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Uncertainty and Vagueness in Knowledge Based Systems

Numerical Methods

verfasst von: Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Verlag: Springer Berlin Heidelberg

Buchreihe : Artificial Intelligence

Enthalten in: Professional Book Archive

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Über dieses Buch

The primary aim of this monograph is to provide a formal framework for the representation and management of uncertainty and vagueness in the field of artificial intelligence. It puts particular emphasis on a thorough analysis of these phenomena and on the development of sound mathematical modeling approaches. Beyond this theoretical basis the scope of the book includes also implementational aspects and a valuation of existing models and systems. The fundamental ambition of this book is to show that vagueness and un certainty can be handled adequately by using measure-theoretic methods. The presentation of applicable knowledge representation formalisms and reasoning algorithms substantiates the claim that efficiency requirements do not necessar ily require renunciation of an uncompromising mathematical modeling. These results are used to evaluate systems based on probabilistic methods as well as on non-standard concepts such as certainty factors, fuzzy sets or belief functions. The book is intended to be self-contained and addresses researchers and practioneers in the field of knowledge based systems. It is in particular suit able as a textbook for graduate-level students in AI, operations research and applied probability. A solid mathematical background is necessary for reading this book. Essential parts of the material have been the subject of courses given by the first author for students of computer science and mathematics held since 1984 at the University in Braunschweig.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. General Considerations of Uncertainty and Vagueness

Abstract

This chapter gives a brief introduction to the field of artificial intelligence and the basic epistemic aspects of modeling in its most general meaning. Furthermore a survey of the concept and organization of the book is provided.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 2. Introduction

Abstract

This chapter provides the set-theoretical notations we use throughout the whole book. In particular, notations for (finite) product spaces are thoroughly introduced. Moreover a simple example is presented, which will be referred to whenever new concepts have to be illustrated. Finally we return to the notions of uncertainty and vagueness, in order to clarify our intuitive understanding of these phenomena.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 3. Vague Data

Abstract

In this chapter we present the theory of L-sets as a tool for handling vagueness. A thorough analysis of the vague data provides us with well defined semantics. It turns out that vagueness has to be conceived as imprecision with respect to different contexts. Due to this interpretation L-sets have to be distinguished from fuzzy sets although there are various formal similarities. Finally we address the evaluation of vague data by the use of the concepts of possibility and necessity, and show how to apply these techniques in practice.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 4. Probability Theory

Abstract

In this chapter we present probability theory as the basic tool for handling uncertainty, i.e. partial beliefs. In order to obtain a mathematically sound representation we always keep in mind a “relative frequency” interpretation or alternatively a degree of confirmation of probabilities. The main shortcoming of many textbooks is that the treatment of imprecise probabilities, which is important for obtaining suitable knowledge representation facilities, is left out. Our approach to this problem relies on the consideration of classes of probabilities.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 5. Random Sets

Abstract

In this chapter we address the problem of aggregating a number of expert opinions which have been expressed by vague data. In our approach a meta-expert attaches nonnegative weights of importance to each of his or her experts θ_i ∈ Θ i = 1,..., n. Therefore we have to discuss the problem of finding suitable weights, integrating pieces of certain knowledge and giving methods for decision making. To clarify our basic idea let us consider a set Ξ (which is for the moment assumed to be finite) containing n “atomic agents” each of which is equally important. If we imagine these atomic agents voting on some motion, then the weight of some “party” A ⊆ E is given by |A|. So selecting agents ξ ∈ Ξ randomly would yield, in a long run, members of A in \( (\tfrac{1}{n}\left| A \right|\bullet 100) \frac{\hbox{$\scriptstyle 0$}}{\hbox{$\scriptstyle 0$}} P(A) = \frac{1}{n}\left| A \right| \) of all cases. Thus we consider \( P(A) = \frac{1}{n}\left| A \right| \) to be the probability of A. Unfortunately, in general not the space E but only a projection Θ is accessible, where φ may denote the point-wise projection mapping, i.e. φ: Ξ → Θ. On the intuitive level 0 represents an assembly of “floor leaders”. Thus the weight of each θ ∈ Θ is given by the “probability” of φ^-(θ) = {ξ ∈ Ξ | φ(ξ) = θ}, which means it is determined by the number of supporters. So we can interpret the importance weights here as a probability (with a classical or frequentistic interpretation).

