Probability theory: the logic of science
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2003
|
| Ausgabe: | 1. publ., repr. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XXIX, 727 S. |
| ISBN: | 0521592712 |
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Datensatz im Suchindex
| _version_ | 1804132789706031104 |
|---|---|
| adam_text | Contents Part I Editor’s foreword Preface Principles and elementary applications 1 Plausible reasoning 1.1 Deductive and plausible reasoning 1.2 Analogies with physical theories 1.3 The thinking computer 1.4 Introducing the robot 1.5 Boolean algebra 1.6 Adequate sets of operations 1.7 The basic desiderata 1.8 Comments 1.8.1 Common language vs. formal logic 1.8.2 Nitpicking 2 The quantitative rules 2.1 The product rule 2.2 The sum rule 2.3 Qualitative properties 2.4 Numerical values 2.5 Notation and finite-sets policy 2.6 Comments 2.6.1 ‘Subjective’ vs. ‘objective’ 2.6.2 Godel’s theorem 2.6.3 Venn diagrams 2.6.4 The ‘Kolmogorov axioms’ 3 Elementary sampling theory 3.1 Sampling without replacement 3.2 Logic vs. propensity 3.3 Reasoning from less precise information 3.4 Expectations 3.5 Other forms and extensions page xvii xix 3 3 6 7 8 9 12 17 19 21 23 24 24 ЗО 35 37 43 44 44 45 47 49 51 52 60 64 66 68
Contents viii Probability as a mathematical tool The binomial distribution Sampling with replacement 3.8.1 Digression: a sermon on reality vs. models Correction for correlations 3.9 3.10 Simplification 3.11 Comments 3.11.1 A look ahead Elementary hypothesis testing Prior probabilities 4.1 Testing binary hypotheses with binary data 4.2 Nonextensibility beyond the binary case 4.3 4.4 Multiple hypothesis testing 4.4.1 Digression on another derivation 4.5 Continuous probability distribution functions Testing an infinite number of hypotheses 4.6 4.6.1 Historical digression Simple and compound (or composite) hypotheses 4.7 Comments 4.8 4.8.1 Etymology 4.8.2 What have we accomplished? Queer uses for probability theory Extrasensory perception 5.1 Mrs Stewart’s telepathic powers 5.2 5.2.1 Digression on the normal approximation 5.2.2 Back to Mrs Stewart Converging and diverging views 5.3 5.4 Visual perception - evolution into Bayesianity? The discovery of Neptune 5.5 5.5.1 Digression on alternative hypotheses 5.5.2 Back to Newton Horse racing and weather forecasting 5.6 5.6.1 Discussion Paradoxes of intuition 5.7 5.8 Bayesian jurisprudence 5.9 Comments 5.9.1 What is queer? Elementary parameter estimation Inversion of the urn distributions 6.1 Both N and R unknown 6.2 Uniform prior 6.3 6.4 Predictive distributions 3.6 3.7 3.8 68 69 72 73 75 81 82 84 86 87 90 97 98 101 107 109 112 115 116 116 117 119 119 120 122 122 126 132 133 135 137 140 142 143 144 146 148 149 149 150 152 154
Contents Truncated uniform priors A concave prior The binomial monkey prior Metamorphosis into continuous parameter estimation Estimation with a binomial sampling distribution 6.9.1 Digression on optional stopping 6.10 Compound estimation problems 6.11 A simple Bayesian estimate: quantitative prior information 6.11.1 From posterior distribution function to estimate Effects of qualitative prior information 6.12 6.13 Choice of a prior 6.14 On with the calculation! 6.15 The Jeffreys prior 6.16 The point of it all 6.17 Interval estimation 6.18 Calculation of variance 6.19 Generalization and asymptotic forms 6.20 Rectangular sampling distribution 6.21 Small samples 6.22 Mathematical trickery 6.23 Comments The central, Gaussian or normal distribution The gravitating phenomenon 7.1 The Herschel-Maxwell derivation 7.2 The Gauss derivation 7.3 Historical importance of Gauss’s result 7.4 The Landon derivation 7.5 Why the ubiquitous use of Gaussian distributions? 7.6 Why the ubiquitous success? 7.7 What estimator should we use? 7.8 Error cancellation 7.9 7.10 The near irrelevance of sampling frequency distributions 7.11 The remarkable efficiency of information transfer 7.12 Other sampling distributions 7.13 Nuisance parameters as safety devices 7.14 More general properties 7.15 Convolution of Gaussians 7.16 The central limit theorem 7.17 Accuracy of computations 7.18 Galton’s discovery 7.19 Population dynamics and Darwinian evolution 7.20 Evolution of humming-birds and flowers 6.5 6.6 6.7 6.8 6.9 ix 157 158 160 163 163 166 167 168 172 177 178 179 181 183 186 186 188 190 192 193 195
