About Updating

Survey of several forms of updating, with a practical illustrative example. We study several updating (conditioning) schemes that emerge naturally from a common scenario to provide some insights into their meaning. Updating is a subtle operation and there is no single method, no single 'good' rule. The choice of the appropriate rule must always be given due consideration. Planchet (1989) presents a mathematical survey of many rules.We focus on the practical meaning of these rules. After summarizing the several rules for conditioning, we present an illustrative example in which the various forms of conditioning can be explained. 1. CONDITIONING RULES FOR BELIEF FUNCTIONS. Let Ω be a finite set with elements ω1, ω2, ...ωn. The ωi are mutually exclusive possible answers to a given question. Let bel: 2Ω→[0,1] be a belief function over Ω, with m its corresponding basic belief mass assignment. For A”Ω, bel(A) quantifies our degree of belief that the true answer to the question is in A. If the ω i are an exhaustive list of possible answers, the closed-world assumption prevails (as no solution exists outside Ω), otherwise the open-world assumption prevails (Smets 1988) Suppose a new piece of evidence that says that the answer is not in A, where A is the set of ωi that are elements of Ω, but not of A. To say that the answer is not in A does not mean that the answer is in A. Under closed-world assumption both statements are equivalent. Under openworld assumption, they are not. Here, the answer might be none of the elements of Ω. When I say "the answer is not in A", I only eliminate the elements of A as possible answer. When I say "the answer is in A", not only I eliminate the elements of A, but I also claim that the answer is an element of A a much stronger claim. This distinction is at the origin of the difference between the open-world and the closed-world assumptions. The updating of bel induced by the information that the answer is not in A can be performed variously. It will depend on the interpretation of the problem that bel is supposed to model. Conditioning C.1. The Unnormalized Dempster's Rule of Conditioning. For all X”Ω, the basic belief masses m(X) given to X is transferred to A∩ X. The basic belief masses assignment mA , the belief function belA and the plausibility function plA obtained after conditioning on A are: mA(B) = ∑ X”A m(B∪X) for B”A mA(B) = 0 for B / ”A belA(B) = bel(B∪A) − bel(A) for B”Ω plA(Β) = pl(Β∩A) for B”Ω This solution is always applicable, even when pl(A) = 0. Notice that mA(Ø) might be non null with this solution. In fact, one has: mA(Ø) = m(Ø) + bel(A). This rule is the one described in the transferable belief model under open world assumption (Smets 1988). Conditioning C.2. The Normalized Dempster's Rule of Conditioning. Masses are transferred as in conditioning C.1, but the result is then proportionally normalized to cope with the masses that would be transferred to the empty set. This avoids ending with a positive mass on Ø, and guarantees that bel(Ω) = 1. After conditioning, one gets: mA(B) = c ∑ X”A m(B∪X ) for B”A, B≠Ø mA(B) = 0 for B / ”A and for B”Ω belA(B) = c (bel(B∪A) − bel(A) ) plA(Β) = c pl(Β∩A) where c-1 = bel(Ω) − bel(A) = pl(A). This solution applies only when pl(A)>0. No solution is provided when pl(A)=0. This rule is the one described in the transferable belief model under closed world assumption. It is the one initially proposed by Shafer (1976a). Conditioning C.2'. Bayesian solution. The classical solution with probability function is, PA(Β) = P(Β∩A) P(A) for B”Ω. This solution applies only if P(A)>0. If P(A)=0, no solution is provided. This is a particular case of conditioning C.2. It is obtained when the belief function bel is a probability function. Conditioning C.3. Yager-Kohlas's Solution. The basic belief masses are transferred as in C.1, but the masses that could be transferred to the empty set are reallocated to A, so: mA(B) = ∑ X”A m(B∪X) ∀B”A,B≠A,B≠Ø mA(A) = ∑ X”A m(A∪X) + bel(A)