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366 15 Combining Information

Consider the diagnosis problem of the patient suffering from a headache

introduced at the beginning of this chapter. D-S theory, not only includes

the three basic diagnoses, concussion, meningitis, and tumor, it also allows

combined diagnoses such as concussion-meningitis. Note that such a com-

bination is a separate point in the probability space, distinct from either con-

cussion or meningitis: it does not represent an event such as (concussion

OR meningitis). Try to imagine a new kind of trauma called a “concussion-

meningitis” or “meningitis-concussion.” Remember that there is only one

way to combine entities in D-S theory, so these two must be regarded as be-

ing the same even though most interpretations would regard them as being

different. The following are two examples of D-S distributions:

Diagnosis Distribution P Distribution Q

concussion 0.5 0.6

meningitis 0 0.2

tumor 0.3 0.2

concussion-meningitis 0.2 0

All entities in D-S theory are of three kinds:elementary, such as concussion

and meningitis; acombination,such as concussion-meningitis; or the “empty

entity,” corresponding to the set-theoretic notion of an empty set. The empty

entity plays a special role in D-S theory. We will say that a D-S distribution is

elementaryif all of its probability is on elementary entities. The distribution

Q above is elementary, while distribution P is not.

The most important contribution of D-S theory is Dempster’s rule of com-

bination which specifies how to combine independent evidence. Unlike the

discrete information combination theorem, Dempster’s rule is postulated as

an axiom; it is not proven based on some underlying theory.

Dempster’s Rule of Combination

LetXandYbe two D-S distributions representing independent evidence for the

same phenomenon. Define a combination distribution∑ Mby the formulaM(C)=

A∩B=CX(A)Y(B), for all nonemptyC. If there exists some nonemptyCsuch

thatM(C)=0, then there exists a D-S distributionZwhich combinesXandY

and is defined by the formula

Z(C)=∑ M(C)

nonemptyDM(D)

,

for every nonemptyC. In other words,Zis obtained fromMby rescaling so that

the probabilities add to 1.