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15.5 Dempster-Shafer Theory 365

random sample of size 200. In this case the distribution of the measurement

has mean (9.93, 15.53) and variance matrix

(

0 .078 0

02. 42

)

. The second way
of combining the information is closer to the true value (10, 16), but both
measurements are within one standard deviation of the true value, so they
are both as accurate as one would expect. Fusing the samples will usually be
more accurate because it makes use of more information about the measure-
ments. The information combination technique uses only the distributions,
not how the distributions were obtained. However, the advantage is small
compared with the estimation error of either technique.

Summary

• PDs can be measured.


• Many standard statistical tests are based on measurements of PDs.


• Independent measurements of PDs can be combined, just like any other
  kinds of measurements.

15.5 Dempster-Shafer Theory

The analogy between the cosine similarity function for the vector space model

in section 6.2 and the information combination theorem is intriguing. In both

cases, one computes the product of weights that occur in the same position in

a vector. In the vector space model the positions in the vector correspond to

terms that can occur in a document. In the information combination theorem

the positions are the possible values of the two random variables. In sec-

tion 6.5 we discussed how the vector space model could be extended to deal

with concept combinations. Surprisingly, a form of concept combination has

been considered for information combination. This interesting development

is called the Dempster-Shafer (D-S) theory of evidence (Shafer 1976).

D-S theory assumes that one has a set of entities which can be assigned

probabilities, and the sum of the probabilities for all entities adds up to 1,

exactly as in ordinary probability theory. The difference is that D-S theory

allows entities to be combinations of other entities. Compared with the con-

cept combinations considered in section 6.5, D-S combinations are very sim-

ple. A D-S combination is the set-theoretic union of its constituents. Thus,

while there are many ways to combine the two terms “test” and “drug,” D-S

theory has only one way to combine them.