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31
Axioms for probability and belieffunction propagation
 Uncertainty in Artificial Intelligence
, 1990
"... 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 ..."
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Cited by 135 (17 self)
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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.
Perspectives on the Theory and Practice of Belief Functions
 International Journal of Approximate Reasoning
, 1990
"... 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 answer ..."
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Cited by 85 (7 self)
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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 belieffunction 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 belieffunction reasoning can also be found in more recent work, by authors
Probabilistic argumentation systems
 Handbook of Defeasible Reasoning and Uncertainty Management Systems, Volume 5: Algorithms for Uncertainty and Defeasible Reasoning
, 2000
"... Different formalisms for solving problems of inference under uncertainty have been developed so far. The most popular numerical approach is the theory of Bayesian inference [42]. More general approaches are the DempsterShafer theory of evidence [51], and possibility theory [16], which is closely re ..."
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Cited by 54 (33 self)
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Different formalisms for solving problems of inference under uncertainty have been developed so far. The most popular numerical approach is the theory of Bayesian inference [42]. More general approaches are the DempsterShafer theory of evidence [51], and possibility theory [16], which is closely related to fuzzy systems.
Propositional information system
, 1996
"... Resolution is an often used method for deduction in propositional logic. Here a proper organization of deduction is proposed which avoids redundant computations. It is based on a generic framework of decompositions and local computations as introduced by Shenoy, Shafer [29]. The system contains the ..."
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Cited by 29 (15 self)
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Resolution is an often used method for deduction in propositional logic. Here a proper organization of deduction is proposed which avoids redundant computations. It is based on a generic framework of decompositions and local computations as introduced by Shenoy, Shafer [29]. The system contains the two basic operations with information, namely marginalization (or projection) and combination; the latter being an idempotent operation in the present case. The theory permits the conception of an architecture of distributed computing. As an important application assumptionbased reasoning is discussed. 1
Local Computation with Valuations from a Commutative Semigroup
 Annals of Mathematics and Artificial Intelligence
, 1996
"... This paper studies a variant of axioms originally developed by Shafer and Shenoy (1988). It is investigated which extra assumptions are needed to perform the local computations in a HUGINlike architecture (Jensen et al. 1990) or in the architecture of Lauritzen and Spiegelhalter (1988). In particul ..."
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Cited by 28 (8 self)
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This paper studies a variant of axioms originally developed by Shafer and Shenoy (1988). It is investigated which extra assumptions are needed to perform the local computations in a HUGINlike architecture (Jensen et al. 1990) or in the architecture of Lauritzen and Spiegelhalter (1988). In particular it is shown that propagation of belief functions can be performed in these architectures. Keywords: articial intelligence, belief function, constraint propagation, expert system, probability propagation, valuationbased system. 1 Introduction An important development in articial intelligence is associated with an abstract theory of local computation known as the Shafer{Shenoy axioms (Shafer and Shenoy 1988; Shenoy and Shafer 1990). These describe in a very general setting how computations can be performed eciently and locally in a variety of problems, just if a few simple conditions are satised. Even though the axioms were developed to formalize computation with belief functions (Shaf...
The DempsterShafer calculus for statisticians
 International Journal of Approximate Reasoning
, 2007
"... The DempsterShafer (DS) theory of probabilistic reasoning is presented in terms of a semantics whereby every meaningful formal assertion is associated with a triple (p, q, r) where p is the probability “for ” the assertion, q is the probability “against” the assertion, and r is the probability of “ ..."
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Cited by 23 (1 self)
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The DempsterShafer (DS) theory of probabilistic reasoning is presented in terms of a semantics whereby every meaningful formal assertion is associated with a triple (p, q, r) where p is the probability “for ” the assertion, q is the probability “against” the assertion, and r is the probability of “don’t know”. Arguments are presented for the necessity of “don’t know”. Elements of the calculus are sketched, including the extension of a DS model from a margin to a full state space, and DS combination of independent DS uncertainty assessments on the full space. The methodology is applied to inference and prediction from Poisson counts, including an introduction to the use of jointree model structure to simplify and shorten computation. The relation of DS theory to statistical significance testing is elaborated, introducing along the way the new concept of “dull ” null hypothesis. Key words: DempsterShafer; belief functions; state space; Poisson model; jointree computation; statistical significance; dull null hypothesis 1
Computation in Valuation Algebras
 IN HANDBOOK OF DEFEASIBLE REASONING AND UNCERTAINTY MANAGEMENT SYSTEMS, VOLUME 5: ALGORITHMS FOR UNCERTAINTY AND DEFEASIBLE REASONING
, 1999
"... Many different formalisms for treating uncertainty or, more generally, information and knowledge, have a common underlying algebraic structure. ..."
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Cited by 22 (4 self)
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Many different formalisms for treating uncertainty or, more generally, information and knowledge, have a common underlying algebraic structure.
A comparison of graphical techniques for decision analysis
 European Journal of Operational Research
, 1994
"... Abstract: Recently, we proposed a new method for representing and solving decision problems based on the framework of valuationbased systems. The new representation is called a valuation network, and the new solution method is called a fusion algorithm. In this paper, we compare valuation networks ..."
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Cited by 19 (10 self)
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Abstract: Recently, we proposed a new method for representing and solving decision problems based on the framework of valuationbased systems. The new representation is called a valuation network, and the new solution method is called a fusion algorithm. In this paper, we compare valuation networks to decision trees and influence diagrams. We also compare the fusion algorithm to the backward recursion method of decision trees and to the arcreversal method of influence diagrams.
Heuristic Algorithms for the Triangulation of Graphs
, 1995
"... Different uncertainty propagation algorithms in graphical structures can be viewed as a particular case of propagation in a joint tree, which can be obtained from different triangulations of the original graph. The complexity of the resulting propagation algorithms depends on the size of the resu lt ..."
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Cited by 19 (3 self)
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Different uncertainty propagation algorithms in graphical structures can be viewed as a particular case of propagation in a joint tree, which can be obtained from different triangulations of the original graph. The complexity of the resulting propagation algorithms depends on the size of the resu lting triangulated graph. The prob lem of obtaining an optimum graph triangu lation is known to be NPcomplete. Thus approximate algorithms which find a good triangulation in reasonable time are of particular interest. This work describes and compares several heuristic algorithms developed for this purpose.