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Measurement Uncertainty and Probability

Book Description

A measurement result is incomplete without a statement of its 'uncertainty' or 'margin of error'. But what does this statement actually tell us? By examining the practical meaning of probability, this book discusses what is meant by a '95 percent interval of measurement uncertainty', and how such an interval can be calculated. The book argues that the concept of an unknown 'target value' is essential if probability is to be used as a tool for evaluating measurement uncertainty. It uses statistical concepts, such as a conditional confidence interval, to present 'extended' classical methods for evaluating measurement uncertainty. The use of the Monte Carlo principle for the simulation of experiments is described. Useful for researchers and graduate students, the book also discusses other philosophies relating to the evaluation of measurement uncertainty. It employs clear notation and language to avoid the confusion that exists in this controversial field of science.

Table of Contents

  1. Coverpage
  2. Measurement Uncertainty and Probability
  3. Title page
  4. Copyright page
  5. Dedication
  6. Contents
  7. Acknowledgements
  8. Introduction
  9. Part I Principles
    1. 1 Foundational ideas in measurement
      1. 1.1 What is measurement?
      2. 1.2 True values and target values
      3. 1.3 Error and uncertainty
      4. 1.4 Identifying the measurand
      5. 1.5 The measurand equation
      6. 1.6 Success – and the meaning of a ‘95% uncertainty interval’
      7. 1.7 The goal of a measurement procedure
    2. 2 Components of error or uncertainty
      1. 2.1 A limitation of classical statistics
      2. 2.2 Sources of error in measurement
      3. 2.3 Categorization by time-scale and by information
      4. 2.4 Remarks
    3. 3 Foundational ideas in probability and statistics
      1. 3.1 Probability and sureness
      2. 3.2 Notation and terminology
      3. 3.3 Statistical models and probability models
      4. 3.4 Inference and confidence
      5. 3.5 Two central limit theorems
      6. 3.6 The Monte Carlo method and process simulation
    4. 4 The randomization of systematic errors
      1. 4.1 The Working Group of 1980
      2. 4.2 From classical repetition to practical success rate
      3. 4.3 But what success rate? Whose uncertainty?
      4. 4.4 Parent distributions for systematic errors
      5. 4.5 Chapter summary
    5. 5 Beyond the ordinary confidence interval 72
      1. 5.1 Practical statistics – the idea of average confidence
      2. 5.2 Conditional confidence intervals
      3. 5.3 Chapter summary
  10. Part II Evaluation of uncertainty
    1. 6 Final preparation
      1. 6.1 Restatement of principles
      2. 6.2 Two important results
      3. 6.3 Writing the measurement model
      4. 6.4 Treating a normal error with unknown underlying variance
    2. 7 Evaluation using the linear approximation 98
      1. 7.1 Linear approximation to the measurand equation
      2. 7.2 Evaluation assuming approximate normality
      3. 7.3 Evaluation using higher moments
      4. 7.4 Monte Carlo evaluation of the error distribution
    3. 8 Evaluation without the linear approximation
      1. 8.1 Including higher-order terms
      2. 8.2 The influence of fixed estimates
      3. 8.3 Monte Carlo simulation of the measurement
      4. 8.4 Monte Carlo simulation – an improved procedure
      5. 8.5 Monte Carlo simulation – variant types of interval
      6. 8.6 Summarizing comments
    4. 9 Uncertainty information fit for purpose
      1. 9.1 Information for error propagation
      2. 9.2 An alternative propagation equation
      3. 9.3 Separate variances – measurement of functionals
      4. 9.4 Worst-case errors – measurement in product testing
      5. 9.5 Time-scales for errors
  11. Part III Related topics
    1. 10 Measurement of vectors and functions
      1. 10.1 Confidence regions
      2. 10.2 Simultaneous confidence intervals
      3. 10.3 Data snooping and cherry picking
      4. 10.4 Curve fitting and function estimation
      5. 10.5 Calibration
    2. 11 Why take part in a measurement comparison?
      1. 11.1 Examining the uncertainty statement
      2. 11.2 Estimation of the Type B error
      3. 11.3 Experimental bias
      4. 11.4 Complicating factors
    3. 12 Other philosophies
      1. 12.1 Worst-case errors
      2. 12.2 Fiducial inference
      3. 12.3 Bayesian inference
      4. 12.4 Measurement without target values
    4. 13 An assessment of objective Bayesian statistics
      1. 13.1 Ignorance and coherence
      2. 13.2 Information and entropy
      3. 13.3 Meaning and communication
      4. 13.4 A presupposition?
      5. 13.5 Discussion
    5. 14 Guide to the Expression of Uncertainty in Measurement
      1. 14.1 Report and recommendation
      2. 14.2 The mixing of philosophies
      3. 14.3 Consistency with other fields of science
      4. 14.4 Righting the Guide
    6. 15 Measurement near a limit – an insoluble problem?
      1. 15.1 Formulation
      2. 15.2 The Feldman–Cousins solution
      3. 15.3 The objective Bayesian solution
      4. 15.4 Purpose and realism
      5. 15.5 Doubts about uncertainty
      6. 15.6 Conclusion
  12. Appendix A The weak law of large numbers
  13. Appendix B The Sleeping Beauty paradox
  14. Appendix C The sum of normal and uniform variates
  15. Appendix D Analysis with one Type A and one Type B error
  16. Appendix E Conservatism of treatment of Type A errors
  17. Appendix F An alternative to a symmetric beta distribution
  18. Appendix G Dimensions of the ellipsoidal confidence region
  19. Appendix H Derivation of the Feldman–Cousins interval
  20. References
  21. Index