Chapter 2: Representing Uncertainty


Overview

Do not expect to arrive at certainty in every subject which you pursue. There are a hundred things wherein we mortals must be content with probability, where our best light and reasoning will reach no farther.

—Isaac Watts

How should uncertainty be represented? This has been the subject of much heated debate. For those steeped in probability, there is only one appropriate model for numeric uncertainty, and that is probability. But probability has its problems. For one thing, the numbers aren't always available. For another, the commitment to numbers means that any two events must be comparable in terms of probability: either one event is more probable than the other, or they have equal probability. It is impossible to say that two events are incomparable in likelihood. Later in this chapter, I discuss some other difficulties that probability has in representing uncertainty.

Not surprisingly, many other representations of uncertainty have been considered in the literature. I examine a number of them here, including sets of probability measures, Dempster-Shafer belief functions, possibility measures, and ranking functions. All these representations are numeric. Later in the chapter I also discuss approaches that end up placing a nonnumeric relative likelihood on events. In particular, I consider plausibility measures, an approach that can be viewed as generalizing all the other notions considered.

Considering so many different approaches makes it easier to illustrate the relative advantages and disadvantages of each approach. Moreover, it becomes possible to examine how various concepts relevant to likelihood play out in each of these representations. For example, each of the approaches I cover in this chapter has associated with it a notion of updating, which describes how a measure should be updated in the light of additional information. In the next chapter I discuss how likelihood can be updated in each of these approaches, with an eye to understanding the commonalities (and thus getting a better understanding of updating, independent of the representation of uncertainty). Later chapters do the same thing for independence and expectation.




Reasoning About Uncertainty
Reasoning about Uncertainty
ISBN: 0262582597
EAN: 2147483647
Year: 2005
Pages: 140

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