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6. DEMPSTER-SHAFER APPROACHA key issue in using certainty factors is whether or not belief can be expressed in a single number. Dempster-Shafer (DS) can handle the narrowing of hypotheses by considering all combinations in the hypothesis set (the power set). The idea is that often the expert finds evidence that bears on a subset of hypotheses rather than just one. The evidence narrows down the likely set but does not bear on just one H. DS includes both Bayesian and CF functions as special cases (Gordon and Shortliffe, 1985a, p. 273). It is based on better mathematical foundations than the CF approach, but the computational requirements are more complex. The frame of discernment [fancy theta] is the set of mutually exclusive and exhaustive hypotheses or ground propositions. So if we have the exclusive and exhaustive set {H1,H2,H3,H4} from the earlier bread baking example, then [fancy theta] = {H1,H2,H3,H4}. Figure 3 shows the powerset that includes the 2[fancy theta] possible subsets of {H1,H2,H3,H4}. DS uses a number between [0,1] to represent the degree of belief in a H, where H could represent any subset of the powerset. The impact of evidence on [fancy theta] is represented by what is called a basic probability assignment (bpa). Remember that a basic probability function would only assign probabilities to H1, H2, H3, and H4 singly. The bpa, on the other hand, assigns a probability m to every subset of [fancy theta] so that [cap sigma]m(x) = 1, x [element of] 2[fancy theta]. Every element in the powerset has a degree of belief assigned to it. By convention, the null set has the bpa = 0. The value m([fancy theta]) refers to that proportion of total belief that remains unassigned to any subset of [fancy theta].
As an example, suppose that evidence supports the hypothesis that the bread did not rise because of a yeast problem (0.7) but does not discern between H1 and H2, bad or omitted/mismeasured yeast. Then m({H1,H2}) = 0.7 and m([fancy theta]) = 0.3. Note that for this to be workable in an expert system context, we would have to have rules that correspond to this notion. For example: if A and B, then H1, H2 (m = 0.7). The belief function Bel can be determined for any H in [fancy theta]. Suppose we want to compute Bel ({H1,H2}). Bel ({H1,H2}) in DS corresponds to the total belief in {H1,H2} and all subsets of {H1,H2}. Thus, Bel ({H1,H2}) = m({H1,H2}) + m({H1}) + m ({H2}). It follows that the Bel([fancy theta]) is always equal to 1. Naturally, in order to use DS in an expert system context, it is also necessary to have a means of combining belief functions to derive a combined belief function when we have more than one observation within the same frame of discernment. Assume again that we have m1({H1,H2}) = 0.8, and we also have m2({H2,H3}) = 0.6. Table 1 shows what Gordon and Shortliffe (1985b) call an intersection tableau that provides a basis for computing the net belief denoted as m1 [direct product] m2. So, the combined belief in {H1,H2} designed Bel1 [direct product] Bel2 is equal to: 7. BAYESIAN BELIEF NETWORKS7.1. WHAT IS A BELIEF NETWORK?Concisely stated, a belief network is a graphical data structure that compactly represents the joint probability distribution (PD) of a problem domain by exploiting conditional dependencies. In what follows, the details involved in explicating the above definition will be expounded. It will also be evident that the belief network captures knowledge of a given problem domain in a natural and efficient way by using causal relationships. Several references provide a good introduction to belief networks (Charniak, 1991; Heckerman, 1995). Pearl (1988) is a comprehensive work on belief networks. The remainder of this section is organized as follows. First, the reasons for using the belief network model are given. Second, the structure of the model is highlighted. Third, the process of building a belief network is enumerated. Fourth, an example of an algorithm that calculates the probabilities desired is discussed. Last, a concrete example is given to illustrate the concepts explained herein.
7.2. WHY USE A BELIEF NETWORK?Belief networks allow one to reason about uncertainty by exploiting conditional dependencies to compactly represent the joint probability distribution, thereby saving computational time and space. What kind of inferences can be made? There are at least four kinds of inferences. Diagnostic inference leads toward finding the probable cause of given evidence in the form of symptoms. Causal inference reasons toward the effect or symptom where the evidence is most likely the cause of the symptom. Intercausal inference reasons between causes of a common effect to find the most likely cause. Finally, mixed inference is a combination of two or more of the above. (Russell and Norvig, 1995).
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