Bayesian reasoning in data analysis
A critical introduction
by Giulio D'Agostini
"Statistics
books must take seriously the need
to teach the foundations of statistical reasoning from the beginning...
D'Agostini's new book does it admirably, building an
edifice od Bayesian statistical reasoning in the physical sciences
on solid foundations.
According to the author, the audience for this
book is practicing physicists and engineers who need to
evaluate uncertainty. To that list, I would like
to add upper-level and graduate students in physics and engineering, statisticians and
statistics graduate students who collaborate with physicists and
engineers, mathematicians who teach statistics courses,
and all Bayesian statisticians"
(J. Am. Stat. Ass.)
| Top | CONTENTS |
hypertexted BIBLIOGRAPHY |
INDEX |
ERRATA | Book reviews |
Author Home Page |
[soft cover (2013) recommended!]

List of contents
Preface
PART 1 - Critical review and outline of the Bayesian alternative
Chapter 1 - Uncertainty in physics and the usual methods of handling it
1.1 Uncertainty in physics [3]
1.2 True value, error and uncertainty [5]
1.3 Sources of measurement uncertainty [6]
1.4 Usual handling of measurement uncertainties [7]
1.5 Probability of observables versus probability of `true values' [9]
1.6 Probability of the causes [11]
1.7 Unsuitability of frequentistic confidence intervals [11]
1.8 Misunderstandings caused by the standard paradigm of hypothesis tests [15]
1.9 Statistical significance versus probability of hypotheses [19]
Chapter 2 - A probabilistic theory of measurement uncertainty
2.1 Where to restart from? [25]
2.2 Concepts of probability [27]
2.3 Subjective probability [29]
2.4 Learning from observations: the `problem of induction' [32]
2.5 Beyond Popper's falsification scheme [34]
2.6 From the probability of the effects to the probability of the causes [34]
2.7 Bayes' theorem for uncertain quantities [36]
2.8 Afraid of `prejudices'? Logical necessity versus frequent practical irrelevance of priors [37]
2.9 Recovering standard methods and short-cuts to Bayesian reasoning [39]
2.10 Evaluation of measurement uncertainty: general scheme [41]
2.10.1 Direct measurement in the absence of systematic errors [41]
2.10.2 Indirect measurements [42]
2.10.3 Systematic errors [43]
2.10.4 Approximate methods [46]
PART 2 - A Bayesian primer
Chapter 3 - Subjective probability and Bayes' theorem
3.1 What is probability? [51]
3.2 Subjective definition of probability [52]
3.3 Rules of probability [55]
3.4 Subjective probability and `objective' description of the physical world [58]
3.5 Conditional probability and Bayes' theorem [60]
3.5.1 Dependence of the probability on the state of information [60]
3.5.2 Conditional probability [61]
3.5.3 Bayes' theorem [63]
3.5.4 `Conventional' use of Bayes' theorem [66]
3.6 Bayesian statistics: learning by experience [68]
3.7 Hypothesis `test' (discrete case) [71]
3.7.1 Variations over a problem to Newton [72]
3.8 Falsificationism and Bayesian statistics [76]
3.9 Probability versus decision [76]
3.10 Probability of hypotheses versus probability of observations [77]
3.11 Choice of the initial probabilities (discrete case) [78]
3.11.1 General criteria [78]
3.11.2 Insufficient reason and Maximum Entropy [81]
3.12 Solution to some problems [82]
3.12.1 AIDS test [82]
3.12.2 Gold/silver ring problem [83]
3.12.3 Regular or double-head coin? [84]
3.12.4 Which random generator is responsible for the observed number? [84]
3.13 Some further examples showing the crucial role of background knowledge [85]
Chapter 4 - Probability distributions (a concise reminder)
4.1 Discrete variables [89]
4.2 Continuous variables: probability and probability density function [92]
4.3 Distribution of several random variables [98]
4.4 Propagation of uncertainty [104]
4.5 Central limit theorem [108]
4.5.1 Terms and role [108]
4.5.2 Distribution of a sample average [111]
4.5.3 Normal approximation of the binomial and of the Poisson distribution [111]
4.5.4 Normal distribution of measurement errors [112]
4.5.5 Caution [112]
4.6 Laws of large numbers [113]
Chapter 5 - Bayesian inference of continuous quantities
5.1 Measurement error and measurement uncertainty [115]
5.1.1 General form of Bayesian inference [116]
5.2 Bayesian inference and maximum likelihood [118]
5.3 The dog, the hunter and the biased Bayesian estimators [119]
5.4 Choice of the initial probability density function [120]
5.4.1 Difference with respect to the discrete case [120]
5.4.2 Bertrand paradox and angels' sex [121]
Chapter 6 - Gaussian likelihood
6.1 Normally distributed observables [123]
6.2 Final distribution, prevision and credibility intervals of the true value [124]
6.3 Combination of several measurements -- Role of priors [125]