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 6. Mass Distributions

Abstract

In this chapter we address once more the common treatment of imprecision and uncertainty. Our approach relies on the attachment of “masses” to vague data. Of course there are extensive similarities with the concept of belief functions [Shafer 1976] and even more with the transferable mass model [Smets 1978], but an important difference remains: we conceive mass distributions (in the case of imprecise data) just as a condensed representation of weighted (random) sets, where sets are used for the representation of imprecise data. Note that this does not necessarily require us to follow the purely frequentistic interpretation of uncertainty presented above, but allows also a “subjective” view. The practical value of the methods we develop in the sequel arises from the fact that they can be applied even if the underlying random set is unknown.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 7. On Graphical Representations

Abstract

In order to provide reasonable notations for dependency relations we summarize firstly some results on graph theory. The study of hypergraphs leads to the consideration of dependency networks, which form a very powerful tool for analyzing qualitative aspects of knowledge representation.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 8. Modeling Aspects

Abstract

The modeling and representation of expert knowledge as well as reasoning and decision making based on this knowledge have often to be done in real-world environments where one is ignorant about certain aspects of an actual problem.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 9. Heuristic Models

Abstract

In this chapter we examine three different “heuristic uncertainty models”. The characteristic feature of heuristic models is that their mathematical foundations are not or only incompletely led back to some sound theory — as given by probability theory, for instance. This is because heuristic approaches aim at avoiding certain “problems” arising from the use of, e.g., probability theory. The reasons that are often mentioned in this context are the amount of data needed (prior and conditional probabilities, joint probability distributions, etc.), the inability to distinguish between absence of belief and doubt, and that it is impossible to represent ignorance.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 10. Fuzzy Set Based Models

Abstract

In Chap. 2 we used L-sets for modeling vague observations such as “The ship is far away from the coast”. The vagueness in this statement arises from the use of the vaguely defined predicate “far away”. A different approach consists in modeling directly the vague concept “far away” instead of modeling observed vague data.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 11. Reasoning with L-Sets

Abstract

In this chapter we develop a propagation algorithm for the case of vague information. In essence the ordinary set calculus described in Sect. 2.2 is extended to the case of L-sets. The final aim is to find an unknown true value ω ₀ ∈ Ω on the basis of a collection of vague data.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 12. Probability Based Models

Abstract

As argued in the preceding parts of this book, probability theory offers a theoretically sound model for representing uncertainty and for considering it in reasoning techniques. Just in the last few years a revival of using probability theory in representing uncertainty has taken place. It was given considerable insight into the application of probability theory and pointed out some misconceptions about its applicability. Also, new theoretical results from statistics and probability theory present arguments for the utility of probabilities for reasoning.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 13. Models Based on the Dempster-Shafer Theory of Evidence

Abstract

Like Bayesian approaches, the Dempster-Shafer theory of evidence aims to model and quantify uncertainty by degrees of belief. But in contrast to Bayesian approaches it permits assignment of degrees of belief to sets of hypotheses rather than to hypotheses in isolation. The underlying idea is that the process of narrowing the hypothesis set with the collection of evidence is better represented in terms of this theory than in terms of Bayesian approaches. For this reason the theory can be viewed as an alternative to Bayesian modeling in probability theory.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 14. Reasoning with Mass Distributions

Abstract

In this chapter we use the methods presented in Chaps. 5 and 6 to establish a reasoning scheme based on the concept of a mass distribution. The qualitative knowledge is again structured by using hypertrees.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Chapter 15. Related Research

Abstract

In this book we restrict our attention to numerical methods that are based on measure-theoretic concepts. Logical approaches to uncertainty and vagueness are not treated, and due to space limitations we shall not present concepts such as probabilistic logic within the book. Although one could argue that probabilistic logic is also a numerical method, we regard it is more as a combination of a calculus for uncertainty handling and a logical calculus, but not as a pure measure-theoretic concept.

Rudolf Kruse, Erhard Schwecke, Jochen Heinsohn

Backmatter

Titel: Uncertainty and Vagueness in Knowledge Based Systems
verfasst von: Rudolf Kruse
Erhard Schwecke
Jochen Heinsohn
Copyright-Jahr: 1991
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-76702-9
Print ISBN: 978-3-642-76704-3
DOI: https://doi.org/10.1007/978-3-642-76702-9