##
**Applied statistical decision theory.
Repr.**
*(English)*
Zbl 0952.62008

Wiley Classics Library. New York, NY: Wiley. xxviii, 356 p. (2000).

[For the review of the original edition from 1968 see Zbl 0181.21801.]

The monograph under review is an introduction to the mathematical analysis of decision making under conditions of uncertainty. The authors emphasize that the Bayesian principles underlying the methods of analysis presented in this book are not in conflict with the principles underlying the traditional decision theory.

The book consists of three parts. Part 1 (Chapters 1-3) describes the general structure of the considered group of analytical methods and indicates how they can be applied to a wide variety of decision problems. In Chapter 1 the authors start by defining the basic data of any decision problem in which experimentation is possible. Next, they describe two basic modes of analysis of these data: the normal form and the extensive form. After proving that both modes must ultimately lead to the same course of action, the authors justify the use of the extensive form in the monograph by its conceptual and technical advantages over the normal form. In Chapter 2 a convenient definition of a sufficient statistic is provided. Next, the concept of marginal sufficiency is used as a basis for discussion of the problems suggested by the words “optimal stopping”. It is shown that after an experiment has already been conducted the experimental data can usually be “sufficiently” described without reference to the way in which the “size” of the experiment was determined. In Chapter 3 the authors take up the problem of assessing prior distributions in a form that expresses the essentials of the decision maker’s judgements about the possible states of nature and at the same time is mathematically tractable.

Part 2 (Chapters 4-6) gives specific analytic results for two specialized classes of problems: (i) problems involving choice among two or more processes when utility is linear in the mean of the chosen process, and (ii) problems of point estimation when utility depends on the difference between the estimate and the true value of the quantity being estimated. Here the class of problems is discussed in which the utility of an entire combination can be decomposed into two additive parts. In Chapter 4 it is pointed out that the assumption of additivity restricts the discussion to problems in which all consequences are monetary. When terminal and sampling utilities are additive, it is usually possible to decompose terminal utility itself into a sum or difference of economically meaningful parts. To allow the reader to gain from the economic heuristics, which such decompositions make possible, the authors are defining and interrelating a number of new economic concepts in this chapter.

In Chapters 5A and 5B the authors further specialize by assuming that the terminal utility of every possible act is linear in the state or some transformation thereof. Chapter 5A is primarily concerned with problems in which the posterior mean is scalar and the act space contains only a finite number of acts. It is shown that under these conditions the expected net gain of many types of experiments can be expressed in terms of a “linear-loss” integral with respect to the “preposterous distribution”. The remainder of Chapter 5A specializes still further to the case where there are only two possible terminal acts and examines the problem of using the results previously obtained for the net gain of a sample of arbitrary size to find the optimal sample size. In Chapter 5B the authors take up the problem of selecting the best of \(r\) “processes” when terminal utility is linear in the mean of the chosen process and sample observations can be taken independently on any or all of the means. The chapter is primarily concerned with the problem of deciding how many observations to take on each process before any terminal decision is reached. In Chapter 6 the authors turn to another special class of problems, of which the most important is the problem of point estimation.

Part 3 (Chapters 7-13) is a systematic compendium of the distribution theory required in Parts 1 and 2, containing definitions of distributions, references to published tables and formulas for moments and other useful integrals.

The monograph under review is an introduction to the mathematical analysis of decision making under conditions of uncertainty. The authors emphasize that the Bayesian principles underlying the methods of analysis presented in this book are not in conflict with the principles underlying the traditional decision theory.

The book consists of three parts. Part 1 (Chapters 1-3) describes the general structure of the considered group of analytical methods and indicates how they can be applied to a wide variety of decision problems. In Chapter 1 the authors start by defining the basic data of any decision problem in which experimentation is possible. Next, they describe two basic modes of analysis of these data: the normal form and the extensive form. After proving that both modes must ultimately lead to the same course of action, the authors justify the use of the extensive form in the monograph by its conceptual and technical advantages over the normal form. In Chapter 2 a convenient definition of a sufficient statistic is provided. Next, the concept of marginal sufficiency is used as a basis for discussion of the problems suggested by the words “optimal stopping”. It is shown that after an experiment has already been conducted the experimental data can usually be “sufficiently” described without reference to the way in which the “size” of the experiment was determined. In Chapter 3 the authors take up the problem of assessing prior distributions in a form that expresses the essentials of the decision maker’s judgements about the possible states of nature and at the same time is mathematically tractable.

Part 2 (Chapters 4-6) gives specific analytic results for two specialized classes of problems: (i) problems involving choice among two or more processes when utility is linear in the mean of the chosen process, and (ii) problems of point estimation when utility depends on the difference between the estimate and the true value of the quantity being estimated. Here the class of problems is discussed in which the utility of an entire combination can be decomposed into two additive parts. In Chapter 4 it is pointed out that the assumption of additivity restricts the discussion to problems in which all consequences are monetary. When terminal and sampling utilities are additive, it is usually possible to decompose terminal utility itself into a sum or difference of economically meaningful parts. To allow the reader to gain from the economic heuristics, which such decompositions make possible, the authors are defining and interrelating a number of new economic concepts in this chapter.

In Chapters 5A and 5B the authors further specialize by assuming that the terminal utility of every possible act is linear in the state or some transformation thereof. Chapter 5A is primarily concerned with problems in which the posterior mean is scalar and the act space contains only a finite number of acts. It is shown that under these conditions the expected net gain of many types of experiments can be expressed in terms of a “linear-loss” integral with respect to the “preposterous distribution”. The remainder of Chapter 5A specializes still further to the case where there are only two possible terminal acts and examines the problem of using the results previously obtained for the net gain of a sample of arbitrary size to find the optimal sample size. In Chapter 5B the authors take up the problem of selecting the best of \(r\) “processes” when terminal utility is linear in the mean of the chosen process and sample observations can be taken independently on any or all of the means. The chapter is primarily concerned with the problem of deciding how many observations to take on each process before any terminal decision is reached. In Chapter 6 the authors turn to another special class of problems, of which the most important is the problem of point estimation.

Part 3 (Chapters 7-13) is a systematic compendium of the distribution theory required in Parts 1 and 2, containing definitions of distributions, references to published tables and formulas for moments and other useful integrals.

Reviewer: Joseph Melamed (Los Angeles)

### MSC:

62C10 | Bayesian problems; characterization of Bayes procedures |

62C05 | General considerations in statistical decision theory |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

91B06 | Decision theory |