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222 Revising judgments in the light of new information
Let us first consider a situation where the geologist is not very confident
about his prior probabilities and where the test drilling is very reliable.
The ‘vaguest’ prior probability distribution that the geologist can put
forward is to assign probabilities of 0.5 to the two events ‘gas exists at
the location’ and ‘gas does not exist at the location’. Any other distribution
would imply that the geologist was confident enough to make
some commitment in one direction. Clearly, if he went to the extreme of
allocating a probability of 1 to one of the events this would imply that he
was perfectly confident in his prediction. Suppose that having put forward
the prior probabilities of 0.5 and 0.5, the result of the test drilling is
received. This indicates that gas is present and the result can be regarded
as 95% reliable. By this we mean that there is only a 0.05 probability that
it will give a misleading indication. (Note that we are assuming, for simplicity,
that the test drilling is unbiased, i.e. it is just as likely to wrongly
indicate gas when it is not there as it is to wrongly indicate the absence
of gas when it is really present.) Figure 8.5 shows the probability tree
and the calculation of the posterior probabilities. It can be seen that these
probabilities are identical to the probabilities of the test drilling giving a
correct or misleading result. In other words, the posterior probabilities
depend only upon the reliability of the new information. The ‘vague’
prior probabilities have had no influence on the result.
A more general view of the relationship between the ‘vagueness’ of
the prior probabilities and the reliability of the new information can
Events
New information
Prior probabilities Conditional probabilities
0.5
Gas exists at location
0.95
Drilling indicates gas
Gas doesnotexist Drilling indicates gas
0.5
0.05
Joint probabilities Posterior probabilities
0.5 × 0.95 = 0.475 0.475
= 0.95
0.5
0.5 × 0.05 = 0.025
0.5
0.025
= 0.05
0.5
Figure 8.5 – The effect of vague prior probabilities and very reliable information