Theory of Probability

12Mathematical Statistics

12.1 The Scope and Limits of the Treatment

12.1.1. A brief account of mathematical statistics within the confines of the final chapter of this book must necessarily offer a limited perspective. Nevertheless, its inclusion serves a very definite purpose.

In Chapter 11, we have already encountered certain of the problems that fall within the purview of the subject matter of mathematical statistics. Specifically, we examined applications of inductive reasoning based on statistical data; that is, data involving a number of observations (possibly a large number) that are, in a certain sense, similar to one another. We have also explained the Bayesian approach to such problems (an approach which constitutes an integral part of the subjectivistic conception), noting that a unified coherent structure cannot be maintained if this is abandoned in favour of other approaches involving a variety of more or less empirical ‘ad hoc’ methods.

In this chapter, we shall attempt to give a more explicit account of the problems with which mathematical statistics is concerned, and of the implications, both for these problems and more generally, which flow from the adoption of one or other of the competing points of view.

12.1.2. In addition to the strictly probabilistic aspects, with which the previous considerations are concerned, we shall have occasion to examine other topics relating to decision theory. There are two basic reasons for this and they correspond to two ...