## Uncertainty and Information: Foundations of Generalized Information TheoryDeal 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. |

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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 |

### 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.