170 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Manager's prior opinion of the product Survey conducted ? Survey result under risk'. If a survey is undertaken, the decision point could be A or D, that is the manager's prior notion and the market test are in agreement. If this is so, the test has contributed to the cost of the decision but not, pre- sumably, to the actual return on the decision, since the same decision for points A and D will likely be made as for points E and F respectively. Consider the remaining decision points B and C. Here the survey results and manager's priors are in conflict and we still have a problem. Is the decision to be made on the basis of the manager's priors, the survey information or perhaps a mixture of both? It would appear the purchase of imperfect information is either costly and useless or leads to a situation equally imponderable with the initial decision! To confuse the issue further, at any particular stage the actual environment is unknown. In other words, even if the manager's prior feeling and the test result are in agreement, they could both be wrong! Bayes theorem gives a way out here. This is a theorem concerning probabilities and may be written (4) p(Z and X) p(z IX') =

DECISION ANALYSIS 171 In words, the probability of an environment being Z, given information X, is the probability of Z and X divided by the probability of obtaining infor- mation X. Since p(Z and X) -- p(X and Z), we have p(Z IX) p(X) = p(X I Z) p(Z). (4) Returning to our problem, the manager has a prior distribution p(Z) for the probability that various environments will prevail, and based solely on this information would make a decision on the basis of that policy X i which maximizes his expected return Y_,Rop(Zj). J Having now received the information X, say, of the survey, presumably the manager can be expected to 'revise' his probability distribution from p(Z) to p(Z [ X) that is the probability of Z given information X. This being the case, the expected return on policy X• is now which using (4) becomes In the particular case under discussion, there are two possible pieces of (1) information X resulting from the survey: X--a favourable environment and therefore advise invest, and X--an unfavourable environment. For each value of X, the policy leading to the greatest expected value will, of course, be selected. This being so, the expression may accordingly be written EV(Xi) = Zlmax ZR•p(X I Zj)p(Zj)} x• t J (1) (2) -- max Y-,RoP(XI Zj)p(Z j) + max ZRup(xIgj)p(gj). i j i j (5) Suppose, for the purpose of demonstration, an analysis of past experience with survey tests indicates they can be expected to give the correct result some 80•o of the time (it is assumed the statement 'correct result' in this context is understood). This being so, we have the following situation.