328 13 Inductive vs. Deductive Reasoning
know how often the symptom occurs when a person has the disease, then
one knowsPr(B|A). Bayes’ law then gives the probability that a person
has the disease when the symptom is observed. In other words, Bayes’ law
gives important information which can be used for the diagnosis of diseases
based on symptoms. Specific examples of the use of Bayes’ law for diagnosis
are given in section 14.2.
As we discussed in section 13.1, The applicability and interpretation of
probability theory has been the subject of both philosophical and mathe-
matical analysis since the time it was originally founded. More recently,
other ways of expressing uncertainty have emerged, as we discussed in sec-
tion 13.2. The question is which of these approaches to uncertainty is the
best.
The current methodology for such comparing approaches to uncertainty
was introduced by De Finetti (De Finetti 1937) who formulated subjective
probability in terms of betting against an adversary. This formulation is
called theDutch bookargument. This made it possible to extend the appli-
cability of probability to questions such as “Will the sun rise tomorrow?” or
“Was there life on Mars?” It also can be used to prove the power of proba-
bility theory in general, and Bayesian analysis in particular. The argument
is that if one knows that an agent consistently follows a non-Bayesian belief
system in a known way, then one can arrange the bets so that the Bayesian
alwayswins (not just on average). If the departure from Bayesian analysis is
inconsistent, then the Bayesian can only win on average.
Although stated in financial terms, the Dutch book argument applies equal-
ly well to any activity which involves some form of utility, whether it is
financial or not, and the associated risk in trying to increase this utility. It fol-
lows that Bayesian analysis is a minimal requirement for rational inference
in experimental science.
There are significant advantages to probability theory as a mechanism for
expressing uncertainty. It is the only approach that is empirically grounded,
and it can be used either empirically or subjectively. Furthermore, Bayesian
analysis will always win over a non-Bayesian analysis whenever one quan-
tifies the risks associated with decisions based on the events in question.
However, probability theory has significant disadvantages. It is much
more computationally complex than the extensional approaches. Specify-
ing a general JPD is a formidable task as the number of random variables
increases. Even for random variables that can take only two values, if there
are 20 random variables, then a joint probability distribution has over 10^6
probabilities. Accordingly, it is very common to assume that the random