6.3.1 Update of estimates in terms of Kalman filter [126]
6.4 Conjugate priors [126]
6.5 Improper priors --- never take models literally! [127]
6.6 Predictive distribution [127]
6.7 Measurements close to the edge of the physical region [128]
6.8 Uncertainty of the instrument scale offset [131]
6.9 Correction for known systematic errors [133]
6.10 Measuring two quantities with the same instrument having an uncertainty of the scale offset [133]
6.11 Indirect calibration [136]
6.12 The Gauss derivation of the Gaussian [137]
Chapter 7 - Counting experiments
7.1 Binomially distributed observables [141]
7.1.1 Observing 0\% or 100\% [145]
7.1.2 Combination of independent measurements [146]
7.1.3 Conjugate prior and many data limit [146]
7.2 The Bayes problem [148]
7.3 Predicting relative frequencies -- Terms and interpretation of Bernoulli's theorem [148]
7.4 Poisson distributed observables [152]
7.4.1 Observation of zero counts [154]
7.5 Conjugate prior of the Poisson likelihood [155]
7.6 Predicting future counts [155]
7.7 A deeper look to the Poissonian case [156]
7.7.1 Dependence on priors --- practical examples [156]
7.7.2 Combination of results from similar experiments [158]
7.7.3 Combination of results: general case [160]
7.7.4 Including systematic effects [162]
7.7.5 Counting measurements in the presence of background [165]
Chapter 8 - Bypassing Bayes' theorem for routine applications
8.1 Maximum likelihood and least squares as particular cases of Bayesian inference [169]
8.2 Linear fit [172]
8.3 Linear fit with errors on both axes [175]
8.4 More complex cases [176]
8.5 Systematic errors and `integrated likelihood' [177]
8.6 Linearization of the effects of influence quantities and approximate formulae [178]
8.7 BIPM and ISO recommendations [181]
8.8 Evaluation of type B uncertainties [183]
8.9 Examples of type B uncertainties [184]
8.10 Comments on the use of type B uncertainties [186]
8.11 Caveat concerning the blind use of approximate methods [189]
8.12 Propagation of uncertainty [191]
8.13 Covariance matrix of experimental results -- more details [192]
8.13.1 Building the covariance matrix of experimental data [192]
8.13.1.1 Offset uncertainty [193]
8.13.1.2 Normalization uncertainty [195]
8.13.1.3 General case [196]
8.14 Use and misuse of the covariance matrix to fit correlated data [197]
8.14.1 Best estimate of the true value from two correlated values [197]
8.14.2 Offset uncertainty [198]
8.14.3 Normalization uncertainty [198]
8.14.4 Peelle's Pertinent Puzzle [202]
Chapter 9 - Bayesian unfolding
9.1 Problem and typical solutions [203]
9.2 Bayes' theorem stated in terms of causes and effects [204]
9.3 Unfolding an experimental distribution [205]
PART 3 - Further comments, examples and applications
Chapter 10 - Miscellanea on general issues in probability and inference
10.1 Unifying role of subjective approach [211]
10.2 Frequentists and combinatorial evaluation of probability [213]
10.3 Interpretation of conditional probability [215]
10.4 Are the beliefs in contradiction to the perceived objectivity of physics? [216]
10.5 Frequentists and Bayesian `sects' [220]
10.5.1 Bayesian versus frequentistic methods [221]
10.5.2 Subjective or objective Bayesian theory? [222]
10.5.3 Bayes' theorem is not everything [226]
10.6 Biased Bayesian estimators and Monte Carlo checks of Bayesian procedures [226]
10.7 Frequentistic coverage [229]
10.7.1 Orthodox teacher versus sharp student - a dialogue by George Gabor [232]
10.8 Why do frequentistic hypothesis tests `often work'? [233]
10.9 Comparing `complex' hypotheses -- automatic Ockham' Razor [239]
10.10 Bayesian networks [241]
10.10.1 Networks of beliefs -- conceptual and practical applications [241]
10.10.2 The gold/silver ring problem in terms of Bayesian networks [242]
Chapter 11 - Combination of experimental results: a closer look
11.1 Use and misuse of the standard combination rule [247]
11.2 `Apparently incompatible' experimental results [249]
11.3 Sceptical combination of experimental results [252]
11.3.1 Application to epsilon'/epsilon [259]
11.3.2 Posterior evaluation of sigma_i [262]
Chapter 12 - Asymmetric uncertainties and nonlinear propagation
12.1 Usual combination of `statistic and systematic errors' [267]
12.2 Sources of asymmetric uncertainties in standard statistical procedures [269]
12.2.1 Asymmetric chi2 and ``Delta_chi2 =1 rule'[269]
12.2.2 Systematic effects [272]
12.2.2.1 Asymmetric beliefs on systematic effects [273]
12.2.2.2 Nonlinear propagation of uncertainties [273]
12.3 General solution of the problem [273]
12.4 Approximate solution [275]
12.4.1 Linear expansion around E[X] [276]
12.4.2 Small deviations from linearity [278]
12.5 Numerical examples [280]
12.6 The non-monotonic case [282]
Chapter 13 Which priors for frontier physics?
