This book provides a multi-level introduction to Bayesian reasoning (as opposed to ''conventional statistics'') and its applications to data analysis. The basic ideas of this ''new'' 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.
Author(s): Giulio D. Agostini
Publisher: World Scientific Publishing Company
Year: 2003
Language: English
Commentary: 42726
Pages: 351
Contents......Page 14
Preface
......Page 8
PART I Critical review and outline of the Bayesian alternative......Page 22
1.1 Uncertainty in physics......Page 24
1.2 True value, error and uncertainty......Page 26
1.3 Sources of measurement uncertainty......Page 27
1.4 Usual handling of measurement uncertainties......Page 28
1.5 Probability of observables versus probability of 'true values'......Page 30
1.7 Unsuitability of frequentistic confidence intervals......Page 32
1.8 Misunderstandings caused by the standard paradigm of hypothesis tests......Page 36
1.9 Statistical significance versus probability of hypotheses......Page 40
2.1 Where to restart from?......Page 46
2.2 Concepts of probability......Page 48
2.3 Subjective probability......Page 50
2.4 Learning from observations: the 'problem of induction'......Page 53
2.6 From the probability of the effects to the probability of the causes......Page 55
2.7 Bayes' theorem for uncertain quantities: derivation from a physicist's point of view......Page 57
2.8 Afraid of 'prejudices'? Logical necessity versus frequent practical irrelevance of the priors......Page 58
2.9 Recovering standard methods and short-cuts to Bayesian reasoning......Page 60
2.10.1 Direct measurement in the absence of systematic errors......Page 62
2.10.2 Indirect measurements......Page 63
2.10.3 Systematic errors......Page 64
2.10.4 Approximate methods......Page 67
PART 2 A Bayesian primer......Page 70
3.1 What is probability?......Page 72
3.2 Subjective definition of probability......Page 73
3.3 Rules of probability......Page 76
3.4 Subjective probability and 'objective' description of the physical world......Page 79
3.5.1 Dependence of the probability on the state of information......Page 81
3.5.2 Conditional probability......Page 82
3.5.3 Bayes' theorem......Page 84
3.5.4 'Conventional' use of Bayes' theorem......Page 87
3.6 Bayesian statistics: learning by experience......Page 89
3.7 Hypothesis 'test' (discrete case)......Page 92
3.7.1 Variations over a problem to Newton......Page 93
3.9 Probability versus decision......Page 97
3.10 Probability of hypotheses versus probability of observations......Page 98
3.11.1 General criteria......Page 99
3.11.2 Insufficient reason and Maximum Entropy......Page 102
3.12.1 AIDS test......Page 103
3.12.2 Gold/silver ring problem......Page 104
3.12.4 Which random generator is responsible for the observed number?......Page 105
3.13 Some further examples showing the crucial role of background knowledge......Page 106
4.1 Discrete variables......Page 110
4.2 Continuous variables: probability and probability density function......Page 113
4.3 Distribution of several random variables......Page 119
4.4 Propagation of uncertainty......Page 125
4.5.1 Terms and role......Page 129
4.5.3 Normal approximation of the binomial and of the Poisson distribution......Page 132
4.5.5 Caution......Page 133
4.6 Laws of large numbers......Page 134
5.1 Measurement error and measurement uncertainty......Page 136
5.1.1 General form of Bayesian inference......Page 137
5.2 Bayesian inference and maximum likelihood......Page 139
5.3 The dog, the hunter and the biased Bayesian estimators......Page 140
5.4.1 Difference with respect to the discrete case......Page 141
5.4.2 Bertrand paradox and angels' sex......Page 142
6.1 Normally distributed observables......Page 144
6.2 Final distribution, prevision and credibility intervals of the true value......Page 145
6.3 Combination of several measurements — Role of priors......Page 146
6.4 Conjugate priors......Page 147
6.6 Predictive distribution......Page 148
6.7 Measurements close to the edge of the physical region......Page 149
6.8 Uncertainty of the instrument scale offset......Page 152
6.10 Measuring two quantities with the same instrument having an uncertainty of the scale offset......Page 154
6.11 Indirect calibration......Page 157
6.12 The Gauss derivation of the Gaussian......Page 158
7.1 Binomially distributed observables......Page 162
7.1.1 Observing 0% or 100%......Page 166
7.1.3 Conjugate prior and many data limit......Page 167
