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Introduction to Imprecise Probabilities

Book Description

In recent years, the theory has become widely accepted and has been further developed, but a detailed introduction is needed in order to make the material available and accessible to a wide audience. This will be the first book providing such an introduction, covering core theory and recent developments which can be applied to many application areas. All authors of individual chapters are leading researchers on the specific topics, assuring high quality and up-to-date contents.

An Introduction to Imprecise Probabilities provides a comprehensive introduction to imprecise probabilities, including theory and applications reflecting the current state if the art. Each chapter is written by experts on the respective topics, including: Sets of desirable gambles; Coherent lower (conditional) previsions; Special cases and links to literature; Decision making; Graphical models; Classification; Reliability and risk assessment; Statistical inference; Structural judgments; Aspects of implementation (including elicitation and computation); Models in finance; Game-theoretic probability; Stochastic processes (including Markov chains); Engineering applications.

Essential reading for researchers in academia, research institutes and other organizations, as well as practitioners engaged in areas such as risk analysis and engineering.

Table of Contents

  1. Cover
  2. WILEY SERIES IN PROBABILITY AND STATISTICS
  3. Title Page
  4. Copyright
  5. Introduction
  6. A brief outline of this book
  7. Guide to the reader
  8. Contributors
  9. Acknowledgements
  10. Chapter 1: Desirability
    1. 1.1 Introduction
    2. 1.2 Reasoning about and with sets of desirable gambles
    3. 1.3 Deriving and combining sets of desirable gambles
    4. 1.4 Partial preference orders
    5. 1.5 Maximally committal sets of strictly desirable gambles
    6. 1.6 Relationships with other, nonequivalent models
    7. 1.7 Further reading
    8. Acknowledgements
  11. Chapter 2: Lower previsions
    1. 2.1 Introduction
    2. 2.2 Coherent lower previsions
    3. 2.3 Conditional lower previsions
    4. 2.4 Further reading
    5. Acknowledgements
  12. Chapter 3: Structural judgements
    1. 3.1 Introduction
    2. 3.2 Irrelevance and independence
    3. 3.3 Invariance
    4. 3.4 Exchangeability
    5. 3.5 Further reading
    6. Acknowledgements
  13. Chapter 4: Special cases
    1. 4.1 Introduction
    2. 4.2 Capacities and <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c04-math-0001.png" alt="c04-math-0001" style="vertical-align:middle;"></img>-monotonicity-monotonicity
    3. 4.3 <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c04-math-0058.png" alt="c04-math-0058" style="vertical-align:middle;"></img>-monotone capacities-monotone capacities
    4. 4.4 Probability intervals on singletons
    5. 4.5 <img xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0" src="images/c04-math-0106.png" alt="c04-math-0106" style="vertical-align:middle;"></img>-monotone capacities-monotone capacities
    6. 4.6 Possibility distributions, p-boxes, clouds and related models
    7. 4.7 Neighbourhood models
    8. 4.8 Summary
  14. Chapter 5: Other uncertainty theories based on capacities
    1. 5.1 Imprecise probability = modal logic + probability
    2. 5.2 From imprecise probabilities to belief functions and possibility theory
    3. 5.3 Discrepancies between uncertainty theories
    4. 5.4 Further reading
  15. Chapter 6: Game-theoretic probability
    1. 6.1 Introduction
    2. 6.2 A law of large numbers
    3. 6.3 A general forecasting protocol
    4. 6.4 The axiom of continuity
    5. 6.5 Doob's argument
    6. 6.6 Limit theorems of probability
    7. 6.7 Lévy's zero-one law
    8. 6.8 The axiom of continuity revisited
    9. 6.9 Further reading
    10. Acknowledgements
  16. Chapter 7: Statistical inference
    1. 7.1 Background and introduction
    2. 7.2 Imprecision in statistics, some general sources and motives
    3. 7.3 Some basic concepts of statistical models relying on imprecise probabilities
    4. 7.4 Generalized Bayesian inference
  17. Chapter 7: Statistical inference
    1. 7.5 Frequentist statistics with imprecise probabilities
    2. 7.6 Nonparametric predictive inference
    3. 7.7 A brief sketch of some further approaches and aspects
    4. 7.8 Data imprecision, partial identification
    5. 7.9 Some general further reading
    6. 7.10 Some general challenges
    7. Acknowledgements
  18. Chapter 8: Decision making
    1. 8.1 Non-sequential decision problems
    2. 8.2 Sequential decision problems
    3. 8.3 Examples and applications
  19. Chapter 9: Probabilistic graphical models
    1. 9.1 Introduction
    2. 9.2 Credal sets
    3. 9.5 Computing with credal networks
    4. 9.6 Further reading
    5. Acknowledgements
  20. Chapter 10: Classification
    1. 10.1 Introduction
    2. 10.2 Naive Bayes
    3. 10.3 Naive credal classifier (NCC)
    4. 10.4 Extensions and developments of the naive credal classifier
    5. 10.5 Tree-based credal classifiers
    6. 10.6 Metrics, experiments and software
    7. 10.7 Scoring the conditional probability of the class
    8. Acknowledgements
  21. Chapter 11: Stochastic processes
    1. 11.1 The classical characterization of stochastic processes
    2. 11.2 Event-driven random processes
    3. 11.3 Imprecise Markov chains
    4. 11.4 Limit behaviour of imprecise Markov chains
    5. 11.5 Further reading
  22. Chapter 12: Financial risk measurement
    1. 12.1 Introduction
    2. 12.2 Imprecise previsions and betting
    3. 12.3 Imprecise previsions and risk measurement
    4. 12.4 Further reading
  23. Chapter 13: Engineering
    1. 13.1 Introduction
    2. 13.2 Probabilistic dimensioning in a simple example
    3. 13.3 Random set modelling of the output variability
    4. 13.4 Sensitivity analysis
    5. 13.5 Hybrid models
    6. 13.6 Reliability analysis and decision making in engineering
    7. 13.7 Further reading
  24. Chapter 14: Reliability and risk
    1. 14.1 Introduction
    2. 14.2 Stress-strength reliability
    3. 14.3 Statistical inference in reliability and risk
    4. 14.4 Nonparametric predictive inference in reliability and risk
    5. 14.5 Discussion and research challenges
  25. Chapter 15: Elicitation
    1. 15.1 Methods and issues
    2. 15.2 Evaluating imprecise probability judgements
    3. 15.3 Factors affecting elicitation
    4. 15.4 Matching methods with purposes
    5. 15.5 Further reading
  26. Chapter 16: Computation
    1. 16.1 Introduction
    2. 16.2 Natural extension
    3. 16.3 Decision making
  27. References
  28. Author index
  29. Subject index
  30. WILEY SERIES IN PROBABILITY AND STATISTICS
  31. End User License Agreement