Book review of Glenn Shafer

Glenn Shafer:
A MATHEMATICAL THEORY OF EVIDENCE (Princeton Univ Press, 1976)
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This book summarizes Dempster-Shafer's theory of evidence that refines Bayes' theory of probabilities. The theory of belief functions relies on two principles: the principle of inferring degrees of belief for one question from subjective probabilities for a related question; and Dempster's rule on how to combine degrees of belief which are based on independent evidence.
In 1968 Arthur Dempster and Glenn Shafer ("A generalization of Bayesian inference") extended Bayes' theory of probabilities by introducing a "belief function" which operates on all subsets of events (not just the single events). In the throwing of a dice, the possible events are only six, but the number of all subsets is 64 (all the combination of two sides, three sides, four sides and five sides). The sum of the probabilities of all subsets is one, but the sum of the probabilities of all the single events is generally less than one.
Therefore, Dempster-Shafer's theory allows one to assign a probability to a group of events, even if the probability of each single event is not known. Indirectly, Dempster-Shafer's theory also allows one to represent "ignorance", as the state in which the belief of an event is not known (while the belief of a set it belongs to is known). Dempster-Shafer's theory does not require a complete probabilistic model of the domain.
An advantage of evidence over probabilities if that its ability to narrow the hypothesis set with the accumulation of evidence.
Shafer, in accordance with Tversky's experiments, thinks that the way we assign probabilities to an event is a mental experiment to build an imaginary situation and the result we obtain depends on the process of construction. People do not have preferences, people build them.