Intuition, Experience and Quantitative Decision-Making:
A Study of the Bayesian Model
James Washington
Economics Statistics I
May 11, 2002
Introduction
A book buyer at amazon.com is facing a difficult situation. There are only 100 copies left of Diana: Her True Story, a keepsake photo album chronicling the life and times of the late Princess Diana. Past sales performance data shows the book has been a best-seller -- evidence which might cause the buyer to place another order, and quickly. But his gut feeling tells him otherwise; he's heard that a definitive biography of Diana is on the way, and moreover, he's guessing that the strong past sales are bound to fade with memory of her untimely death. He doesn't want to risk overstocking, but he doesn't want amazon.com to be caught empty-handed if more orders come in. How does the buyer make his decision -- based on the facts alone, or by gambling with his gut instinct?
Quantitative decision-making is thought of most often as an objective exercise based only on the cold analysis of verifiable hard facts. Intuition and even experience tends to be excluded from quantitative decision-making on the grounds that such information is subjective in character, and, thus, has no role in quantitative analysis.
But decisions based solely on the analysis of objective data alone are often less effective at predicting future events than those based on a mixture of facts and "hunches" or informed guesses.1 Classical statistics are concerned only with the analysis of sampled data, which permits the researcher to make inferences concerning total populations, with the exclusion of any personal judgment or opinions. By contrast, statistics modeled on Bayes' Theorem purposefully incorporate informed judgments into the analyses of data. Informed judgments are based on sound experience. While such judgments may be termed intuition, they are not irrational.
This paper examines the use of Bayesian statistics, which include informed judgments based on experience into quantitative decision-making, and its application in the business environment.
Bayesian Analysis
Bayesian statistics are based on Bayes' theorem, which is concerned with the derivation of a conditional probability 2. A conditional probability is an alteration of the probability found through the application of classical statistical analysis.
Bayes' theorem was developed by Thomas Bayes, an English clergyman and mathematician of the eighteenth century 3. In its most simple form, the theorem is a variation on the general formula for conditional probability. The significance of Bayes' theorem is that it is applied in the context of sequential events, and further, that the computational version of the formula provides the basis for determining the conditional probability of an event having occurred in the first sequential position given that a particular event has been observed in the second sequential position4.
Bayesian statistics has been expanded to incorporate the expected value criterion, which is often referred to as the Bayesian criterion5. Use of the Bayesian criterion is a distinct procedure (involving the use of expected values) from that of the Bayesian theoremóinvolving the revision of prior probability values.
The principal techniques of classical inference are interval estimation and hypothesis testing. By contrast, the primary concern of Bayesian decision analysis is the choice of a decision act6. Although classical statistical techniques are directly concerned with estimating values or with the testing of hypotheses concerning population parameters, the results of these procedures are related to alternative courses of action or to alternative decisions. As an example, the acceptance of a null hypothesis (or the inability to reject a null hypothesis) in classical statistical analysis which indicated that the sales level for a product will be below the breakeven point for that product typically would be associated also with a decision not to market the product. Bayesian statistical analysis also is concerned with the selection of the best decision.
The primary differences between classical statistical analysis and Bayesian statistical analysis are the use of subjective information in the Bayesian analyses, and the evaluation of alternative decision actions within the context of economic consequences7. In Bayesian statistical analysis, economic consequences may be formulated as either conditional values (payoffs) or as conditional opportunity losses (regrets).
Essentially, the choice between probability levels of Type I and Type II errors is the basis on which the relative importance of two alternative types of mistakes are assessed in hypothesis testing in classical statistical analysis8. The use of the opportunity loss concept in Bayesian statistical analysis represents a similar approach to evaluation, but does so in a more explicit manner. Whereas classical statistical analysis decision procedures are based entirely on the analysis of data collected through a random sampling of the total population, the decision procedures associated with Bayesian statistical analysis may include the analysis of sample data, but are not dependent solely upon the availability of such data.
From the perspective of practicality, an important plus for Bayesian statistical analysis, as opposed to classical statistical analysis, is that decision analysis begins with an identification of managerial judgments, which may then be included in the analysis (Peebles 24). This approach to quantitative analysis means that statistical analysis personnel must work in close coordination with the managerial personnel within an organizationóa plus for the procedure in and of itself. By contrast, the exclusive orientation of classical statistical analysis towards sample data does not provide managerial personnel with the opportunity to present their informed judgments for inclusion in statistical decision analyses, or to gain the feeling that informed judgments on their part are viewed as significant by statistical analysis personnel.
One of the principal applications of Bayesian statistical analysis is the derivation of alternative probabilities to those which may be observed in sample data9. Bayes' theorem is used for the purpose of altering probabilities associated with an entire set of probable events, or states, in a decision situation. In the context of such a decision situation, prior probability distribution is that probability distribution which is applicable before any sample data are collected. In Bayesian decision analysis, this type of probability distribution is often subjective, in that the data upon which it is based is itself based on the judgments of individuals. In Bayesian statistical analysis, however, the prior probability distribution also may be based on historical data.
The posterior probability distribution is the probability distribution which is applicable subsequent to the observation of sample data, and subsequent to the use of these data to revise the prior probability distribution through application of the Bayes' theorem10. In order to apply the Bayes' theorem, the prior probability of an uncertain event and the conditional probability of the sample result must be known. Typically, the conditional probabilities are determined by the application of some standard probability distribution according to the character of the sampling situation11. Bayesian formulas for the determination of posterior distribution probabilities differ according to the conditions of the prior distribution probability12.
