Bayesian probability theory: applications in the physical sciences
Gespeichert in:
| Hauptverfasser: | , , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2014
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | IX, 637 Seiten Illustrationen, Diagramme |
| ISBN: | 9781107035904 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804152437646295040 |
|---|---|
| adam_text | Titel: Bayesian probability theory
Autor: Linden, Wolfgang von der
Jahr: 2014
Contents
Preface page xi
PARTI INTRODUCTION 1
1 The meaning of probability 3
1.1 Classical definition of probability 3
1.2 Statistical definition of probability 9
1.3 Bayesian understanding of probability 10
2 Basic definitions for frequentist statistics and Bayesian inference 15
2.1 Definition of mean, moments and marginal distribution 15
2.2 Worked example: The three-urn problem 24
2.3 Frequentist statistics versus Bayesian inference 28
3 Bayesian inference 33
3.1 Propositions 33
3.2 Selected examples 34
3.3 Ockham s razor 43
4 Combinatorics 47
4.1 Preliminaries 47
4.2 Partitions, binomial and multinomial distributions 52
4.3 Occupation number problems 59
4.4 Geometrie and hypergeometric distributions 62
4.5 The negative binomial distribution 66
5 Random walks 71
5.1 First return 71
5.2 First lead 75
5.3 Random walk with absorbing wall 80
6 Limit theorems 83
6.1 Stirling s formula 83
6.2 de Moivre-Laplace theorem/local limit theorem 84
6.3 Bernoulli s law of large numbers 86
6.4 Poisson s law 87
vi Contents
7 Continuous distributions 92
7.1 Continuous propositions 92
7.2 Distribution function and probability density functions 93
7.3 Application in Statistical physics 96
7.4 Definitions for continuous distributions 98
7.5 Common probability distributions 99
7.6 Order statistic 118
7.7 Transformation of random variables 121
7.8 Characterisüc function 124
7.9 Error propagation 131
7.10 Helmert transformation 136
8 The central limit theorem 139
8.1 The theorem 139
8.2 Stable distributions 142
8.3 Proof of the central limit theorem 143
8.4 Markov chain Monte Carlo (MCMC) 144
8.5 The multivariate case 146
9 Poisson processes and waiting times 147
9.1 Stochastic processes 147
9.2 Three ways to generate Poisson points 150
9.3 Waiting time paradox 154
9.4 Order statistic of Poisson processes 156
9.5 Various examples 157
PART II ASSIGNING PROBABILITIES 163
10 Prior probabilities by transformation invariance 165
10.1 Bertrand s paradox revisited 167
10.2 Prior for scale variables 169
10.3 The prior for a location variable 171
10.4 Hyperplane priors 171
10.5 The invariant Riemann measure (Jeffreys prior) 176
11 Testable Information and maximum entropy 178
11.1 Discrete case 178
11.2 Properties of the Shannon entropy 182
11.3 Maximum entropy for continuous distributions 194
12 Quantified maximum entropy 201
12.1 The entropic prior 201
12.2 Derivation of the entropic prior 202
12.3 Saddle-point approximation for the normalization 203
12.4 Posterior probability density 204
Contents vii
12.5 Regularization and good data 205
12.6 A technical trick 209
12.7 Application to ill-posed inversion problems 210
13 Global smoothness 215
13.1 A primer on cubic splines 216
13.2 Second derivative prior 219
13.3 First derivative prior 221
13.4 Fisher Information prior 221
PART III PARAMETER ESTIMATION 225
14 Bayesian parameter estimation 227
14.1 The estimation problem 227
14.2 Loss and risk function 227
14.3 Confidence intervals 231
14.4 Examples 231
15 Frequentist parameter estimation 236
15.1 Unbiased estimators 236
15.2 The maximum likelihood estimator 237
15.3 Examples 237
15.4 Stopping criteria for experiments 241
15.5 Is unbiasedness desirable at all? 245
15.6 Least-squares fitting 246
16 The Cramer-Rao inequality 248
16.1 Lower bound on the variance 248
16.2 Examples 249
16.3 Admissibility of the Cramer-Rao limit 251
PART IV TESTING HYPOTHESES 253
17 The Bayesian way 255
17.1 Some illustrative examples 256
17.2 Independent measurements with Gaussian noise 262
18 The frequentist approach 276
18.1 Introduction 276
18.2 Neyman-Pearson lemma 281
19 Sampling distributions 284
19.1 Mean and median of i.i.d. random variables 284
19.2 Mean and variance of Gaussian samples 294
19.3 z-Staüsüc 297
19.4 Student s f-statistic 299
19.5 Fisher-Snedecor F-statistic 302
viii Contents
19.6 Chi-squared in case of missing parameters 305
