Introduction

Abstract

The advance of science and technology is due to the desire to improve existing knowledge and to address new problems that have arisen. At any stage in the history of science, solutions to new problems require new concepts and new tools. Mathematics is called upon to formulate precisely the new concepts involved and to provide the new tools required for solutions. The mathematical theory of probability is a typical example. It serves as the backbone for the theory of statistics. Statistical techniques invade other fields, such as engineering. In engineering problems in which uncertainty is often an important factor, one borrows many statistical techniques, these being the primary body of mathematical tools available for use in uncertainty. The uncertainty involved might be due to errors in measurements, or to the lack of certain and complete knowledge of the system under consideration. But in applying these techniques, one has to be in the domain of applicability of probability theory. The success with problems such as stochastic control, identification of systems, pattern recognition, filtering, and so on, is due to the fact that these areas are assumed to be in the domain of applicability of probability theory. The knowledge that comes from uncertain information is modeled by probability measures, and the techniques are those from statistical decision theory. And it should be noted that the logic used is always the classical two-valued logic.