Posterior analysis in Bayesian analysis is the process by which the value of sample data is established before the sample data are collected13. The basic procedure involved is that of considering all of the possible sample outcomes, determining the estimated value in the decision process of each of the possible sample outcomes, and, finally, determining the expected values of the sample data by weighting each of these different values by the probability that the associated sample outcome will actually occur. It can be seen, thus, that Bayesian statistical analysis is the principal theoretical basis for decision-tree analysis, as well as for most of the other quantitative decision-rule analysis procedures14.
The use of Bayesian decision analysis in cases where normal distributions apply requires that only two decision acts be evaluated at one time and that the payoff functions associated with these two alternatives are linear 15. The prior probability distribution is descriptive of the uncertainty which is associated with the decision maker's estimate of the probability of occurrence of a random event. It is not the event which follows the probability distribution. Rather, it is the estimate of the event. Since the estimate of the event is based on an informed judgment, there is no mathematical theorem which would justify the use of the normal distribution in respect to such a judgment in any specific situation. For judgment situations in which an informed decision maker is aware of a number of uncertain factors which could influence the value of the eventual outcome in one direction or the other, however, the use of the normal distribution has been found to be a satisfactory approximation of the uncertainty inherent in the estimate16. A normal distribution is defined by identifying the mean and the standard deviation of the distribution. The mean of the prior distribution may be obtained by asking the decision maker to identify the most likely value of the random event or by asking the decision maker for that value which reflects a 50.0 percent chance that the actual value will be lower and a 50.0 percent chance that the actual value will be higher. The first approach is, essentially, a request for the mode of the probability distribution, while the second approach is a request for the median of the distribution. The mean, median, and mode are all at the same point for a normally distributed variable.
The existence of a linear payoff function is an indication that the expected payoff value associated with a specific act is a linear function with respect to the uncertain level of the state17. The point at which two linear payoff functions cross indicates the point of indifference in respect to making a choice between the two alternative courses of action, because the conditional values of the acts are equal at that point. This point of indifference is called the breakeven point in Bayesian statistical decision-analysis. The specific value of the random variable may be established quantitatively (algebraically) by setting the two equations (for alternative decision choices) equal to one another and solving for the random variable.
Once having established the point of indifference (the breakeven point), the best decision (act) may be determined graphically by observing whether the prior mean is above or below the point of indifference, and by choosing the decision which is optimal on the preferred side of the breakeven point18. The best decision (act) also may be identified by substituting the value of the prior mean in each payoff function, and choosing the decision with the highest expected value. It is the latter procedure which is most often employed on Bayesian statistical decision-analysis.
Applying Bayesian Analysis to Decision-Making
Returning, then, to our amazon.com book buyer's dilemma, we can apply Bayesian analysis to bring both data and informed experience to play.
The buyer knows the probability distribution of a state variable, which, for purposes of this example, is assumed to be demand for the Diana book in units per day. The inventory problem is to make an optimal point estimate for the demand when there exist certain regrets, or cost consequences, which are associated with either an overestimation or an underestimation of actual demand. The point estimate is used as the order quantity, and, for each unsold book, there is a cost involved (loss of a perishable product as an example), and, for each book not available for sale to amazon.com buyers, there is a cost involved (for example, loss of marginal profit). The total expected regret (cost) is minimized when the incremental expected regret from overestimating equals the incremental expected regret from underestimating. Thus, when the loss per unit of overestimation is equal to the loss per unit of underestimation, the median is the best Bayesian estimate. If the probability distribution of the random variable is symmetrical, the median will coincide with the mean.
A practical approach of Bayesian estimation is a criterion that risk willingness should be proportional. Thus, if the cost per unit of underestimation is four times greater than is the unit cost of overestimation, then the buyer should be only one-quarter as willing to risk an underestimate as he is willing to risk an overestimation. In such a scenario, the book buyer would go ahead and place the order for more copies because the consequences of not having the Diana book in stock is four times greater than the consequences of having left-over inventory.
Bayesian statistical analysis is an alternative to classical statistical analysis, which provides for the incorporation into the analysis of informed judgments. As such, Bayesian statistical analysis is not suitable for experimental analysis. Bayesian statistical analysis, however, is useful in business and political applications involving actual and opportunity costs where decisions often must be made under uncertain conditions.
Summary and Conclusion
This research examined the role of intuition and experience in quantitative decision-making. The examination showed that, in the form of informed judgment, experience, or intuition, can play a positive role in quantitative decision-making. The application of informed judgment to the process of quantitative decision-making is accomplished through the performance of Bayesian statistical analysis which is concerned with the derivation of a conditional probability of a future event.
Works Cited
Davis, W. W. "Bayesian Analysis of the Linear Model Subject to Linear Inequality Constraints." Journal of the American Statistical Association 73 (September 1982): 573-579.
Emory, C. W. Business Research Methods, 3rd ed. Chicago: Richard D. Irwin, Inc., 1989.
Markowitz, H. M., and Xu, G. L. "Data Mining Corrections." Journal of Portfolio Management 21 (Fall 1998): 60-69.
Peebles, P. Z., Jr. Probability, Random Variables, and Random Signal Principles, 3rd ed. New York: McGraw-Hill, Inc., 1997.
Pfaffenberger, R. C., and Patterson, J. H. Statistical Methods For Business and Economics, 5th ed. Chicago: Richard D. Irwin, Inc., 1999.
Shao, S. P. Mathematics and Quantitative Methods. Cincinnati: South-Western Publishing Company, 1980.
Summers, G. W., Peters, W. S., and Armstrong, C. P. Basic Statistics in Business and Economics, 5th ed. Belmont, California: Wadsworth Publishing Company, 1994.
Zellner, A. An Introduction to Bayesian Inference in Econometrics. New York: John Wiley & Sons, 1991.
