2.9 Choosing a Representation

Before concluding this chapter, a few words are in order regarding the problem of modeling a real-world situation. It should be clear that, whichever approach is used to model uncertainty, it is important to be sensitive to the implications of using that approach. Different approaches are more appropriate for different applications.

Probability has the advantage of being well understood. It is a powerful tool; many technical results have been proved that facilitate its use, and a number of arguments suggest that, under certain assumptions (whose reasonableness can be debated), probability is the only "rational" way to represent uncertainty.
Sets of probability measures have many of the advantages of probability but may be more appropriate in a setting where there is uncertainty about the likelihood.
Belief functions may prove useful as a model of evidence, especially when combined with Dempster's Rule of Combination.
In Chapter 8, it is shown that possibility measures and ranking functions deal well with default reasoning and counterfactual reasoning, as do partial preorders.
Partial preorders on possible worlds may be also more appropriate in setting where no quantitative information is available.
Plausibility measures provide a general approach that subsumes all the others considered and thus are appropriate for proving general results about ways of representing uncertainty.

In some applications, the set of possible worlds is infinite. Although I have focused on the case where the set of possible worlds is finite, it is worth stressing that all these approaches can deal with an infinite set of possible worlds with no difficulty, although occasionally some additional assumptions are necessary. In particular, it is standard to assume that the algebra of sets is closed under countable union, so that it is a σ-algebra. In the case of probability, it is also standard to assume that the probability measure is countably additive. The analogue for possibility measures is the assumption that the possibility of the union of a countable collection of disjoint sets is the sup of the possibility of each one. (In fact, for possibility, it is typically assumed that the possibility of the union of an arbitrary collection of sets is the sup of the possibility of each one.) Except for the connection between belief functions and mass functions described in Theorem 2.4.3, the connection between possibility measures and mass functions described in Theorem 2.5.4, and the characterization result for ≽^s in Theorem 2.7.6, all the results in the book apply even if the set of possible worlds is infinite. The key point here is that the fact that the set of possible worlds is infinite should not play a significant role in deciding which approach to use in modeling a problem.

See Chapter 12 for more discussion of the choice of the representation.