358 15 Combining Information
criteria for making diagnoses. As a result, the diagnoses would not usually
be independent.
For a more extreme example, suppose that the first doctor concludes that
the probabilities are 0.9, 0.0, and 0.1; and the second doctor gives the prob-
abilities as 0.0. 0.9, and 0.1. The combined distribution will conclude that
the tumor has probability 1.0, while the other two diagnoses are impossible
(Zadeh 1984). This seems to be wrong. However, it makes perfectly good
sense. In the words of Sherlock Holmes in “The Blanched Soldier”, “When
you have eliminated all which is impossible, then whatever remains, how-
ever improbable, must be the truth.” Each of the doctors has concluded that
one of the diagnoses is impossible, so the third possibility must be the truth.
On the other hand, one can question whether such totally different diagnoses
would happen independently. In other words, it is unlikely that the doctors
are independently observing the same phenomenon. Such observations are
said to beincompatible.
However, there are other circumstances for which observations that are
incompatible are actually independent and therefore fusable. For example,
if one distribution represents the probability of occurrence of a rare disease,
and another distribution represents the observation that a particular patient
definitely has the disease, then the combination of the two distributions is
simple: the patient has the disease.
An even more extreme example would be two observations in which all of
the possibilities have been declared to be impossible in one or the other ob-
servation. The discrete information combination theorem gives no combined
distribution in this case because the hypotheses are not satisfied. One says
that such observations areinconsistent.Summary
- The discrete information combination theorem gives the formula for fus-
ing independent discrete random variables that measure the same phe-
nomenon. - Incompatible PDs can be combined but care must be taken to interpret the
combined distribution properly. - Inconsistent PDs cannot be combined at all.