Verfügbarkeit: Probability theory

Probability theory: the logic of science

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Bibliographische Detailangaben
1. Verfasser:	Jaynes, Edwin T. 1922-1998 (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Cambridge [u.a.] Cambridge Univ. Press 2010
Ausgabe:	7. print.
Schlagworte:	Wahrscheinlichkeitsrechnung Wahrscheinlichkeitstheorie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Literaturverz. S. 683 - 720. - Hier auch später erschienene, unveränderte Nachdrucke
Beschreibung:	XXIX, 727 S. graph. Darst.
ISBN:	9780521592710 0521592712

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Datensatz im Suchindex

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adam_text	Contents Editor s foreword page xvii Preface xix Part I Principles and elementary applications 1 Plausible reasoning 3 1.1 Deductive and plausible reasoning 3 1.2 Analogies with physical theories 6 1.3 The thinking computer 7 1.4 Introducing the robot 8 1.5 Boolean algebra 9 1.6 Adequate sets of operations 12 1.7 The basic desiderata 17 1.8 Comments 19 1.8.1 Common language vs. formal logic 21 1.8.2 Nitpicking 23 2 The quantitative rules 24 2.1 The product rule 24 2.2 The sum rule 30 2.3 Qualitative properties 35 2.4 Numerical values 37 2.5 Notation and finite-sets policy 43 2.6 Comments 44 2.6.1 Subjective vs. Objective 44 2.6.2 Gödel s theorem 45 2.6.3 Venn diagrams 47 2.6.4 The Kolmogorov axioms 49 3 Elementary sampling theory 51 3.1 Sampling without replacement 52 3.2 Logic vs. propensity 60 3.3 Reasoning from less precise information 64 3.4 Expectations 66 3.5 Other forms and extensions 68 viii Contents 3.6 Probability as a mathematical tool 3.7 The binomial distribution 3.8 Sampling with replacement 3.8.1 Digression: a sermon on reality vs. models 3.9 Correction for correlations 3.10 Simplification 3.11 Comments 3.11.1 A look ahead 4 Elementary hypothesis testing 4.1 Prior probabilities 4.2 Testing binary hypotheses with binary data 4.3 Nonextensibility beyond the binary case 4.4 Multiple hypothesis testing 4.4.1 Digression on another derivation 4.5 Continuous probability distribution functions 4.6 Testing an infinite number of hypotheses 4.6.1 Historical digression 4.7 Simple and compound (or composite) hypotheses 4.8 Comments 4.8.1 Etymology 4.8.2 What have we accomplished? 5 Queer uses for probability theory 5.1 Extrasensory perception 5.2 Mrs Stewart s telepathic powers 5.2.1 Digression on the normal approximation 5.2.2 Back to Mrs Stewart 5.3 Converging and diverging views 5.4 Visual perception - evolution into Bayesianity? 5.5 The discovery of Neptune 5.5.1 Digression on alternative hypotheses 5.5.2 Back to Newton 5.6 Horse racing and weather forecasting 5.6.1 Discussion 5.7 Paradoxes of intuition 5.8 Bayesian jurisprudence 5.9 Comments 5.9.1 What is queer? 6 Elementary parameter estimation 6.1 Inversion of the urn distributions 6.2 Both N and R unknown 6.3 Uniform prior 6.4 Predictive distributions Contents ix 6.5 Truncated uniform priors 157 6.6 A concave prior 158 6.7 The binomial monkey prior 160 6.8 Metamorphosis into continuous parameter estimation 163 6.9 Estimation with a binomial sampling distribution 163 6.9.1 Digression on optional stopping 166 6.10 Compound estimation problems 167 6.11 A simple Bayesian estimate: quantitative prior information 168 6.11.1 From posterior distribution function to estimate 172 6.12 Effects of qualitative prior information 177 6.13 Choice of a prior 178 6.14 On with the calculation ! 179 6.15 The Jeffreys prior 181 6.16 The point of it all 183 6.17 Interval estimation 186 6.18 Calculation of variance 186 6.19 Generalization and asymptotic forms 188 6.20 Rectangular sampling distribution 190 6.21 Small samples 192 6.22 Mathematical trickery 193 6.23 Comments 195 The central, Gaussian or normal distribution 198 7.1 The gravitating phenomenon 199 7.2 The Herschel-Maxwell derivation 200 7.3 The Gauss derivation 202 7.4 Historical importance of Gauss s result 203 7.5 The Landon derivation 205 7.6 Why the ubiquitous use of Gaussian distributions? 207 7.7 Why the ubiquitous success? 