198 199 200 202 203 205 207 210 211 213 215 216 218 219 220 221 222 224 227 229 231
Contents 7.21 Application to economics 7.22 The great inequality of Jupiter and Saturn 7.23 Resolution of distributions into Gaussians 7.24 Hermite polynomial solutions 7.25 Fourier transform relations 7.26 There is hope after all 7.27 Comments 7.27.1 Terminology again Sufficiency, ancillarily, and all that 8.1 Sufficiency 8.2 Fisher sufficiency 8.2.1 Examples 8.2.2 The Blackwell-Rao theorem 8.3 Generalized sufficiency 8.4 Sufficiency plus nuisance parameters 8.5 The likelihood principle 8.6 Ancillarity 8.7 Generalized ancillary information 8.8 Asymptotic likelihood: Fisher information 8.9 Combining evidence from different sources 8.10 Pooling the data 8.10.1 Fine-grained propositions 8.11 Sam’s broken thermometer 8.12 Comments 8.12.1 The fallacy of sample re-use 8.12.2 A folk theorem 8.12.3 Effect of prior information 8.12.4 Clever tricks and gamesmanship Repetitive experiments: probability and frequency 9.1 Physical experiments 9.2 The poorly informed robot 9.3 Induction 9.4 Are there general inductive rules? 9.5 Multiplicity factors 9.6 Partition function algorithms 9.6.1 Solution by inspection 9.7 Entropy algorithms 9.8 Another way of looking at it 9.9 Entropy maximization 9.10 Probability and frequency 9.11 Significance tests 9.11.1 Implied alternatives 233 234 235 236 238 239 240 240 243 243 245 246 247 248 249 250 253 254 256 257 260 261 262 264 264 266 267 267 270 271 274 276 277 280 281 282 285 289 290 292 293 296
Contents Comparison of psi and chi-squared The chi-squared test Generalization Halley’s mortality table Comments 9.16.1 Theirrationalists 9.16.2 Superstitions 10 Physics of ‘random experiments’ An interesting correlation 10.1 Historical background 10.2 How to cheat at coin and die tossing 10.3 10.3.1 Experimental evidence Bridge hands 10.4 General random experiments 10.5 Induction revisited 10.6 But what about quantum theory? 10.7 Mechanics under the clouds 10.8 More on coins and symmetry 10.9 10.10 Independence of tosses 10.11 The arrogance of the uninformed Advanced applications 11 Discrete prior probabilities: the entropy principle A new kind of prior information 11.1 Minimum pj 11.2 Entropy: Shannon’s theorem 11.3 The Wallis derivation 11.4 An example 11.5 Generalization: a more rigorous proof 11.6 Formal properties of maximum entropy 11.7 distributions Conceptual problems - frequency correspondence 11.8 Comments 11.9 12 Ignorance priors and transformation groups What are we trying to do? 12.1 Ignorance priors 12.2 Continuous distributions 12.3 Transformation groups 12.4 12.4.1 Location and scale parameters 12.4.2 A Poisson rate 12.4.3 Unknown probability for success 12.4.4 Bertrand’s problem Comments 12.5 9.12 9.13 9.14 9.15 9.16 Part II xi 300 302 304 305 310 310 312 314 314 315 317 320 321 324 326 327 329 331 335 338 343 343 345 346 351 354 355 358 365 370 372 372 374 374 378 378 382 382 386 394
