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

We present PULCinella and its use in comparing uncertainty theories. PULCinella is a general tool for Propagating Uncertainty based on the Local Computation technique of Shafer and Shenoy. It may be specialized to different uncertainty theories: at the moment, Pulcinella can propagate probabilities, belief functions, Boolean values, and possibilities. Moreover, Pulcinella allows the user to easily define his own specializations. To illustrate Pulcinella, we analyze two examples by using each of the four theories above. In the first one, we mainly focus on intrinsic differences between theories. In the second one, we take a knowledge engineer viewpoint, and check the adequacy of each theory to a given problem. 1. INTRODUCTION A new interest has grown up recently in the uncertainty management community. Moving from consideration of efficiency, ease of representation, and generality, a number of techniques for representing and propagating uncertainty in networks have been proposed (e.g....

ii I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy.

This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of

An approximated simulation algorithm is presented for the propagation of convex sets of probabilities. It is assumed that the graph is such that an exact probabilistic propagation is feasible. The algorithm is a simulated annealing procedure, which randomly selects probability distributions among the possible ones, performing at the same time an exact probabilistic propagation. The algorithm can be applied to general directed acyclic graphs and is carried out on a tree of cliques. Some experimental tests are shown. 1. Introduction One of the main problems with probabilistic propagation algoritms on graphical structures is the introduction of the initial exact conditional probabilities. A number of authors have tried to overcome this difficulty by allowing the use of intervals on the specified probabilities [5, 10, 11, 3, 14, 13, 19, 23, 2]. Some of these works [10, 11, 3, 13, 23] focus on the use of convex sets of probabilities. Convex sets are a more general tool for representing un...

Abstract: Recently, we proposed a new method for representing and solving decision problems based on the framework of valuation-based 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 arc-reversal method of influence diagrams.

This paper examines the applicability of belief functions methodology in three reasoning tasks: (1) representation of incomplete knowledge, (2) belief updating, and (3) evidence pooling. We find that belief functions have difficulties representing incomplete knowledge, primarily knowledge expressed in conditional sentences. In this context, we also show that the prevailing practices of encoding if-then rules as belief function expressions are inadequate, as they lead to counterintuitive con-clusions under chaining, contraposition, and reasoning by cases. Next, we examine the role of belief functions in updating states of belief and find that, if partial knowledge is encoded and updated by belief function methods, the updating pro-cess violates basic patterns of plausibility and the resulting beliefs cannot serve as a basis for rational decisions. Finally, assessing their role in evidence pooling, we find that belief functions offer a rich language for describing the evidence gath-ered, highly compatible with the way people summarize observations. However, the methods available for integrating evidence into a coherent state of belief ca-

An Evolution Program is presented to propagate convex sets of probabilities. This algorithm is useful when the number of extreme points in the 'a posteriori' convex set for a variable is too high and a single probabilistic propagation is feasible. We have tested the algorithm in a random causal network with a random number of conditional probabilities in each one of the variables. The experimental evaluation show that the resulting intervals obtained for the cases of a variable are similar to those obtained using an exact method of propagation. 1 Introduction Graphical structures have been used to represent and manipulate independence relationships [16]. Propagation algorithms were first developed for the probabilistic case [11, 17]. These independence relationships can be used to obtain factorizations of uncertainty representations given by means of other formalisms (see Shafer and Shenoy [19, 18]). One of the particularizations of the Shafer and Shenoy system was given by Cano, Mora...

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