15.1 Combining Discrete Information 357
combined random variableZis given by
Pr(Z=v)=Pr(X=v|X=Y)=Pr(X=vandX=Y)
Pr(X=Y).
Now(X=vandX=Y)is logically equivalent to(X=vandY=v). Since
XandYare independent,Pr(X=vandY=v)=Pr(X=v)Pr(Y =v).
Therefore,
Pr(Z=v)=Pr(X=v)Pr(Y=v)
Pr(X=Y).
The result then follows.
Information combination depends on two important criteria: the obser-
vations must be independent and they must be measuring the same phe-
nomenon. As we saw in chapters 2 and 4, the essence of semantics is the
determination of when two entities are the same. Information combination
is also determined by sameness. Indeed, information combination can be
regarded as the basis for the semantics of uncertainty.
Another important assumption of the information combination theorem
above is that the event representing the sameness of two random variables
is(X=Y), i.e., where the two observations are exactly the same. However,
it is conceivable that in some situations the sameness relationship could be
more complicated. This is especially true when the values being observed
are not sharply distinguishable. For example, not everyone uses the same
criteria to characterize whether a person is obese or overweight. As a result,
independent observations can be calibrated differently. Such observations
should not be combined unless they can be recalibrated. In section 15.5, we
consider a more general event for representing the sameness of two discrete
random variables that addresses these concerns to some degree.
To illustrate how the information combination theorem can be applied,
suppose that a patient is complaining of a severe headache. For simplicity,
assume that the only possible diagnoses are concussion, meningitis, and tu-
mor. One doctor concludes that the probabilities of the diagnoses are 0.7,
0.2, and 0.1, respectively. Another doctor concludes that the probabilities
are 0.5, 0.3, and 0.2, respectively. Combining these two yields the probabil-
ities 0.81, 0.14, and 0.05, respectively. Note that the most likely diagnosis
becomes more likely, while the least likely one becomes less likely. The rea-
son for this is that the diagnoses have been assumed to beindependent.In
practice, diagnoses will be based on tests and symptoms that are observed
by both doctors. In addition, doctors have similar training and use the same