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. 1.

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 Johann-Heinrich 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

In the last decade, several architectures have been proposed for exact computation of marginals using local computation. In this paper, we compare three architectures---Lauritzen-Spiegelhalter, Hugin, and Shenoy-Shafer---from the perspective of graphical structure for message propagation, message-passing scheme, computational efficiency, and storage efficiency. 1 INTRODUCTION In the last decade, several architectures have been proposed in the uncertain reasoning literature for exact computation of marginals of multivariate discrete probability distributions. One of the pioneering architectures for computing marginals was proposed by Pearl [1986]. Pearl's architecture applies to singly connected Bayes nets. For multiply connected Bayes nets, Pearl [1986] proposed the method of conditioning to reduce a multiply connected Bayes net to several singly connected Bayes nets. In 1988, Lauritzen and Spiegelhalter [1988] proposed an alternative architecture for computing marginals that applies...

Abstract not found

The main objective of this paper is to describe how Dempster-Shafer’s (DS) theory of belief functions fits in the framework of valuation-based systems (VBS). Since VBS serve as a framework for managing uncertainty in expert systems, this facilitates the use of DS belief-function theory in expert systems. Keywords: Dempster-Shafer’s theory of belief functions, valuation-based systems, expert systems 1.

In this paper, we review recent applications of Dempster-Shafer theory (DST) of belief functions to auditing and business decision-making. We show how DST can better map uncertainties in the application domains than Bayesian theory of probabilities. We review the applications in auditing around three practical problems that challenge the effective application of DST, namely, hierarchical evidence, versatile evidence, and statistical evidence. We review the applications in other business decisions in two loose categories: judgment under ambiguity and business model combination. Finally, we show how the theory of linear belief functions, a new extension of DST, can provide an alternative solution to a wide range of business problems. 1.

This paper describes an abstract framework, called valuation network (VN), for representing and solving discrete optimization problems. In VNs, we represent information in an optimization problem using functions called valuations. Valuations represent factors of an objective function. Solving a VN involves using two operators called combination and marginalization. The combination operator tells us how to combine the factors of the objective function to form the global objective function (also called joint valuation). Marginalization is either maximization or minimization. Solving a VN can be described simply as finding the marginal of the joint valuation for the empty set. We state some simple axioms that combination and marginalization need to satisfy to enable us to solve a VN using local computation. We describe a fusion algorithm for solving a VN using local computation. For optimization problems, the fusion algorithm reduces to non-serial description of the dynamic programming method, and the axioms can be viewed as conditions that permit the use of dynamic programming. Subject classification: Dynamic programming: theory, algorithm. 1

copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. The author reserves all other publication and other rights in association with the copyright in the thesis, and except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatever without the author’s prior written permission.

The use of discrete optimization (DO) models and algorithms makes it possible to solve many practical problems in scheduling theory, network optimization, routing in communication networks, facility location, optimization in enterprise resource planning, and logistics (in particular, in supply chain