## Uncertainty and Information: Foundations of Generalized Information TheoryDeal with information and uncertainty properly and efficientlyusing tools emerging from generalized information theory Uncertainty and Information: Foundations of Generalized InformationTheory contains comprehensive and up-to-date coverage of resultsthat have emerged from a research program begun by the author inthe early 1990s under the name "generalized information theory"(GIT). This ongoing research program aims to develop a formalmathematical treatment of the interrelated concepts of uncertaintyand information in all their varieties. In GIT, as in classicalinformation theory, uncertainty (predictive, retrodictive,diagnostic, prescriptive, and the like) is viewed as amanifestation of information deficiency, while information isviewed as anything capable of reducing the uncertainty. A broadconceptual framework for GIT is obtained by expanding theformalized language of classical set theory to include moreexpressive formalized languages based on fuzzy sets of varioustypes, and by expanding classical theory of additive measures toinclude more expressive non-additive measures of varioustypes. This landmark book examines each of several theories for dealingwith particular types of uncertainty at the following fourlevels: * Mathematical formalization of the conceived type ofuncertainty * Calculus for manipulating this particular type ofuncertainty * Justifiable ways of measuring the amount of uncertainty in anysituation formalizable in the theory * Methodological aspects of the theory With extensive use of examples and illustrations to clarify complexmaterial and demonstrate practical applications, generoushistorical and bibliographical notes, end-of-chapter exercises totest readers' newfound knowledge, glossaries, and an Instructor'sManual, this is an excellent graduate-level textbook, as well as anoutstanding reference for researchers and practitioners who dealwith the various problems involving uncertainty and information. AnInstructor's Manual presenting detailed solutions to all theproblems in the book is available from the Wiley editorialdepartment. |

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### Contents

1 | |

2 Classical PossibilityBased Uncertainty Theory | 26 |

3 Classical ProbabilityBased Uncertainty Theory | 61 |

4 Generalized Measures and Imprecise Probabilities | 101 |

5 Special Theories of Imprecise Probabilities | 143 |

6 Measures of Uncertainty and Information | 196 |

7 Fuzzy Set Theory | 260 |

8 Fuzzification of Uncertainty Theories | 315 |

Appendix B Uniqueness of Generalized Hartley Measure in the DempsterShafer Theory | 430 |

Appendix C Correctness of Algorithm 61 | 437 |

Appendix D Proper Range of Generalized Shannon Entropy | 442 |

Appendix E Maximum of GSa in Section 69 | 447 |

Appendix F Glossary of Key Concepts | 449 |

Appendix G Glossary of Symbols | 455 |

458 | |

487 | |

9 Methodological Issues | 355 |

10 Conclusions | 415 |

Appendix A Uniqueness of the UUncertainty | 425 |

### Other editions - View all

Uncertainty and Information: Foundations of Generalized Information Theory George J. Klir No preview available - 2005 |

Uncertainty and Information: Foundations of Generalized Information Theory George J. Klir No preview available - 2005 |

### Common terms and phrases

Â Â a-cut application Assume Axiom basic probability assignment binary relation bodies of evidence calculate capacity of order Cartesian product Choquet capacity Choquet integral convex sets crisp sets cutworthy defined by Eq denote determine equation example expressed formalized formula fuzzy intervals fuzzy numbers fuzzy relations fuzzy set theory given Hartley functional Hartley measure imprecise probabilities inequalities information theory Klir l-measures log log log lower and upper lower probability marginal marginal probability membership function minimum uncertainty Möbius representation monotone measures noninteractive nonspecificity obtained Œ Â Œ Œ ŒP(X possibility profile possibility theory principle of maximum probability distribution function probability measures probability theory properties real numbers respectively sets of probability Shannon entropy shown in Figure standard fuzzy subadditivity subsets Table Theorem tion tuple uncertainty theories universal set upper probability functions values variable xX Œ

### Popular passages

Page 1 - I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.

Page 3 - ... conquered lies in the fact that these problems, as contrasted with the disorganized situations with which statistics can cope, show the essential feature of organization. We will therefore refer to this group of problems as those of organized complexity.

Page 1 - In physical science a first essential step in the direction of learning any subject is to find principles of numerical reckoning and methods for practically measuring some quality connected with it. I often say that when you can measure what you are speaking about and express it in numbers, you know something about it ; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory...

Page 18 - Membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of degree.

Page 3 - ... and left untouched a great middle region. The importance of this middle region, moreover, does not depend primarily on the fact that the number of variables involved is moderate — large compared to two, but small compared to the number of atoms in a pinch of salt. The problems in this middle region, in fact, will often involve a considerable number of variables.

Page 10 - A is contained in B, then A is said to be a subset of B, and B is said to be a superset of A.