13.1 Frontier physics measurements at the limit to the detector sensitivity [285]
13.2 Desiderata for an optimal report of search results [286]
13.3 Master example: Inferring the intensity of a Poisson process in the presence of background [287]
13.4 Modelling the inferential process [288]
13.5 Choice of priors [288]
13.5.1 Uniform prior [289]
13.5.2 Jeffreys' prior [290]
13.5.3 Role of priors [292]
13.5.4 Priors reflecting the positive attitude of researchers [292]
13.6 Prior-free presentation of the experimental evidence [295]
13.7 Some examples of R-function based on real data [298]
13.8 Sensitivity bound versus probabilistic bound [299]
13.9 Open versus closed likelihood [302]
PART 4 - Conclusion
Chapter 14 - Conclusions and bibliography
14.1 About subjective probability and Bayesian inference] 307}
14.2 Conservative or realistic uncertainty evaluation? [308]
14.3 Assessment of uncertainty is not a mathematical game [310]
14.4 Bibliographic note [310]
| Top | CONTENTS |
hypertexted BIBLIOGRAPHY |
INDEX |
ERRATA | Book reviews |
Author Home Page |
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R. Jeffrey, ``Subjective Probability (The Real Thing)'', 2002
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| Top | CONTENTS |
hypertexted BIBLIOGRAPHY |
INDEX |
ERRATA | Book reviews |
Author Home Page |
Index
"Delta chi^2 = 1" rule, 170, 269--271
"Delta ln L =1/2" rule, 269
3 sigma rule, 236
AIDS test problem
formulation, 20
solution, 82
Anderson C.D., 218
approximate methods, 39, 41, 46
arbitrariness and subjectivism, 30
Aristoteles, 229
Astone P., 166
asymmetric uncertainty, 267
average, 32, 90, 93
distribution, 111
probability, 9
axioms of probability, 29
background in counting experiments, 165
Basu S., 233
Bayes factor, 72, 82, 239, 296
Bayes T., 52
Bayes' theorem, 32, 36, 63--65, 99
criticism of `conventional use', 67
Bayesian, 32
estimators, 226
inference, 34, 116
networks, 241, 242, 244, 246
statistics, 68
belief
degree of, 29, 32, 53
networks, 241, 242, 244, 246
Bernardo J.M., 223
Bernoulli J., 52
Bernoulli's theorem, 113, 148--150
Berry D.A., 218, 232
Bertrand paradox, 121, 122
bet
and probability, 29
coherent, 54
coherent bet, 29
odds, 54
beta distribution, 97
betting odds, 55
binomial
distribution, 91
likelihood, 41, 141
BIPM, 181
recommendation INC-1, 31
black hole, 218
blending factor, 126
bound
probabilistic, 299, 301
sensitivity, 299, 301
branching ratio, 141
CELLO, 199
central limit theorem, 110--113, 183, 187
terms, 108
chi-square
$Delta \chi^2 = 1$ rule, 170, 269--271
distribution, 96
minimization (fit), 170, 269--271
test, 236
coherence, 29, 55, 213, 216, 226
coherent bet, 54
Coletti G., 225
combination of results
binomial model, 146
Gaussian model, 125, 247
outliers, 247
Poisson model, 158--162
sceptical combination, 252--254
combinatorial `definition' of probability, 30
complete class, 58
conditional inference, 44
conditional probability, 60--62, 215, 216
conditioning, 32
confidence (probability), 73
confidence interval (frequentistic), 11, 232
conjugate prior, 126
binomial likelihood, 146
Gaussian likelihood, 126
Poisson likelihood, 155
contradiction
proof by, 15
correlation
between events, 63
coefficient, 99
due to common systematics, 192
due to systematics, 133, 134, 179--181, 193, 195
correlation matrix, 101
counting experiments, 141
covariance, 99
matrix
misuse in fits, 197
of experimental results, 192, 193, 195
covariance matrix, 101
coverage, 229--231, 301
Cox R.T., 57
credibility interval, 124
cumulative function, 90, 93
Cygnus X-1, 218
de Finetti B., 3, 26, 27, 51, 55, 60, 88, 142, 150, 225
