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224 Revising judgments in the light of new information
his belief, or where the prior probability is very small, implying that he
strongly believes that gas will not be found. In the latter case, so strong
is his disbelief that he severely restricts the effect of the disconfirming
evidence from the test drilling.
At the extreme, if your prior probability of an event occurring is
zero then the posterior probability will also be zero. Whatever new
information you receive, no matter how reliable it is, you will still
refuse to accept that the event is possible. In general, assigning prior
probabilities of zero or one is unwise. You may think that it is impossible
that San Marino will become a nuclear power by the year 2020. However,
if you hear that seismic instruments have detected signs of nuclear testing
there then you should allow yourself some chance of being persuaded
by this information that the event might just be possible. Assigning a
very small but non-zero prior probability might therefore be wiser.
Ironically, if the new information has less than a 0.5 chance of being
reliable its result is of interest and the more unreliable it is, the greater
the effect it will have on the prior probability. For example, if the test
drilling is certain to give the wrong indication then you can be sure that
the opposite of what has been indicated is the true situation!
Applying Bayes’ theorem to a decision problem
We will now consider the application of Bayes’ theorem to a decision
problem: a process which is sometimes referred to as posterior analysis.
This simply involves the use of the posterior probabilities, rather than
the prior probabilities, in the decision model.
A retailer has to decide whether to hold a large or a small stock of a
product for the coming summer season. A payoff table for the courses
of action and outcomes is shown below:
(Profits)
Decision Low sales High sales
Hold small stocks $80 000 $140 000
Hold large stocks $20 000 $220 000
The following table shows the retailer’s utilities for the above sums of
money (it can be assumed that money is the only attribute which he is
concerned about):