xii Contents Decision theory, historical background 13.1 Inference vs. decision 13.2 Daniel Bernoulli’s suggestion 13.3 The rationale of insurance 13.4 Entropy and utility 13.5 The honest weatherman 13.6 Reactions to Daniel Bernoulli and Laplace 13.7 Wald ’s decision theory 13.8 Parameter estimation for minimum loss 13.9 Reformulation of the problem 13.10 Effect of varying loss functions 13.11 General decision theory 13.12 Comments 13.12.1 Objectivity’ of decision theory 13.12.2 Loss functions in human society 13.12.3 A new look at the Jeffreys prior 13.12.4 Decision theory is not fundamental 13.12.5 Another dimension? 14 Simple applications of decision theory 14.1 Definitions and preliminaries 14.2 Sufficiency and information 14.3 Loss functions and criteria of optimum performance 14.4 A discrete example 14.5 How would our robot do it? 14.6 Historical remarks 14.6.1 The classical matched filter 14.7 The widget problem 14.7.1 Solution for Stage 2 14.7.2 Solution for Stage 3 14.7.3 Solution for Stage 4 14.8 Comments 15 Paradoxes of probability theory 15.1 How do paradoxes survive and grow? 15.2 Summing a series the easy way 15.3 Nonconglomerability 15.4 The tumbling tetrahedra 15.5 Solution for a finite number of tosses 15.6 Finite vs. countable additivity 15.7 The Borel-Kolmogorov paradox 15.8 The marginalization paradox 15.8.1 On to greater disasters 13 397 397 398 400 402 402 404 406 410 412 415 417 418 418 421 423 423 424 426 426 428 430 432 437 438 439 440 443 445 449 450 451 451 452 453 456 459 464 467 470 474
Contents Discussion 15.9.1 The DSZ Example #5 15.9.2 Summary 15.10 A useful result after all? 15.11 How to mass-produce paradoxes 15.12 Comments 16 Orthodox methods: historical background 16.1 The early problems 16.2 Sociology of orthodox statistics 16.3 Ronald Fisher, Harold Jeffreys, and JerzyNeyman 16.4 Pre-data and post-data considerations 16.5 The sampling distribution for an estimator 16.6 Pro-causal and anti-causal bias 16.7 What is real, the probability or the phenomenon? 16.8 Comments 16.8.1 Communication difficulties 17 Principles and pathology of orthodox statistics 17.1 Information loss 17.2 Unbiased estimators 17.3 Pathology of an unbiased estimate 17.4 The fundamental inequality of the sampling variance 17.5 Periodicity: the weather in Central Park 17.5.1 The folly of pre-filtering data 17.6 A Bayesian analysis 17.7 The folly of randomization 17.8 Fisher: common sense at Rothamsted 17.8.1 The Bayesian safety device 17.9 Missing data 17.10 Trend and seasonality in time series 17.10.1 Orthodox methods 17.10.2 The Bayesian method 17.10.3 Comparison of Bayesian and orthodox estimates 540 17.10.4 An improved orthodox estimate 17.10.5 The orthodox criterion of performance 17.11 The general case 17.12 Comments 18 The Ap distribution and rule of succession 18.1 Memory storage for old robots 18.2 Relevance 18.3 A surprising consequence 18.4 Outer and inner robots 15.9 xiii 478 480 483 484 485 486 490 490 492 493 499 500 503 505 506 507 509 510 511 516 518 520 521 527 531 532 532 533 534 535 536 541 544 545 550 553 553 555 557 559
Contents xiv 18.5 18.6 18.7 18.8 18.9 18.10 18.11 19 20 21 An application Laplace’s rule of succession Jeffreys’ objection Bass or carp? So where does this leave the rule? Generalization Confirmation and weight of evidence 18.11.1 Is indifference based on knowledge or ignorance? 18.12 Carnap’s inductive methods 18.13 Probability and frequency in exchangeable sequences 18.14 Prediction of frequencies 18.15 One-dimensional neutron multiplication 18.15.1 The frequentist solution 18.15.2 The Laplace solution 18.16 The de Finetti theorem 18.17 Comments Physical measurements 19.1 Reduction of equations of condition 19.2 Reformulation as a decision problem 19.2.1 Sermon on Gaussian error distributions 19.3 The underdetermined case: К is singular 19.4 The overdetermined case: К can be made nonsingular 19.5 Numerical evaluation of the result 19.6 Accuracy of the estimates 19.7 Comments 19.7.1 A paradox Model comparison 20.1 Formulation of the problem 20.2 The fair judge and the cruel realist 20.2.1 Parameters known in advance 20.2.2 Parameters unknown 20.3 But where is the idea of simplicity? 20.4 An example: linear response models 20.4.1 Digression: the old sermon still another time 20.5 Comments 20.5.1 Final causes Outliers and robustness 21.1 The experimenter’s dilemma 21.2 Robustness 21.3 The two-model model 21.4 Exchangeable selection 21.5 The general Bayesian solution 561 563 566 567 568 568 571 573 574 576 576 579 579 581 586 588 589 589 592 592 594 595 596 597 599 599 601 602 603 604 604 605 607 608 613 614 615 615 617 619 620 622