representation theorem, 28
de Rujula A., 238
decision theory, 76
deduction, 34
degree of belief, 29, 32, 53
Descartes' cogito, 26
deterministic law, 5
DIN, 115
dog-hunter, 12, 119
Dose V., 253
edge of physical region
measurement at the, 12, 128
efficiencies, 42
Efron B., 221
Einstein A., 25, 26, 30
empirical law of chance, 150
entropy, 82
equiprobability, 52
error, 4, 5, 115, 116
normal distributed, 41
statistical, 8
systematic, 8, 43
events, 53
evidence, 239, 240
exchangeability, 28, 142
expected gain, 76
expected value, 90, 93, 99
exponential distribution, 95
falsification scheme, 34
falsificationism, 76
Fermi E., 171
Feynman R., 51, 220
final probability, 65
finite partition, 58
Fisher R.A., 172
fit, 172--174
Franklin A., 217
fuzzy logic, 26
Gabor G.
a Socratic exchange, 232, 233
Galilei G., 211
Galison P., 218
gamma distribution, 95
Gauss K.F., 52, 137--139, 172
Gaussian
distribution, 94
distribution of error, 112
Gauss derivation, 137--139
likelihood, 41, 123
gold/silver ring problem
Bayesian network solution, 242, 244, 246
formulation, 67
solution, 83
good sense, 52
gravitational constant, 59
Hawking S., 218
HERA `events', 21, 220
hidden variables, 30
Higgs boson, 78
claim of discovery, 23
Howson C., 11
Hume D., 25, 26, 28, 33
hunter-dog, 12, 119
hypotheses
probability of, 19
hypothesis
test (frequentistic), 15, 233
implication, 58
improper priors, 127
independence, 63
indifference principle, 81
indirect measurements, 42
induction, 32, 34
influence quantity, 43, 116, 117
information entropy, 82
initial probability, 65
innovation, 126
input quantity, 273
intersubjectivity, 31
ISO, 5, 115, 181
Guide, 5, 31, 53, 308, 310
Jacobian, 105
JavaBayes, 242, 243
Jaynes E., 57
Jaynes' robot, 223
Jeffreys H., 57, 221
priors, 223, 290--293
Kalman filter, 126
Kant I., 25, 26
kurtosis, 185, 279
Lad F., 225
Laplace P.-S., 30, 51, 52, 141, 225
recursive formula, 144
rule of succession, 144
laws of large numbers, 113
learning by experience, 68
least squares, 39, 40, 170
likelihood, 35, 36
$Delta \chi^2 = 1$ rule, 269
binomial, 41
closed, 285, 302
Gaussian, 41, 123
maximumemph {see} mximum likelihood, 170
open, 285, 302
Poisson, 41
principle, 170, 229, 236
Linden W. von der, 253
linear fit, 172--174
linearization, 178
logical product, 58
logical sum, 58
lower limit
binomial model, 145
Mach E., 26
marginalization, 98
maximum bounds, 8
Maximum Entropy, 57, 82
maximum entropy, 223
maximum likelihood, 39, 40, 118, 122, 169, 170
mean, 90, 93
measurand, 6
Millikan R., 252
mode, 32
moments, 91
central, 91
Moore D., 222
multinomial distribution, 40, 103
negative mass, 12, 128
Newton I., 72
non-informative priors, 223
normal, see {Gaussian}{94}
null hypothesis, 15
null observation, 154
objective Bayesian theory, 222
objective inference, 223
objectivity of physics, 216--218
Ockham' Razor, 239, 240
odds in betting, 54
operational subjectivism, 225
Orear J., 171
outliers, 247
output quantity, 273
p-value, 16
Particle Data Group (PDG), 213, 221, 248, 259, 260, 268, 269
particle identification, 66
Pauli W., 51
Pearson chi-square, 40, 96, 103
Peelle's Pertinent Puzzle, 202
penalization, 55
Pizzella G., 166
Poincar'e H., 11, 25, 30, 38, 115, 238
Poisson
distribution, 92
likelihood, 41, 152
process, 95, 287
Polya G., 213
Popper K.R., 34, 76
positive attitude (of researchers), 131, 293
posterior, 65
predictive distribution
binomial likelihood, 151
Gaussian likelihood, 127
Poisson likelihood, 155
prevision, 17, 124
prior knowledge, 33
priors, 36--38, 40, 65, 120, 125, 126
conjugate, 126
logical necessity, 37, 238
motivated by `positive attitude', 131
