Deal with information and uncertainty properly and efficiently
using tools emerging from generalized information theory
Uncertainty and Information: Foundations of Generalized Information
Theory contains comprehensive and up-to-date coverage of results
that have emerged from a research program begun by the author in
the early 1990s under the name "generalized information theory"
(GIT). This ongoing research program aims to develop a formal
mathematical treatment of the interrelated concepts of uncertainty
and information in all their varieties. In GIT, as in classical
information theory, uncertainty (predictive, retrodictive,
diagnostic, prescriptive, and the like) is viewed as a
manifestation of information deficiency, while information is
viewed as anything capable of reducing the uncertainty. A broad
conceptual framework for GIT is obtained by expanding the
formalized language of classical set theory to include more
expressive formalized languages based on fuzzy sets of various
types, and by expanding classical theory of additive measures to
include more expressive non-additive measures of various
types.
This landmark book examines each of several theories for dealing
with particular types of uncertainty at the following four
levels:
* Mathematical formalization of the conceived type of
uncertainty
* Calculus for manipulating this particular type of
uncertainty
* Justifiable ways of measuring the amount of uncertainty in any
situation formalizable in the theory
* Methodological aspects of the theory
With extensive use of examples and illustrations to clarify complex
material and demonstrate practical applications, generous
historical and bibliographical notes, end-of-chapter exercises to
test readers' newfound knowledge, glossaries, and an Instructor's
Manual, this is an excellent graduate-level textbook, as well as an
outstanding reference for researchers and practitioners who deal
with the various problems involving uncertainty and information. An
Instructor's Manual presenting detailed solutions to all the
problems in the book is available from the Wiley editorial
department.
Autorentext
GEORGE J. KLIR, PhD, is currently Distinguished Professor of Systems Science at Binghamton University, SUNY. Since immigrating to the U.S. in 1966, he has held positions at UCLA, Fairleigh Dickinson University, and Binghamton University. He is a Life Fellow of IEEE, IFSA, and the Netherlands Institute for Advanced Studies. He has served as president of SGSR, IFSR, NAFIPS, and IFSA. He has published over 300 research papers and sixteen books, and has edited ten books. He has also served as Editor in Chief of the International Journal of General Systems since 1974 and of the IFSR International Book Series on Systems Science and Engineering since 1985. He has received numerous professional awards, including five honorary doctoral degrees, Bernard Bolzano's Gold Medal, Arnold Kaufmann's Gold Medal, and the SUNY Chancellor's Award for "Exemplary Contributions to Research and Scholarship." He is listed in Who's Who in America and Who's Who in the World. His current research interests include intelligent systems, soft computing, generalized information theory, systems modeling and design, fuzzy systems, and the theory of generalized measures. He has guided twenty-nine successful doctoral dissertations in these areas. Some of his research has been funded by grants from NSF, ONR, the United States Air Force, NASA, Sandia Labs, NATO, and various industries.
Klappentext
Deal with information and uncertainty properly and efficiently using tools emerging from generalized information theory
Uncertainty and Information: Foundations of Generalized Information Theory contains comprehensive and up-to-date coverage of results that have emerged from a research program begun by the author in the early 1990s under the name "generalized information theory" (GIT). This ongoing research program aims to develop a formal mathematical treatment of the interrelated concepts of uncertainty and information in all their varieties. In GIT, as in classical information theory, uncertainty (predictive, retrodictive, diagnostic, prescriptive, and the like) is viewed as a manifestation of information deficiency, while information is viewed as anything capable of reducing the uncertainty. A broad conceptual framework for GIT is obtained by expanding the formalized language of classical set theory to include more expressive formalized languages based on fuzzy sets of various types, and by expanding classical theory of additive measures to include more expressive non-additive measures of various types.
This landmark book examines each of several theories for dealing with particular types of uncertainty at the following four levels:
- Mathematical formalization of the conceived type of uncertainty
- Calculus for manipulating this particular type of uncertainty
- Justifiable ways of measuring the amount of uncertainty in any situation formalizable in the theory
- Methodological aspects of the theory
With extensive use of examples and illustrations to clarify complex material and demonstrate practical applications, generous historical and bibliographical notes, end-of-chapter exercises to test readers' newfound knowledge, glossaries, and an Instructor's Manual, this is an excellent graduate-level textbook, as well as an outstanding reference for researchers and practitioners who deal with the various problems involving uncertainty and information.
Inhalt
Preface xiii
Acknowledgments xvii
1 Introduction 1
1.1. Uncertainty and Its Significance 1
1.2. Uncertainty-Based Information 6
1.3. Generalized Information Theory 7
1.4. Relevant Terminology and Notation 10
1.5. An Outline of the Book 20
Notes 22
Exercises 23
2 Classical Possibility-Based Uncertainty Theory 26
2.1. Possibility and Necessity Functions 26
2.2. Hartley Measure of Uncertainty for Finite Sets 27
2.2.1. Simple Derivation of the Hartley Measure 28
2.2.2. Uniqueness of the Hartley Measure 29
2.2.3. Basic Properties of the Hartley Measure 31
2.2.4. Examples 35
2.3. Hartley-Like Measure of Uncertainty for Infinite Sets 45
2.3.1. Definition 45
2.3.2. Required Properties 46
2.3.3. Examples 52
Notes 56
Exercises 57
3 Classical Probability-Based Uncertainty Theory 61
3.1. Probability Functions 61
3.1.1. Functions on Finite Sets 62
3.1.2. Functions on Infinite Sets 64
3.1.3. Bayes' Theorem 66
3.2. Shannon Measure of Uncertainty for Finite Sets 67
3.2.1. Simple Derivation of the Shannon Entropy 69
3.2.2. Uniqueness of the Shannon Entropy 71
3.2.3. Basic Properties of the Shannon Entropy 77
3.2.4. Examples 83
3.3. Shannon-Like Measure of Uncertainty for Infinite Sets 91
Notes 95
Exercises 97
4 Generalized Measures and Imprecise Probabilities 101
4.1. Monotone Measures 101
4.2. Choquet Capacities 106
4.2.1. Möbius Representation 107
4.3. Imprecise Probabilities: General Principles 110
4.3.1. Lower and Upper Probabilities 112
4.3.2. Alternating Choquet Capacities 115
4.3.3. Interaction Representation 116
4.3.4. Möbius Representation 119
4.3.5. Joint and Marginal Imprecise Probabilities 121
4.3.6. Conditional Imprecise Probabilities 122
4.3.7. Noninteraction of Imprecise Probabilities 123
4.4. Arguments for Imprecise Probabilities 129
4.5. Choquet Integral 133
4.6. Unifying Features of Imprecise Probabilities…