. A rapidly growing number of user and student modeling systems have employed numerical techniques for uncertainty management. The three major paradigms are those of Bayesian networks, the Dempster-Shafer theory of evidence, and fuzzy logic. In this overview, each of the first three main sections focuses on one of these paradigms. It first introduces the basic concepts by showing how they can be applied to a relatively simple user modeling problem. It then surveys systems that have applied techniques from the paradigm to user or student modeling, characterizing each system within a common framework. The final main section discusses several aspects of the usability of these techniques for user and student modeling, such as their knowledge engineering requirements, their need for computational resources, and the communicability of their results. Key words: numerical uncertainty management, Bayesian networks, Dempster-Shafer theory, fuzzy logic, user modeling, student modeling 1. Introdu...

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

: Belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the Kolmogorov axioms defining probability functions. We argue that there are (at least) two useful and quite different ways of understanding belief functions. The first is as a generalized probability function (which technically corresponds to the inner measure induced by a probability function). The second is as a way of representing evidence. Evidence, in turn, can be understood as a mapping from probability functions to probability functions. It makes sense to think of updating a belief if we think of it as a generalized probability. On the other hand, it makes sense to combine two beliefs (using, say, Dempster's rule of combination) only if we think of the belief functions as representing evidence. Many previous papers have pointed out problems with the belief function approach; the claim of this paper is that these problems can be explained as a consequence of confounding the...

By analyzing the relationships among chance, weight of evidence and degree ofbelief, it is shown that the assertion \chances are special cases of belief functions " and the assertion \Dempster's rule can be used to combine belief functions based on distinct bodies of evidence " together lead to an inconsistency in Dempster-Shafer theory. To solve this problem, some fundamental postulates of the theory must be rejected. A new approach for uncertainty management is introduced, which shares many intuitive ideas with D-S theory, while avoiding this problem. 1

A logic is defined that allows to express information about statistical probabilities and about degrees of belief in specific propositions. By interpreting the two types of probabilities in one common probability space, the semantics given are well suited to model the in uence of statistical information on the formation of subjective beliefs. Cross entropy minimization is a key element in these semantics, the use of which is justified by showing that the resulting logic exhibits some very reasonable properties.

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Degrees of belief are formed using observed evidence and statistical background information. In this paper we examine the process of how prior degrees of belief derived from the evidence are combined with statistical data to form more specific degrees of belief. A statistical model for this process then is shown to vindicate the cross-entropy minimization principle as a rule for probabilistic default-inference. 1 Introduction A knowledge based system incorporating reasoning with uncertain information gives rise to quantitative statements of two different kinds: statements expressing statistical information and statements of degrees of belief. "10% of applicants seeking employment at company X who are invited to an interview will get a job there" is a statistical statement. "The likelihood that I will be invited for an interview if I apply for a job at company X is about 0.6" expresses a degree of belief. In this paper, both of these kinds of statements are regarded as probabilistic, i...

Belief updating schemes in artificial intelligence may be viewed as three dimensional languages, consisting of a syntax (e.g. probabilities or certainty factors), a calculus (e.g. Bayesian or CF combination rules), and a semantics (i.e. cognitive interpretations of competing formalisms). This paper studies the rational scope of those languages on the syntax and calculus grounds. In particular, the paper presents an endomorphism theorem which highlights the limitations imposed by the conditional independence assumptions implicit in the CF calculus. Implications of the theorem to the relationship between the CF and the Bayesian languages and the Dempster-Shafer theory of evidence are presented. The paper concludes with a discussion of some implications on rule-based knowledge engineering in uncertain domains. 1.

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