The Oil Drilling Case
An oil company is attempting to decide whether to drill for oil on a particular site. There are three possible states of nature:
1 No oil (state of nature S1, which we will denote as none)
2 Some oil (state of nature S2, which we will denote as some)
3 Much oil (state of nature S3, which we will denote as much)
Based on experience and knowledge concerning the site’s geological characteristics, the oil company feels that the prior probabilities of these states of nature are as follows:
In order to obtain more information about the potential drilling site, the oil company can perform a seismic experiment, which has three readings—low, medium, and high. Moreover, information exists concerning the accuracy of the seismic experiment. The company’s historical records tell us that
1 Of 100 past sites that were drilled and produced no oil, 4 sites gave a high reading. Therefore,
2 Of 400 past sites that were drilled and produced some oil, 8 sites gave a high reading. Therefore,
3 Of 300 past sites that were drilled and produced much oil, 288 sites gave a high reading. Therefore,
Intuitively, these conditional probabilities tell us that sites that produce no oil or some oil seldom give a high reading, while sites that produce much oil often give a high reading. Figure 19.1(a) shows a tree diagram that illustrates the prior probabilities of no, some, and much oil and the above conditional probabilities. This figure also gives the conditional probabilities for medium and low readings of the seismic experiment given each of the states of nature (none, some, or much).
Figure 19.1: A Tree Diagram and Probability Revision Tables for Bayes’ Theorem in the Oil Drilling Example
Now, suppose that when the company performs the seismic experiment on the site in question, it obtains a high reading. The previously given conditional probabilities suggest that, given this new information, the company might feel that the likelihood of much oil is higher than its prior probability P(much) = .1, and that the likelihoods of some oil and no oil are lower than the prior probabilities P(some) = .2 and P(none) = .7. To be more specific, we wish to revise the prior probabilities of no, some, and much oil to what we call posterior probabilities. We can do this by using Bayes’ Theorem as follows.
If we wish to compute P(none | high), we first calculate
Then Bayes’ Theorem says that
These calculations are summarized in part (b) of Figure 19.1. This table, which is called a probability revision table, also contains the calculations of the revised probabilities P(some | high) = .03125 and P(much | high) = .75. The revised probabilities in part (b) of Figure 19.1 tell us that, given that the seismic experiment gives a high reading, the revised probabilities of no, some, and much oil are .21875, .03125, and .75, respectively.
Since the posterior probability of much oil is .75, we might conclude that we should drill on the oil site. However, this decision should also be based on economic considerations, and we will see in Section 19.3 how to combine Bayesian posterior probabilities with economic considerations to arrive at reasonable decisions.
In this section we have only introduced Bayes’ Theorem. There is an entire subject called Bayesian statistics, which uses Bayes’ Theorem to update prior belief about a probability or population parameter to posterior belief. The use of Bayesian statistics is controversial in the case where the prior belief is largely based on subjective considerations, because many statisticians do not believe that we should base decisions on subjective considerations. Realistically, however, we all do this in our daily lives. For example, how each of us viewed the evidence in the O. J. Simpson trial had a great deal to do with our prior beliefs about both O. J. Simpson and the police.
Exercises for Section 19.1