Bayesian Analysis A New Approach to Statistical Decision-Making

Abstract

Introduction A small revolution is going on in statistics today as the emphasis is slowly shifting from description to inference to decision-making. The newest branch of statistics, grouped generally under terms such as "statistical decision theory" or "Bayesian statistics", had its beginnings many years ago in ideas expounded by Bayes, with more recent contributions from Savage, Wald, Raiffa and Schlaifer. Rather than contradict, these new ideas extend "classical" statistics, particularly in the areas of significance tests and confidence-interval estimates, by introducing concepts of personal probabilities and economic gains and losses directly into the statistical calculations. These new approaches are quite powerful and also quite controversial. This paper will seek to illustrate some points of agreement and differences between the two approaches, with an illustration of the use of Bayesian statistics in a very simplified decision problem. An Example Decision Problem An oil company is considering an exploration program in a geologic region. Wells cost $250,000 each, and the firm believes that the exploration program will be approximately a breakeven proposition (or that the firm will earn its minimum rate of return) if at least 20 per cent of the wells are successful. This implies that "success" on any one well is defined for this firm as at least a present-value $1,250,000 well $1,250,000 (.20) = $250,000. For simplification only, this present value is assumed to be a constant. The variable is the success ratio P. Should the firm begin the drilling program? Success or failure depends on whether the unknown parameter P is less than, equal to, or greater than .20. The Classical Approach Two Hypotheses In classical statistical terms, the problem could be stated as a decision between two competing hypotheses on which evidence may be brought to bear: H(o): P greater than .20; and H(1): P less than .20.Actually, there are many possible values for P, ranging from 0 to 1.0, and the true value of P is unknown. The value P = .20 can be thought of as the breakeven value P(b) for which the firm would be about indifferent between drilling or not drilling. H(o) could be labeled the "null" hypothesis, inasmuch as rejection of this hypothesis when it is true may be regarded by this firm as more serious than acceptance of it when it is false. H(1), therefore, becomes the "alternate" hypothesis, incidentally, some operators might prefer to relabel these, feeling that drilling when P less than .20 is more serious, but most oil men probably feel that missing out on a potentially productive area is more serious than spending money on a nonproductive area. Errors Now if the firm rejects H, when it is actually true (in this case the firm abandons the program when, actually, it is potentially profitable), it is said to be an error of the "first kind", Type I. if the firm accepts H(o) when it is actually false (in this case, the firm spends money when the program is actually not profitable), then the firm is said to incur an error of the "second kind", Type II. This is set forth in Table 1.What shall the firm do? if no one in the organization is willing to express any feeling (probability) one way or another as to whether either of the two hypotheses is more likely to be true than the other, then the firm is in danger of the same fate as Buridan's ass who starved to death between two bales of hay because they were equidistant. Sampling to Formulate Decision Rules At this point, the classical statistician would probably suggest a sample, i.e., drill a few wells. And, based on the sample, the statistician would talk about an inference concerning the true value of P by choosing between the two hypotheses about this value. He generally would focus his attention on the choice between rejection and acceptance of the supposedly more serious null hypothesis. Of course, such acceptance or rejection does lead to a "terminal" action which, in this example, would either begin or not begin an expanded drilling program. TABLE 1 - STATE OF THE WORLD p .20 p .20 H(o): p .20 Correct Type II H(1): p .20 Type I Correct JPT P. 603^

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