# Guiding sums for uncertain futures

Introduction to Statistical Decision Theory
May 24, 1996

This book has been long in the making, but John Pratt, Howard Raiffa and Robert Schlaifer have produced an excellent introduction to Bayesian statistical decision theory. It is largely self-contained, providing much of the necessary background in probability theory and statistics. There are many exercises, examples and case studies, which makes for a good textbook for mathematically advanced undergraduates or graduates.

Statistical decision theory is about choosing strategies under uncertainty to arrive at a desired outcome, designing experiments and utilising the resulting information in an optimal way. The concept of a lottery is used as a key tool in the book in order to bring the notion of uncertainty back to canonical relations and preferences, or utility. Fundamental to the analysis of a decision process is the so-called decision tree. These concepts are well explained by the case studies and examples, which are primarily directed towards business economists. One example is a wildcatter in search of oil, who may or may not employ a geologist to do experiments in order to find out the structure beneath the surface. On the basis of this information the wildcatter may or may not drill, and if he does drill, may or may not find oil.

The Bayesian approach is subjective. This means that prior judgements about the outcome of a process are incorporated in the statistical evaluation of experiments that are undertaken to learn about this process. When the experiments do not display much information about the parameters of the process, the prior judgement will have a strong impact on the final, or posterior, assessment. However, when the data from the experiments do give much information, the prior beliefs may have a much smaller posterior impact. In order to determine which experiment is optimal, a preposterior analysis investigates the various impacts of the possible results of different experiments on the final action taken.

By the Bayesian incorporation of prior beliefs or knowledge in the statistical analysis of experiments, and by formulating objectives, like minimising costs or maximising profits, techniques are developed that allow the decision maker optimally to choose from, and design experiments. For example, if a business that contacts its clients by means of a mailing list wants to sample from its client stock in order to get an idea about the demand of a certain new product, the optimal sample size can be determined, given cost and revenue structures, and given the objective of the firm.

The strength of the book is its clear exposition of abstract concepts, which are well illustrated by tables, figures and schemes. The necessary background in probability theory and statistics, which is required to understand decision theory as it is developed, is not presented at once, but is provided in chapters just before it is needed. This keeps the focus on the development at hand, but does lead to some repetition.

As this is an introductory course, the main focus is on univariate processes. The particular processes considered are those of success/failure (Bernoulli), the number of certain events (Poisson) and processes determining an unknown physical quantity (Normal). In all these cases, the issues relating to the development of the statistical decision theory are treated in the same manner. For the Normal process, the theory for the multivariate case is also developed, and the book concludes with a chapter on the multiple linear regression model.

An interesting issue is the distinction between Bayesian and classical, or objective, statistics. One chapter discusses the classical principles and contrasting them with Bayesian principles. From this, the Bayesian approach to decision theory is depicted as the natural one, although the authors admit that "many people do not accept the axioms on which the book is based". This highlights a problem for the teaching of this material, as most undergraduate students are trained in the classical statistical theory. For these students, adopting the Bayesian framework may be difficult. However, as the authors have written the book with such clear exposition, this step is made a lot easier.

Frank Windmeijer is human capital and mobility fellow, department of economics, University College London.

## Introduction to Statistical Decision Theory

Author - John W. Pratt, Howard Raiffa, and Robert Schlaifer
ISBN - 0 262 16144 3
Publisher - MIT Press
Price - £55.50
Pages - 874