Thinking and Deciding

134 NORMATIVE THEORY OF PROBABILITY
With a tree like this, it is easy to calculate the probability of a given set of symptoms,
as long as each line (branch) on the tree has been assigned a probability and as
long as we know the prior probability of the disease. To get some idea of how this
is done, consider only the probabilities labeled a, b, and c in the tree diagram; these
are, respectively, p(P1|D1), p(S1|P1), and p(S2|P1). Considering only this part of
the whole tree, the probability of S1 and S2 together, given D1, would be a · b · c.
This is because the probability of the pathstate P1 given D1 is a, and if the pathstate
is present, the probabilities of the symptoms are b and c, respectively. Because the
symptoms are independent, given the pathstate, we can multiply their probabilities.
(For more details, see Duda, Hart, and Nilsson, 1976; Kelly and Barclay, 1973; Pearl,
1982; and Schwartz, Baron, and Clarke, 1988.)
Computers make complex calculations easy, and it may seem as though all we
need to do, in order to construct an expert system, is develop the best possible normative
theory and then program a computer with it. We must remember, however,
that expert systems are truly systems. They are not only computer programs: rather,
they are computer programs, plus their justifications, plus (most important) procedures
for eliciting probabilities from people (experts). Experts must be given the
best opportunity to use their knowledge, and, if necessary, they must be helped to
think about it in the most useful way. Thus, expert systems are prescriptive solutions
to practical problems. They are not necessarily just realizations of a normative
model. In particular, we need to do more descriptive research, to find out what kinds
of structures are most suited for elicitation of personal probabilities from experts and
to determine the extent to which expert opinion and frequency data can each be relied
upon. The design of expert systems depends as much on the psychology of experts
as on the mathematical possibilities.
Conclusion
The theory of probability describes the ideal way of making quantitative judgments
about belief. Even when we simply judge which of two beliefs is stronger, without
assigning numbers to either belief, the one we judge to be stronger should be the
one that is more probable according to the rules of probability (as applied to our
other beliefs). Let us now turn to the psychological research in this area to find out
how people ordinarily make judgments of these types. The difference between the
normative theory (described in this chapter) and the descriptive theory (Chapter 6)
will suggest some prescriptive improvements.
Answers to exercises:
1. p(neg|ca) =.208, p(neg|ben) =.904, p(ca) =.01
p(ca|neg) =
p(neg|ca)p(ca)
p(neg|ca)p(ca)+p(neg|ben)p(ben)
=
(.208)(.01)
= .0023
(.208)(.01) + (.904)(.99)
The lesson here is that negative results can be reassuring.