Glenn Shafer

Axioms for probability and belief-function propagation

In this paper, we describe an abstract framework and axioms under which exact local computation of marginals is possible. The primitive objects of the framework are variables and valuations. The primitive operators of the framework are combination and marginalization. These operate on valuations. We state three axioms for these operators and we derive the possibility of local computation from the axioms. Next, we describe a propagation scheme for computing marginals of a valuation when we have a factorization of the valuation on a hypertree. Finally we show how the problem of computing marginals of joint probability distributions and joint belief functions fits the general framework.

Perspectives on the theory and practice of belief functions

Abstract The theory of belief functions is a generalization of the Bayesian theory of subjective probability judgement. The authors 1976 book, A Mathematical Theory of Evidence, is still a standard reference for this theory, but it is concerned primarily with mathematical foundations. Since 1976, considerable work has been done on interpretation and implementation of the theory. This article reviews this work, as well as newer work on mathematical foundations. It also considers the place of belief functions within the broader topic of probability and the place of probability within the larger set of formalisms used by artificial intelligence.

Implementing Dempster's rule for hierarchial evidence

This article gives an algorithm for the exact implementation of Dempster’s rule in the case of hierarchical evidence. This algorithm is computationally efficient, and it makes the approximation suggested by Gordon and Shortliffe unnecessary. The algorithm itself is simple, but its derivation depends on a detailed understanding of the interaction of hierarchical evidence.

Propagating belief functions in qualitative Markov trees

Abstract This article is concerned with the computational aspects of combining evidence within the theory of belief functions. It shows that by taking advantage of logical or categorical relations among the questions we consider, we can sometimes avoid the computational complexity associated with brute-force application of Dempsters rule. The mathematical setting for this article is the lattice of partitions of a fixed overall frame of discernment. Different questions are represented by different partitions of this frame, and the categorical relations among these questions are represented by relations of qualitative conditional independence or dependence among the partitions. Qualitative conditional independence is a categorical rather than a probabilistic concept, but it is analogous to conditional independence for random variables. We show that efficient implementation of Dempsters rule is possible if the questions or partitions for which we have evidence are arranged in a qualitative Markov tree—a tree in which separations indicate relations of qualitative conditional independence. In this case, Dempsters rule can be implemented by propagating belief functions through the tree.

Languages and Designs for Probability Judgment

Theories of subjective probability are viewed as formal languages for analyzing evidence and expressing degrees of belief. This article focuses on two probability language, the Bayesian language and the language of belief functions [199]. We describe and compare the semantics (i.e., the meaning of the scale) and the syntax (i.e., the formal calculus) of these languages. We also investigate some of the designs for probability judgment afforded by the two languages.

Allocations of Probability

This paper studies belief functions, set functions which are normalized and monotone of order 8. The concepts of continuity and condensability are defined for belief functions, and it is shown how to extend continuous or condensable belief functions from an algebra of subsets to the corresponding power set. The main tool used in this extension is the theorem that every belief function can be represented by an allocation of probability i.e., by a n-homomorphism into a positive and completely additive probability algebra. This representation can be deduced either from an integral representation due to Choquet or from more elementary work by Revuz and Honeycutt.

Belief Functions and Parametric Models

The theory of belief functions assesses evidence by fitting it to a scale of canonical examples in which the meaning of a message depends on chance. In order to analyse parametric statistical problems within the framework of this theory, we must specify the evidence on which the parametric model is based. This article gives several examples to show how the nature of this evidence affects the analysis. These examples also illustrate how the theory of belief functions can deal with problems where the evidence is too weak to support a parametric model.

The Combination of Evidence

This article provides a historical and conceptual perspective on the contrast between the Bayesian and belief function approaches to the probabilistic combination of evidence. It emphasizes the simplest example of non‐Bayesian belief‐function combination of evidence, which was developed by Hooper in the 1680s.

Belief-Function Formulas for Audit Risk

This article relates belief functions to the structure of audit risk and provides formulas for audit risk under certain simplifying assumptions. These formulas give plausibilities of error in the belief-function sense.

The Sources of Kolmogorov’s Grundbegriffe

Andrei Kolmogorovs Grundbegriffe der Wahrscheinlichkeits-rechnung put probabilitys modern mathematical formalism in place. It also provided a philosophy of probability--an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe--the work of the earlier scholars whose ideas Kolmogorov synthesized.