Introduction to Statistical Decision Theory: Utility Theory and Causal Analysis

Book Preface

Statistics is the discipline that deals with collection and scientific treatment of data. Statistics transforms data into information and, subsequently, into knowledge. Whenever the knowledge is used for decision-making purposes, the decision theory frame is involved. Statistical decision theory is the new scientific discipline that comes from merging statistics and decision theory. Many authors [12, 45, 199] state that statistical inference and statistical decision theory must be considered as distinct disciplines, but other authors (see, for instance, [40, 144, 145, 147, 190, 217]) consider statistical decision theory as a natural and necessary generalization of statistical inference. Furthermore, the decision-making approach, through the combination of various theories of statistical inference, avoids the dogmatisms that can lead to paradoxical situations, is free from logical errors, is more effective in practical contexts, and can successfully address a wider range of problems than the traditional approaches.

The aim of this book is to provide the theoretical background to approach decision theory from a statistical perspective, taking into account both the traditional approaches in terms of value theory and expected utility theory, other than the generalized utility theories, and the recent developments in terms of causal inference. The integration among decision theory, classical and Bayesian statistical inference, and causal inference is illustrated step-by-step, showing how sample evidence, prior information on the states of nature, and effects of actions of the decision-maker affect the formalization and the solution of a (statistical) decisional problem. Furthermore, some examples and case studies that rely on actual decisional problems are illustrated throughout the book.

The book is specifically designed to appeal to students who intend to acquire a broader knowledge of statistics with respect to what is usually taught in the upper level university curricula. It provides a rethink of statistical science from the decisional point of view. The book is also devoted to practitioners worldwide involved in decision-making tasks within several fields of knowledge. For such target populations, the book will be a reference work that will provide both the theoretical framework behind decision-making and, through illustrative examples and case studies, a practical guide to carry out decisional processes supported by a quantitative approach. Case studies aim at showing the potentialities of the statistical approach in some unconventional decisional contexts (e.g., banking sector, management of a public service) and some of them rely on actual work and study experiences of the two authors of this book.

The prerequisites to properly understand the content of this book are at the level of a conventional full-year mathematics course, whereas no previous course in probability and statistics is required, as a synthetic introduction to probability and statistical inference is provided in Chapter 2. However, the reader who is totally unfamiliar with statistics might find it helpful to consult other introductory statistical books, such as volumes by Mood, Graybill and Boes [158], Casella and Berger [32], and Olive [161].

The book is organized into six chapters. Chapter 1 addresses preliminary aspects related to decision theory and key concepts that will be employed in the subsequent chapters. After a brief illustration of the historical origins of decision theory, the fundamentals of normative theory, descriptive theory, and prescriptive theory are introduced, together with basic elements of modern decision theory, that is, concepts of value function and utility function. Next, the distinction of decisions under certainty, risk, and uncertainty is illustrated, in accordance with the informational background available to the decision maker. Particular attention is then devoted to decisional criteria under uncertainty (i.e., Wald’s criterion, max-max criterion, Hurwicz’s criterion, Savage’s criterion, Laplace’s criterion). The remainder of the chapter deals with the relationship between decision theory and (descriptive and, mainly,) inferential statistics, which in turn distinguishes in classical and Bayesian statistical inference. To conclude, a clarification is provided to discern among classical decision theory, Bayesian decision theory, classical statistical decision theory, and Bayesian statistical decision theory.

Chapter 2 examines the basic concepts of probability theory and statistical inference and it is designed mainly for readers lacking a solid statistical background; it can be skipped by other readers. The first part of the chapter focuses on Kolmogorov’s axioms and some relevant theorems with special attention to Bayes’ rule. Attention is also devoted to the concept of random variable and to the detailed description of some relevant families of random variables. In the second part of the chapter, we deal with sample distributions and the concept of likelihood function. Then, we focus on the typical inferential problems of point and interval estimation and hypothesis testing, which are investigated both from the frequentist and the Bayesian perspectives. The last part of the chapter carries out a thorough investigation into regression and structural equation modeling.

Chapter 3 is aimed at in-depth illustration of normative decision theories that concern how decisions should be taken by ideally rational individuals. We first introduce the problem of decisions under certainty (i.e., when consequences of actions are known), describing a set of axioms of rational behavior and the concept of value function. Then, decisions under risk (i.e., when consequences of actions are not known, but prior information is available) are investigated. In detail, we speak about decisions under risk when prior probabilities of states of nature are given (objectively or subjectively defined). In the former case, we refer to the expected utility theory formulated by von Neumann and Morgenstern, while in the latter case we explicitly refer to the subjective expected utility theory formulated by Savage. In both cases, we illustrate the rational behavior axioms and demonstrate the existence and uniqueness (upto a positive linear transformation) of the utility function, which is a function defined along a quantitative scale that allows representing the decision maker’s preferences. The remainder of the chapter is devoted to describing several empirical failures of the basic axioms of expected utility theory. Then, alternative utility theories are illustrated that are based on weak formulations of some of the basic axioms, with a special focus on the rank-dependent utility theory and the prospect and cumulative prospect theories.

Chapter 4 focuses on the practical issue concerning the elicitation of the utility function. First, the different attitudes towards risk are introduced, distinguishing among risk aversion, risk seeking, and risk neutrality, and the relationship with the shape (i.e., convex, concave, S-shaped, inverted S-shaped) of the utility function is investigated. Next, the main approaches to assess the utility function are described. The classical paradigm is first illustrated, which is based on using complete information for the utility elicitation. In such a setting, two main types of approaches are distinguished: the standard gamble methods (e.g., lottery equivalent method, certainty equivalent method, probability equivalent method) and the paired gamble methods (e.g., preference comparisons method, probability equivalent-paired method, tradeoff method). Then, alternative elicitation procedures are described that rely on partial preference information, such as approaches based on the concepts of Bayesian elicitation and double expected utility, as well as on clustering of utility functions on the basis of some easily observable characteristics of decision makers. Hints are also provided for the problem of elicitation when the decision maker is a collective body consisting of a multitude of individuals.