Catálogo de publicaciones - libros

We begin by clarifying terms. is grounds for belief—an imprecise concept. There must be many valid reasons for believing and hence many ways of making the concept of evidence precise. Most of our beliefs are held because mother—or someone else we trust—told us so. The law trusts sworn testimony. Scientific and statistical evidence are other, different grounds for conclusion, supposedly particularly reliable kinds.

It is a historical fact that geometry was crucial in the development of the modern view of mathematics and the axiomatic method. David Hilbert (1862–1943) judged that the invention of non-Euclidean geometry was “the most suggestive and notable achievement of the last century,” a very strong statement, considering the advance of science during the period. Hilbert meant that the concepts of truth and knowledge, and how to discover truth and acquire knowledge, all this was changed by the invention of non-Euclidean geometry. This chapter reviews some of that historical development.

Evidence for the special success of science can be seen everywhere: in the automobiles we drive, the television we view, the increase in life expectancy, and the fact that man has walked on the moon. We may question the values, or lack thereof, of science, but surely science has been singularly successful in many of the enterprises which it has attempted. It is therefore a little surprising that what constitutes meaningful scientific work is one of the most hotly debated issues of our time. The nature of scientific process and method, the aims or goals of science, the nature and objectivity of scientific knowledge—all are controversial. Still, there is much agreement. This chapter discusses the controversy and attempts to find points of agreement.

Like mathematics, statistics, too, is a handmaiden of science. See the preface in Senn (2003) in this regard. But, perhaps unlike mathematics, statistics derives much of its character and meaning from related disciplines. In addition to logic, mathematics, and scientific method, there are other bodies of thought to which statistics can look for guidance and with which it need come to terms. Further important guides are law, learning theory, and economics. Here we mention the first two; later, in discussing the price interpretation of probability, we touch on economics.

Gambling is the origin of the mathematical theory of probability. Cardano, who died in 1576, wrote a 15-page “gambler’s manual” which treated dice and other problems; however, the effective beginning of the subject was in the year 1654. The Chevalier de Mere was concerned over the following problem of “points”: A game between two persons is won by the player who first scores three points. Each of the participants places at stake 32 pistoles, the winner taking the entire stake of 64 pistoles. If the two leave off playing when the game is only partly finished, then how should the stakes be divided? For example, if the players have one and two points, respectively, and their chances for winning each point are equal, then what should be the division of stakes? The Chevalier consulted Blaise Pascal, who solved the problem and communicated his solution to Fermat. In the ensuing correspondence, the two mathematicians initiated the study of probability theory. In Chapter 6 we shall return to the modern theory of gambling.

Consider again the origins of our subject. The formulation of the de Mere-Pascal-Fermat problem of points was not in terms of equally likely or frequency or personal degree of belief but as a question concerning the value of a player’s position in the game. The issue in general is, if one is to receive a prize or payment, but the amount of that payment is uncertain, then how does that uncertainty affect the value of the prize? A modern development is that this question has been answered in terms of a personal price interpretation of probability. This chapter is our version of the details.

The idea of a probabilistic experiment or chance situation was introduced by example in Section 5.2. There are several attitudes concerning the indeterminism which we call chance: (i) chance is the opposite of determinism; (ii) the situation is in principle completely determinable but the initial conditions have been incompletely specified; (iii) in some respects the world is fundamentally random; and (iv) the indeterminism is located in the mind(s) of the observer(s) rather than in the exterior world. We shall immediately discuss (i) and (ii). The third attitude may be correct, but it begs for an explanation which we cannot provide. Discussion of attitude (iv) was begun in Chapter 6 and will continue.

We begin our discussion of statistical evidence by clarifying terms; while it won’t quite serve as a definition, that aspect of statistics which we consider is concerned with models of inductive rationality. We will be concerned with rational reasoning processes which transform statements about the outcomes of particular experiments into general statements about how similar experiments may be expected to turn out. The purpose of statistical theory is to and statistical methods.

Rubin (1984) describes statistical inference to be Bayesian, if known as well as unknown quantities are treated as random variables—knowns having been observed but unknowns unobserved—and conclusions are drawn about unknowns by calculating their conditional distribution given knowns from a specified joint distribution.

As a consequence of Wald’s (1950) powerful work, statistics was for a time defined as the art and science of making decisions in the face of uncertainty. The decision problem assumes the questionable true value model of Section 8.1 and contemplates deciding, on the basis of data, between various possible actions when the state of nature is unknown. It is anticipated that the data will be helpful in choosing an action since the probabilities of data depend on the state. The task of the decision maker is to choose a decision rule specifying the action d(y) to be taken if data y is observed. The theory assumes known numerical losses (a, ) in taking each action a when each state “obtains.”