19.7 Common hypothesis tests 308
20 Comparison of Bayesian and frequentist hypothesis tests 324
20.1 Prior knowledge is prior data 324
20.2 Dependence on the stopping criterion 325
PARTV REAL-WORLD APPLICATIONS 331
21 Regression 333
21.1 Linear regression 334
21.2 Models with nonlinear parameter dependence 350
21.3 Errors in all variables 353
22 Consistent inference on inconsistent data 364
22.1 Erroneously measured uncertainties 364
22.2 Combining incompatible measurements 380
23 Unrecognized Signal contributions 396
23.1 The nuclear fission cross-section 239Pu (n, f) 396
23.2 Electron temperature in a tokamak edge plasma 399
23.3 Signal-background Separation 403
24 Change point problems 409
24.1 The Bayesian change point problem 409
24.2 Change points in a binary image 415
24.3 Neural network modelling 420
24.4 Thin film growth detected by Auger analysis 427
25 Function estimation 431
25.1 Deriving trends from observations 432
25.2 Density estimation 439
26 Integral equations 451
26.1 Abel s integral equation 452
26.2 The Laplace transform 456
26.3 The Kramers-Kronig relations 459
26.4 Noisy kerneis 463
26.5 Deconvolution 465
27 Model selection 470
27.1 Inelastic electron scattering 473
27.2 Signal-background Separation 474
27.3 Spectral line broadening 478
27.4 Adaptive choice of pivots 481
27.5 Mass spectrometry 484
Contents ix
28 Bayesian experimental design 491
28.1 Overview of the Bayesian approach 491
28.2 Optimality criteria and Utility functions 492
28.3 Examples 493
28.4 N-step-ahead designs 504
28.5 Experimental design: Perspective 504
PART VI PROBABILISTIC NUMERICAL TECHNIQUES 507
29 Numerical Integration 509
29.1 The deterministic approach 509
29.2 Monte Carlo Integration 515
29.3 Beyond the Gaussian approximation 531
30 Monte Carlo methods 537
30.1 Simple sampling 537
30.2 Variance reduction 542
30.3 Markov chain Monte Carlo 544
30.4 Expectation value of the sample mean 555
30.5 Equilibration 560
30.6 Variance of the sample mean 561
30.7 Taming rugged PDFs by tempering 564
30.8 Evidence integral and partition function 568
31 Nested sampling 572
31.1 Motivation 572
31.2 The theory behind nested sampling 579
31.3 Application to the classical ideal gas 584
31.4 Statistical uncertainty 592
31.5 Concluding remarks 594
Appendix A Mathematical compendium 595
Appendix B Selected proofs and derivations 611
AppendixC Symbols and notation 619
References 620
Index 631
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| spelling | Linden, Wolfgang von der Verfasser aut Bayesian probability theory applications in the physical sciences Wolfgang von der Linden ; Volker Dose ; Udo von Toussaint Cambridge Cambridge University Press 2014 IX, 637 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Bayes-Verfahren (DE-588)4204326-8 gnd rswk-swf Bayes-Verfahren (DE-588)4204326-8 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Dose, Volker 1940- Verfasser (DE-588)126114730 aut Toussaint, Udo von 1970- Verfasser (DE-588)122400186 aut DE-601 pdf/application http://www.gbv.de/dms/bowker/toc/9781107035904.pdf Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027462066&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4079013-7 (DE-588)4204326-8 |
| title | Bayesian probability theory applications in the physical sciences |
| title_auth | Bayesian probability theory applications in the physical sciences |
| title_exact_search | Bayesian probability theory applications in the physical sciences |
| title_full | Bayesian probability theory applications in the physical sciences Wolfgang von der Linden ; Volker Dose ; Udo von Toussaint |
| title_fullStr | Bayesian probability theory applications in the physical sciences Wolfgang von der Linden ; Volker Dose ; Udo von Toussaint |
| title_full_unstemmed | Bayesian probability theory applications in the physical sciences Wolfgang von der Linden ; Volker Dose ; Udo von Toussaint |
| title_short | Bayesian probability theory |
| title_sort | bayesian probability theory applications in the physical sciences |
| title_sub | applications in the physical sciences |
| topic | Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Bayes-Verfahren (DE-588)4204326-8 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Bayes-Verfahren |
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