40 Years of Dempster-Shafer Theory

Originally published in The Reasoner Volume 10, Number 12– December 2016

The International Journal of Approximate Reasoning is celebrating 40 years of Dempster–Shafer theory with a very interesting special issue.

The opening sentence of the Editorial, by Thierry Denoeux, sets the tone

Among the many books published each year, some are good and a few are very good, but only exceptionally does a book propose a radically different way of approaching a scientific question, and start a new research field. “A Mathematical Theory of Evidence” by Glenn Shafer, which appeared in 1976, is one of those.


This special issue includes three previously unpublished papers which Shafer wrote in the 80s “Dempster’s rule of combination”, “The problem of dependent evidence” and “Constructive decision theory”. In addition to an introduction to those, the editorial contains a list of 177 papers on Dempster-Shafer theory which appeared on IJAR only. This clearly gives a measure of the impact of Shafer’s work on the AI-related uncertain reasoning community.

The three papers are put into perspective by Shafer’s intellectual auto-biography, titled A mathematical theory of evidence turns 40, which opens the collection. Together with a number of anecdotes from his academic life Shafer recounts the origin and development of his contributions to the wider field of uncertain reasoning. With one particularly a nice image, Shafer describes the statistical culture of the early 70s as a pendulum swinging back and forth between the strict subjectivist view of, among others, de Finetti, Savage, and Lindley, and the objectivist methods of the Neyman-Pearson philosophy. Neither of them satisfied Shafer entirely, for various reasons. In particular he was led to believe that the Kolmogorov axioms were adequate for frequencies and other objective notions of chance, not so for a subjective measure of uncertainty. For those latter he resolved not to use the very term probability. Building on work by A. Dempster, Shafer’s doctoral dissertation provided an analysis of what properties were desirable for subjective measures of uncertainty, which he termed Belief Functions. That was the beginning of Dempster-Shafer theory.

In retrospect, Shafer makes it clear that he sees his theory as one of a number of potentially useful measures of uncertainty. In particular he pleads agnostic concerning the prescriptive role of the theory, a point discussed in the third paper. Commenting on this he says

Perhaps the belief-function calculus is like another tool in the toolbox or another medicine in the pharmacy. A decision analyst might prescribe it for a particular problem but it still might fail.

One very interesting aspect which emerges from Shafer’s intellectual autobiography is the crucial influence of his early interest in the history and philosophy of probability. Uncertain as to wether he wanted, as a young student, to pursue philosophy or mathematics, Shafer describes himself several times as always having been interested in the very meaning of probability. As to the role of history he concludes by saying

After studying probability and partial belief for 45 years, my sturdiest belief about the enterprise is that the most enduring advances will draw on history.