Uncertainty and information : foundations of generalized information theory Cover Table of Contents Preface Acknowledgments 1 Introduction 1.1. Uncertainty and Its Significance 1.2. Uncertainty Based Information 1.3. Generalized Information Theory 1.4. Relevant Terminology and Notation 1.5. An Outline of the Book Notes Exercises 2 Classical Possibility Based Uncertainty Theory 2.1. Possibility and Necessity Functions 2.2. Hartley Measure of Uncertainty for Finite Sets 2.2.1. Simple Derivation of the Hartley Measure 2.2.2. Uniqueness of the Hartley Measure 2.2.3. Basic Properties of the Hartley Measure 2.2.4. Examples 2.3. Hartley Like Measure of Uncertainty for Infinite Sets 2.3.1. Definition 2.3.2. Required Properties 2.3.3. Examples Notes Exercises 3 Classical Probability Based Uncertainty Theory 3.1. Probability Functions 3.1.1. Functions on Finite Sets 3.1.2. Functions on Infinite Sets 3.1.3. Bayes p( s Theorem 3.2. Shannon Measure of Uncertainty for Finite Sets 3.2.1. Simple Derivation of the Shannon Entropy 3.2.2. Uniqueness of the Shannon Entropy 3.2.3. Basic Properties of the Shannon Entropy 3.2.4. Examples 3.3. Shannon Like Measure of Uncertainty for Infinite Sets Notes Exercises 4 Generalized Measures and Imprecise Probabilities 4.1. Monotone Measures 4.2. Choquet Capacities 4.2.1. Ms Representation 4.3. Imprecise Probabilities: General Principles 4.3.1. Lower and Upper Probabilities 4.3.2. Alternating Choquet Capacities 4.3.3. Interaction Representation 4.3.4. Ms Representation 4.3.5. Joint and Marginal Imprecise Probabilities 4.3.6. Conditional Imprecise Probabilities 4.3.7. Noninteraction of Imprecise Probabilities 4.4. Arguments for Imprecise Probabilities 4.5. Choquet Integral 4.6. Unifying Features of Imprecise Probabilities Notes Exercises 5 Special Theories of Imprecise Probabilities 5.1. An Overview 5.2. Graded Possibilities 5.2.1. Ms Representation 5.2.2. Ordering of Possibility Profiles 5.2.3. Joint and Marginal Possibilities 5.2.4. Conditional Possibilities 5.2.5. Possibilities on Infinite Sets 5.2.6. Some Interpretations of Graded Possibilities 5.3. Sugeno l Measures 5.3.1. Ms Representation 5.4. Belief and Plausibility Measures 5.4.1. Joint and Marginal Bodies of Evidence 5.4.2. Rules of Combination 5.4.3. Special Classes of Bodies of Evidence 5.5. Reachable Interval Valued Probability Distributions 5.5.1. Joint and Marginal Interval Valued Probability Distributions 5.6. Other Types of Monotone Measures Notes Exercises 6 Measures of Uncertainty and Information 6.1. General Discussion 6.2. Generalized Hartley Measure for Graded Possibilities 6.2.1. Joint and Marginal U Uncertainties 6.2.2. Conditional U Uncertainty 6.2.3. Axiomatic Requirements for the U Uncertainty 6.2.4. U Uncertainty for Infinite Sets 6.3. Generalized Hartley Measure in Dempster£Shafer Theory 6.3.1. Joint and Marginal Generalized Hartley Measures 6.3.2. Monotonicity of the Generalized Hartley Measure 6.3.3. Conditional Generalized Hartley Measures 6.4. Generalized Hartley Measure for Convex Sets of Probability Distributions 6.5. Generalized Shannon Measure in Dempster Shafer Theory 6.6. Aggregate Uncertainty in Dempster£Shafer Theory 6.6.1. General Algorithm for Computing the Aggregate Uncertainty 6.6.2. Computing the Aggregated Uncertainty in Possibility Theory 6.7. Aggregate Uncertainty for Convex Sets of Probability Distributions 6.8. Disaggregated Total Uncertainty 6.9. Generalized Shannon Entropy 6.10. Alternative View of Disaggregated Total Uncertainty 6.11. Unifying Features of Uncertainty Measures Notes Exercises 7 Fuzzy Set Theory 7.1. An Overview 7.2. Basic Concepts of Standard Fuzzy Sets 7.3. Operations on Standard Fuzzy Sets 7.3.1. Complementation Operations 7.3.2. Intersection and Union Operations 7.3.3. Combinations of Basic Operations 7.3.4. Other Operations 7.4. Fuzzy Numbers and Intervals 7.4.1. Standard Fuzzy Arithmetic 7.4.2. Constrain