Contents 21.6 Pure outliers 21.7 One receding datum 22 Introduction to communication theory 22.1 Origins of the theory 22.2 The noiseless channel 22.3 The information source 22.4 Does the English language have statistical properties? 22.5 Optimum encoding: letter frequencies known 22.6 Better encoding from knowledge of digram frequencies 22.7 Relation to a stochastic model 22.8 The noisy channel Appendix A Other approaches to probability theory A. 1 The Kolmogorov system of probability A.2 The de Finetti system of probability A.3 Comparative probability A.4 Holdouts against universal comparability A.5 Speculations about lattice theories Appendix В Mathematical formalities and style B.l Notation and logical hierarchy B.2 Our ‘cautious approach’ policy B.3 Willy Feller on measure theory B.4 Kronecker vs. Weierstrasz B.5 What is a legitimate mathematical function? B.5.1 Delta-functions B.5.2 Nondifferentiable functions B.5.3 Bogus nondifferentiable functions B.6 Counting infinite sets? B.7 The Hausdorff sphere paradox and mathematical diseases B.8 What am I supposed to publish? B.9 Mathematical courtesy Appendix C Convolutions and cumulants С. 1 Relation of cumulants and moments C.2 Examples References Bibliography Author index Subject index XV 624 625 627 627 628 634 636 638 641 644 648 651 651 655 656 658 659 661 661 662 663 665 666 668 668 669 671 672 674 675 677 679 680 683 705 721 724
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 1. publ., repr. |
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| id | DE-604.BV019357068 |
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| indexdate | 2024-07-09T19:58:24Z |
| institution | BVB |
| isbn | 0521592712 |
| language | English |
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| spelling | Jaynes, Edwin T. 1922-1998 Verfasser (DE-588)123687519 aut Probability theory the logic of science E. T. Jaynes. Ed. by G. Larry Bretthorst 1. publ., repr. Cambridge [u.a.] Cambridge Univ. Press 2003 XXIX, 727 S. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Probabilités Probabilités ram théorie probabilité inriac Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 1\p DE-604 Bretthorst, G. Larry Sonstige oth Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012820928&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Jaynes, Edwin T. 1922-1998 Probability theory the logic of science Probabilités Probabilités ram théorie probabilité inriac Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4064324-4 (DE-588)4079013-7 |
| title | Probability theory the logic of science |
| title_auth | Probability theory the logic of science |
| title_exact_search | Probability theory the logic of science |
| title_full | Probability theory the logic of science E. T. Jaynes. Ed. by G. Larry Bretthorst |
| title_fullStr | Probability theory the logic of science E. T. Jaynes. Ed. by G. Larry Bretthorst |
| title_full_unstemmed | Probability theory the logic of science E. T. Jaynes. Ed. by G. Larry Bretthorst |
| title_short | Probability theory |
| title_sort | probability theory the logic of science |
| title_sub | the logic of science |
| topic | Probabilités Probabilités ram théorie probabilité inriac Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Probabilités théorie probabilité Wahrscheinlichkeitsrechnung Wahrscheinlichkeitstheorie |
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