show that our approach is more general than the pessimistic and the optimistic decision criteria proposed Dubois et al. Also, the use of binary utility and the way utility is combined with uncertainty set our approach apart from proposals for decision theory with non-additive probability. In many of those proposals, it is implicitly assumed that utility should be combined with uncertainty through multiplication. We argue that this assumption is not always justifiable. We apply our decision theory for the problem of statistical inference. We suggest a novel solution for the statistical decision problem. By allowing prior information to be expressed by partial consonant belief functions, our solution generalizes the Bayesian solution.