We describe the transferable belief model, a model for representing quantified beliefs based on belief functions. Beliefs can be held at two levels: (1) a credal level where beliefs are entertained and quantified by belief functions, (2) a pignistic level where beliefs can be used to make decisions and are quantified by probability functions. The relation between the belief function and the probability function when decisions must be made is derived and justified. Four paradigms are analyzed in order to compare Bayesian, upper and lower probability, and the transferable belief approaches.

Many AI problems, when formalized, reduce to evaluating the probability that a propositional expression is true. In this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability. We consider various methods used in approximate reasoning such as computing degree of belief and Bayesian belief networks, as well as reasoning techniques such as constraint satisfaction and knowledge compilation, that use approximation to avoid computational difficulties, and reduce them to model-counting problems over a propositional domain. We prove that counting satisfying assignments of propositional languages is intractable even for Horn and monotone formulae, and even when the size of clauses and number of occurrences of the variables are extremely limited. This should be contrasted with the case of deductive reasoning, where Horn theories and theories with binary clauses are distinguished by the e...

We generalize the Bayes ’ theorem within the transferable belief model framework. The Generalized Bayesian Theorem (GBT) allows us to compute the belief over a space Θ givenanobservationx⊆Xwhen one knows only the beliefs over X for every θi ∈ Θ. We also discuss the Disjunctive Rule of Combination (DRC) for distinct pieces of evidence. This rule allows us to compute the belief over X from the beliefs induced by two distinct pieces of evidence when one knows only that one of the pieces of evidence holds. The properties of the DRC and GBT and their uses for belief propagation in directed belief networks are analysed. The use of the discounting factors is justfied. The application of these rules is illustrated by an example of medical diagnosis.

The objective of this paper is to describe the potential oered by the Dempster–Shafer theory (DST) of evidence as a promising improvement on ‘‘traditional’ ’ approaches to decision analysis. Dempster–Shafer techniques originated in the work of Dempster on the use of probabilities with upper and lower bounds. They have subsequently been popularised in the literature on Artificial Intelligence (AI) and Expert Systems, with particular emphasis placed on combining evidence from dierent sources. In the paper we introduce the basic concepts of the DST of evidence, briefly mentioning its origins and comparisons with the more traditional Bayesian theory. Following this we discuss recent developments of this theory including analytical and application areas of interest. Finally we discuss developments via the use of an example incorporating DST with the Analytic Hierarchy Process

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In this paper, a probabilistic relational model is presented which combines relational algebra with probabilistic retrieval. Based on certain independence assumptions, the operators of the relational algebra are redefined such that the probabilistic algebra is a generalization of the standard relational algebra. Furthermore, a special join operator implementing probabilistic retrieval is proposed. When applied to typical document databases, queries can not only ask for documents, but for any kind of object in the database. In addition, an implicit ranking of these objects is provided in case the query relates to probabilistic indexing or uses the probabilistic join operator. The proposed algebra is intended as a standard interface to combined database and IR systems, as a basis for implementing user-friendly interfaces. 1 Introduction The fields of databases (DB) and information retrieval (IR) have been coexisting for a very long time, but with little influence on each other. IR peop...

We describe a new approach for reasoning with belief functions in this paper. This approach is fundamentally unrelated to probabilities and is consistent with Shafer and Tversky's canonical examples.

The mathematical theory of evidence has been introduced by Glenn Shafer in 1976 as a new approach to the representation of uncertainty. This theory can be represented under several distinct but more or less equivalent forms. Probabilistic interpretations of evidence theory have their roots in Arthur Dempster's multivalued mappings of probability spaces. This leads to random set and more generally to random lter models of evidence. In this probabilistic view evidence is seen as more or less probable arguments for certain hypotheses and they can be used to support those hypotheses to certain degrees. These degrees of support are in fact the reliabilities with which the hypotheses can be derived from the evidence. Alternatively, the mathematical theory of evidence can be founded axiomatically on the notion of belief functions or on the allocation of belief masses to subsets of a frame of discernment. These approaches aim to present evidence theory as an extension of probability theory. Evidence theory has been used to represent uncertainty in expert systems, especially in the domain of diagnostics. It can be applied to decision analysis and it gives a new perspective for statistical analysis. Among its further applications are image processing, project planing and scheduling and risk analysis. The computational problems of evidence theory

Abstract. This paper outlines an application of belief functions to forecasting the demand for a new service in a new category, based on new technology. Forecasting demand for a new product or service is always diÆcult. It is more so when the product category itself is new, and so unfamiliar to potential consumers, and the quality of service of the product is dependent upon a new technology whose actual performance quality is not known in advance. In such a situation, market research is often unreliable, and so the beliefs of key stakeholders regarding the true values of underlying variables typically vary considerably. Belief functions provide a means of representing and combining these varied beliefs which is more expressive than traditional point probability estimates. 1

A simple Monte-Carlo algorithm can be used to calculate Dempster-Shafer belief very efficiently unless the conflict between the evidences is very high. This paper introduces and explores Markov Chain Monte-Carlo algorithms for calculating Dempster-Shafer belief that can also work well when the conflict is high.