non-informative, 223
objective, 223
probabilistic law, 4
probability
``does not exists'', 60
axioms, 29, 55
basic rules, 29, 55
classical, 52
classical view, 27
combinatorial definition, 51
concepts, 27
conditional, 60--62
density function, 32, 92
distribution
beta, 97
binomial, 91
bivariate Gaussian, 101
chi-square, 96
conditional, 98
Erlang, 96
exponential, 95
gamma, 95
Gaussian, 94, 185, 187
marginal, 98
multinomial, 40, 103
normal, see {Gaussian}{94}
Poisson, 92
triangular, 97, 185, 187
triangular asymmetric, 185, 187
uniform, 94, 185, 187
favorable over possible cases, 51
frequentistic definition, 51
frequentistic view, 27
function, 32, 90
inversion, 10, 14
Laplace `definition', 30, 52
logical view, 27
objective, 28
of causes, 11, 34
of hypotheses, 19, 77
of observations, 77
standard `definitions', 30
statistical view, 27
subjective, 28, 29, 52, 58
proof by contradiction, 15
propagation of uncertainties, 8, 42, 44, 104--108, 191
linearization, 178--181, 276--278
non-monotonic case, 282
nonlinear, 273, 278, 279
proportions, 42
quantum mechanics, 29
random variables, 31, 89
relative belief updating ratio, 296
repeatability, 7
reproductive property, 111
Salvini G., 237
Schrödinger E., 53, 57, 60, 142, 225
Scozzafava R., 225
sensitivity
analysis, 71, 188, 303
bound, 299, 301
coefficient, 108
sets
properties, 56
versus events, 58
shape distortion function, 296
signal to noise ratio, 66
significance
level, 15
probabilities, 16
statistical, 19
skewness, 185, 279
smearing, 204
Smith A.F.M., 221, 223
standard deviation, 32, 91
standard statistical methods
recovering, 39
statistical effects, 7
statistical error, 8
statistical significance, 19
supersymmetric particles, 59
systematic effects, 7
systematic error, 8, 43
correlation, 133, 134
linearization, 178--181
normalization, 195
of known size, 133
offset, 131, 193
Poisson model, 162
tail, probability of, 22
Thorne K., 218
top quark, 59
Tribus M., 82
true value, 5--7
probability of, 9
type A uncertainty, 133, 181
type B uncertainty, 133, 181--184, 186
uncertain numbers, 31, 89
uncertainty, 4, 5, 115, 116
`standard' treatment, 7
asymmetric, 267
propagation, 8, 42, 44, 104--108, 191, 268
source, 3
sources (ISO), 6
uncertainty and probability, 29
unfolding, 203--206, 208
uniform distribution, 94
upper limit
binomial model, 145
Poisson model, 154, 157--164, 166
Urbach P., 11
variance, 32, 90, 99
Venn diagrams, 56
von Mises R., 67, 213
Weise K., 182
Wittgenstein L., 13
Zech G., 229
Zellner A., 221
Zeno's paradox, 92
Errata
(These corrections have been applied to the second reprint,
released in summer 2005)
Other typos (also in second reprint):
- In denominator of Eq. (8.20) the average of x should be squared.
Book reviews
- Journal of the American Statistical Association (by G. Woodworth),
2004, vol. 99, no. 468, pp. 1201 - 1202
"Statistics
books must take seriously the need
to teach the foundations of statistical reasoning from the beginning...
D'Agostini's new book does it admirably, building an
edifice od Bayesian statistical reasoning in the physical sciences
on solid foundations.
According to the author, the audience for this
book is practicing physicists and engineers who need to
evaluate uncertainty. To that list, I would like
to add upper-level and graduate students in physics and engineering, statisticians and
statistics graduate students who collaborate with physicists and
engineers, mathematicians who teach statistics courses,
and all Bayesian statisticians"
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ERRATA | Book reviews |
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