Comments on shafer's "perspectives on the theory and practice of belief functions"

The theory of belief functions provides one way to use mathematical probability in subjective judgment. It is a generalization of the Bayesian theory of subjective probability. When we use the Bayesian theory to quantify judgments about a question, we must assign probabilities to the possible answers to that question. The theory of belief functions is more flexible; it allows us to derive degrees of belief for a question from probabilities for a related question. These degrees of belief may or may not have the mathematical properties of probabilities; how much they differ from probabilities will depend on how closely the two questions are related. Examples of what we would now call belief-function reasoning can be found in the late seventeenth and early eighteenth centuries, well before Bayesian ideas were developed. In 1689, George Hooper gave rules for combining testimony that can be recognized as special cases of Dempster's rule for combining belief functions (Shafer 1986a). Similar rules were formulated by Jakob Bernoulli in his Ars Conjectandi, published posthumously in 1713, and by JohannHeinrich Lambert in his Neues Organon, published in 1764 (Shafer 1978). Examples of belief-function reasoning can also be found in more recent work, by authors who were unaware of the seventeenth and eighteenth century work. For example, Per Olof Ekelöf, a Swedish legal scholar, reinvented Hooper's rules in the early 1960s (Shafer 1986a, Gärdenfors et al. 1983). In its present generality, however, the theory of belief functions is due to Arthur P. Dempster and myself. Dempster set out the basic ideas of the theory in a series of articles in the 1960s. I developed the theory further and established its terminology and notation in 1976, in A Mathematical Theory of Evidence. Because of the role of Dempster and myself, the theory is sometimes called the “Dempster-Shafer theory.” My 1976 monograph is still the most comprehensive source of information on belief functions. Unfortunately, this monograph says little about interpretation, implementation, or computation. It is concerned primarily with mathematical foundations. During the past fifteen years, a great deal has been learned about the interpretation, implementation, and computation of belief functions, and fresh progress has been made on the mathematical foundations as well. This work is scattered in journals in a wide variety of fields, including statistics, psychology, philosophy, engineering, accounting, and artificial intelligence. No one has drawn the new work together in a comprehensive way. This article is primarily a summary of my own current views. It pulls together some of the strands in the scattered literature, but it falls short of the comprehensive review that is needed. Though the bibliography is lengthy, it is not comprehensive. My failure to mention particular contributions should not be taken as an indication of disinterest or disagreement. I begin with some opinions about the place of belief functions in the larger tool-drawer of probabilistic methods and about the place of this whole tool-drawer in artificial intelligence (Sections 1 and 2). These opinions are personal. I expect that few readers will agree with them fully. They may serve, nonetheless, to dispel some misunderstandings about the scope of the theory of belief functions.