Bayesian reasoning in data analysis a critical introduction - D'Agostini, G., (Giulio) (World Scientific, 2003)
Includes bibliographical references (p. 313-323) and index
A multi-level introduction to Bayesian reasoning (as opposed to "conventional statistics") and its applications to data analysis. The basic ideas of this approach to the quantification of uncertainty are presented using examples from research and everyday life. Applications covered include: parametric inference; combination of results; treatment of uncertainty due to systematic errors and background; comparison of hypotheses; unfolding of experimental distributions; upper/lower bounds in frontier-type measurements. Approximate methods for routine use are derived and are shown often to coincide - under well-defined assumptions - with "standard" methods, which can therefore be seen as special cases of the more general Bayesian methods. In dealing with uncertainty in measurements, modern metrological ideas are utilized, including the ISO classification of uncertainty into type A and type B. These are shown to fit well into the Bayesian framework
Pt. 1 Critical review and outline of the Bayesian alternative. 1. Uncertainty in physics and the usual methods of handling it. 1.1. Uncertainty in physics. 1.2. True value, error and uncertainty. 1.3. Sources of measurement uncertainty. 1.4. Usual handling of measurement uncertainties. 1.5. Probability of observables versus probability of 'true values'. 1.6. Probability of the causes. 1.7. Unsuitability of frequentistic confidence intervals. 1.8. Misunderstandings caused by the standard paradigm of hypothesis tests. 1.9. Statistical significance versus probability of hypotheses --
- 2. A probabilistic theory of measurement uncertainty. 2.1. Where to restart from? 2.2. Concepts of probability. 2.3. Subjective probability. 2.4. Learning from observations: the 'problem of induction'. 2.5. Beyond Popper's falsification scheme. 2.6. From the probability of the effects to the probability of the causes. 2.7. Bayes' theorem for uncertain quantities. 2.8. Afraid of 'prejudices'? Logical necessity versus frequent practical irrelevance of priors. 2.9. Recovering standard methods and short-cuts to Bayesian reasoning. 2.10. Evaluation of measurement uncertainty: general scheme --
- pt. 2. A Bayesian primer. 3. Subjective probability and Bayes' theorem. 3.1 What is probability? 3.2. Subjective definition of probability. 3.3. Rules of probability. 3.4. Subjective probability and 'objective' description of the physical world. 3.5. Conditional probability and Bayes' theorem. 3.6. Bayesian statistics: learning by experience. 3.7. Hypothesis 'test' (discrete case). 3.8. Falsificationism and Bayesian statistics. 3.9. Probability versus decision. 3.10. Probability of hypotheses versus probability of observations. 3.11. Choice of the initial probabilities (discrete case). 3.12. Solution to some problems. 3.13. Some further examples showing the crucial role of background knowledge -- 4. Probability distributions (a concise reminder). 4.1. Discrete variables. 4.2. Continuous variables: probability and probability density function. 4.3. Distribution of several random variables. 4.4. Propagation of uncertainty. 4.5. Central limit theorem. 4.6 Laws of large numbers --