7.3 Predicting relative frequencies — Terms and interpretation of Bernoulli's theorem......Page 169
7.4 Poisson distributed observables......Page 173
7.4.1 Observation of zero counts......Page 175
7.6 Predicting future counts......Page 176
7.7.1 Dependence on priors — practical examples......Page 177
7.7.2 Combination of results from similar experiments......Page 179
7.7.3 Combination of results: general case......Page 181
7.7.4 Including systematic effects......Page 183
7.7.5 Counting measurements in the presence of background......Page 186
8.1 Maximum likelihood and least squares as particular cases of Bayesian inference......Page 190
8.2 Linear fit......Page 193
8.3 Linear fit with errors on both axes......Page 196
8.4 More complex cases......Page 197
8.5 Systematic errors and 'integrated likelihood'......Page 198
8.6 Linearization of the effects of influence quantities and approximate formulae......Page 199
8.7 BIPM and ISO recommendations......Page 202
8.8 Evaluation of type B uncertainties......Page 204
8.9 Examples of type B uncertainties......Page 205
8.10 Comments on the use of type B uncertainties......Page 207
8.11 Caveat concerning the blind use of approximate methods......Page 210
8.12 Propagation of uncertainty......Page 212
8.13.1 Building the covariance matrix of experimental data......Page 213
8.14.1 Best estimate of the true value from two correlated values......Page 218
8.14.3 Normalization uncertainty......Page 219
8.14.4 Peelle's Pertinent Puzzle......Page 223
9.1 Problem and typical solutions......Page 224
9.2 Bayes' theorem stated in terms of causes and effects......Page 225
9.3 Unfolding an experimental distribution......Page 226
PART 3 Further comments, examples and applications......Page 230
10.1 Unifying role of subjective approach......Page 232
10.2 Frequentists and combinatorial evaluation of probability......Page 234
10.3 Interpretation of conditional probability......Page 236
10.4 Are the beliefs in contradiction to the perceived objectivity of physics?......Page 237
10.5 Frequentists and Bayesian 'sects'......Page 241
10.5.1 Bayesian versus frequentistic methods......Page 242
10.5.2 Subjective or objective Bayesian theory?......Page 243
10.6 Biased Bayesian estimators and Monte Carlo checks of Bayesian procedures......Page 247
10.7 Frequentistic coverage......Page 250
10.7.1 Orthodox teacher versus sharp student - a dialogue by George Gabor......Page 253
10.8 Why do frequentistic hypothesis tests 'often work'?......Page 254
10.9 Comparing 'complex' hypotheses — automatic Ockham' Razor......Page 260
10.10.1 Networks of beliefs - conceptual and practical applications......Page 262
10.10.2 The gold/silver ring problem in terms of Bayesian networks......Page 263
11.1 Use and misuse of the standard combination rule......Page 268
11.2 'Apparently incompatible' experimental results......Page 270
11.3 Sceptical combination of experimental results......Page 273
11.3.1 Application to έ/ε......Page 280
11.3.2 Posterior evaluation of σι......Page 283
12.1 Usual combination of 'statistic and systematic errors'......Page 288
12.2.1 Asymmetric χ2 and '∆χ2 = 1 rule'......Page 290
12.2.2 Systematic effects......Page 293
12.3 General solution of the problem......Page 294
12.4 Approximate solution......Page 296
12.4.1 Linear expansion around E(X)......Page 297
12.4.2 Small deviations from linearity......Page 299
12.5 Numerical examples......Page 301
12.6 The non-monotonic case......Page 303
13.1 Frontier physics measurements at the limit to the detector sensitivity......Page 306
13.2 Desiderata for an optimal report of search results......Page 307
13.3 Master example: Inferring the intensity of a Poisson process in the presence of background......Page 308
13.5 Choice of priors......Page 309
13.5.1 Uniform prior......Page 310
13.5.2 Jeffreys' prior......Page 311
13.5.4 Priors reflecting the positive attitude of researchers......Page 313
13.6 Prior-free presentation of the experimental evidence......Page 316
13.7 Some examples of R-function based on real data......Page 319
13.8 Sensitivity bound versus probabilistic bound......Page 320
13.9 Open versus closed likelihood......Page 323
PART 4 Conclusion......Page 326
14.1 About subjective probability and Bayesian inference......Page 328
14.2 Conservative or realistic uncertainty evaluation?......Page 329
14.4 Bibliographic note......Page 331
Bibliography......Page 334
Index......Page 346
