CONTENTS
1.1Introduction
1.2Decision theory
1.3Value theory and utility theory
1.4Decisions and informational background
1.5Statistical inference and decision theory
1.6The decision-making approach to statistics
1.1Introduction
This chapter provides a general overview of topics related to decision theory and statistics that will be in-depth analyzed throughout the subsequent chapters.
We first introduce some basic concepts in the field of decision theory, distinguishing between the normative and the descriptive approaches. We also outline how a prescriptive approach may represent a satisfactory compromise to compound theoretical rules of rational behavior, which are fundamentals of normative decision theory, with empirical evidence that characterizes the descriptive decision theory. We also point out the focus of this book for individual and rational decisions as opposed to group and moral decisions.
Next, attention focuses on the fundamentals of modern decision theory, also known as value theory or utility theory. The elements that are involved in any decisional setting are defined, that is, actions, states of nature, and consequences or outcomes that depend on the interaction between actions and states of nature. In addition, concepts of value function and utility function are introduced to represent the preferences of a decision-maker for alternative actions. Finally, the decision table and the decision tree are introduced as instruments to describe a decisional problem.
The relation between decisions and information is then investigated, distinguishing the decisional criteria according to the information that the decision maker possesses about the states of nature. More precisely, we introduce the distinction among decisions under certainty, when the state of nature is known; decisions under risk, when a probability distribution about the states of nature is given or may be subjectively elicited by the decision maker; decisions under uncertainty, when no information about the plausibility of the states of nature is given. Special attention is devoted to decisional criteria under uncertainty, that is, the Wald’s criterion, the max-max criterion, the Hurwicz’s criterion, the Savage’s criterion, and the Laplace’s criterion.
The remainder of the chapter discusses the relations between statistics and decision theory. First, an overview of the statistics discipline is provided, distinguishing between descriptive and inferential statistical methods and between classical and Bayesian approaches. Second, a classification of decision theory is provided according to the information available about the states of nature (i.e., no information, prior information, sample information, and posterior information). We distinguish among classical decision theory, when no information is given, Bayesian decision theory when only prior information is available, classical statistical decision theory when only sample evidence is available, and Bayesian statistical decision theory when both sample evidence and prior information are explicitly taken into account. In conclusion, the opportunity of a decision-making approach to statistics is outlined to favor the transformation of raw data into useful information finalized to take decisions.
1.2Decision theory
Decision theory is concerned with processes of decision-making. Through the analysis of the behavior of actors (individuals or groups) involved in a decisional process, it is possible to examine and study how decision makers take or should take decisions.
Making decisions is an everyday activity in many professions and sciences that involves challenging aspects, and this process is of keen interest to researchers from different scientific fields: philosophy and logic, mathematics and statistics, psychology and sociology, economics, and so on.
Applications of the theory range from abstract speculations, relating to ideally rational individuals, to the resolution of specific decision-making problems. Decision theorists investigate the logical consequences of different decision-making rules or explore the logical-mathematical aspects of different descriptions of rational behaviors. On the contrary, applied researchers are interested in analyzing how decisional processes take place in practical situations.
In this perspective, we may distinguish decision theory in normative decision theory and descriptive decision theory1. Normative decision analysts define behavioral rules concerning how decisions should be taken to maximize the well-being of the decision maker. Descriptive decision analysts study how decisions are actually taken in practical contexts. This distinction is useful but rather artificial, being the actual way of making decisions certainly relevant for setting theoretical rules, and, in turn, theoretical rules represent an essential element to evaluate the observed behavior of decision makers.
Throughout this book attention will be focused on the essential elements of normative decision theory, whereas descriptive decision theory will be of no direct relevance, as it is the subject of specific disciplines, such as psychology, sociology and, in some respects, economics. Moreover, a series of constraints and conditioning that emerge from the analysis of actual decisional processes will be also taken into account (see Sections 3.5 and 3...