210 7.8 What estimator should we use? 211 7.9 Error cancellation 213 7.10 The near irrelevance of sampling frequency distributions 215 7.11 The remarkable efficiency of information transfer 216 7.12 Other sampling distributions 218 7.13 Nuisance parameters as safety devices 219 7.14 More general properties 220 7.15 Convolution of Gaussians 221 7.16 The central limit theorem 222 7.17 Accuracy of computations 224 7.18 Galton s discovery 227 7.19 Population dynamics and Darwinian evolution 229 7.20 Evolution of humming-birds and flowers 231 Contents 7.21 Application to economics 233 7.22 The great inequality of Jupiter and Saturn 234 7.23 Resolution of distributions into Gaussians 235 7.24 Hermite polynomial solutions 236 7.25 Fourier transform relations 238 7.26 There is hope after all 239 7.27 Comments 240 7.27.1 Terminology again 240 Sufficiency, ancillarity, and all that 243 8.1 Sufficiency 243 8.2 Fisher sufficiency 245 8.2.1 Examples 246 8.2.2 The Blackwell-Rao theorem 247 8.3 Generalized sufficiency 248 8.4 Sufficiency plus nuisance parameters 249 8.5 The likelihood principle 250 8.6 Ancillarity 253 8.7 Generalized ancillary information 254 8.8 Asymptotic likelihood: Fisher information 256 8.9 Combining evidence from different sources 257 8.10 Pooling the data 260 8.10.1 Fine-grained propositions 261 8.11 Sam s broken thermometer 262 8.12 Comments 264 8.12.1 The fallacy of sample re-use 264 8.12.2 A folk theorem 266 8.12.3 Effect of prior information 267 8.12.4 Clever tricks and gamesmanship 267 Repetitive experiments: probability and frequency 270 9.1 Physical experiments 271 9.2 The poorly informed robot 274 9.3 Induction 276 9.4 Are there general inductive rules? 277 9.5 Multiplicity factors 280 9.6 Partition function algorithms 281 9.6.1 Solution by inspection 282 9.7 Entropy algorithms 285 9.8 Another way of looking at it 289 9.9 Entropy maximization 290 9.10 Probability and frequency 292 9.11 Significance tests 293 9.11.1 Implied alternatives 296 Contents 10 Part II Ad 11 12 9.12 Comparison of psi and chi-squared 300 9.13 The chi-squared test 302 9.14 Generalization 304 9.15 Halley s mortality table 305 9.16 Comments 310 9.16.1 The irrationalists 310 9.16.2 Superstitions 312 Physics of random experiments 314 10.1 An interesting correlation 314 10.2 Historical background 315 10.3 How to cheat at coin and die tossing 317 10.3.1 Experimental evidence 320 10.4 Bridge hands 321 10.5 General random experiments 324 10.6 Induction revisited 326 10.7 But what about quantum theory? 327 10.8 Mechanics under the clouds 329 10.9 More on coins and symmetry 331 10.10 Independence of tosses 335 10.11 The arrogance of the uninformed 338 meed ; applications Discrete prior probabilities: the entropy principle 343 11.1 A new kind of prior information 343 11.2 Minimum Σ pf 345 11.3 Entropy: Shannon s theorem 346 11.4 The Wallis derivation 351 11.5 An example 354 11.6 Generalization: a more rigorous proof 355 11.7 Formal properties of maximum entropy distributions 358 11.8 Conceptual problems - frequency correspondence 365 11.9 Comments 370 Ignorance priors and transformation groups 372 12.1 What are we trying to do? 372 12.2 Ignorance priors 374 12.3 Continuous distributions 374 12.4 Transformation groups 378 12.4.1 Location and scale parameters 378 12.4.2 A Poisson rate 382 12.4.3 Unknown probability for success 382 12 A A Bertrand s problem 386 12.5 Comments 394 Contents 13 Decision theory, historical background 397 13.1 Inference vs. decision 397 13.2 Daniel Bernoulli s suggestion 398 13.3 The rationale of insurance 400 13.4 Entropy and utility 402 13.5 The honest weatherman 402 13.6 Reactions to Daniel Bernoulli and Laplace 404 13.7 Wald s decision theory 406 13.8 Parameter estimation for minimum loss 410 13.9 Reformulation of the problem 412 13.10 Effect of varying loss functions 415 13.11 General decision theory 417 13.12 Comments 418 13.12.1 Objectivity of decision theory 418 13.12.2 Loss functions in human society 421 13.12.3 A new look at the Jeffreys prior 423 13.12.4 Decision theory is not fundamental 423 13.12.5 Another dimension? 424 14 Simple applications of decision theory 426 14.1 Definitions and preliminaries 426 14.2 Sufficiency and information 428 14.3 Loss functions and criteria of optimum performance 430 14.4 A discrete example 432 14.5 How would our robot do it? 437 14.6 Historical remarks 438 14.6.1 The classical matched filter 439 14.7 The widget problem 440 14.7.1 Solution for Stage 2 443 14.7.2 Solution for Stage 3 445 14.7.3 Solution for Stage 4 449 14.8 Comments 450 15 Paradoxes of probability theory 451 15.1 How do paradoxes survive and grow? 451 15.2 Summing a series the easy way 452 15.3 Nonconglomerability 453 15.4 The tumbling tetrahedra 456 15.5 Solution for a finite number of tosses 459 15.6 Finite vs. countable additivity 464 15.7 The Borel-Kolmogorov paradox 467 15.8 The marginalization paradox 470 15.8.1 On to greater disasters 474 Contents xiii 15.9 Discussion 478 15.9.1 The DSZ Example #5 480 15.9.2 Summary 483 15.10 A useful result after all? 484 15.11 How to mass-produce paradoxes 485 15.12 Comments 486 16 Orthodox methods: historical background 490 16.1 The early problems 490 16.2 Sociology of orthodox statistics 492 16.3 Ronald Fisher, Harold Jeffreys, and Jerzy Neyman 493 16.4 Pre-data and post-data considerations 499 16.5 The sampling distribution for an estimator 500 16.6 Pro-causal and anti-causal bias 503 16.7 What is real, the probability or the phenomenon? 505 16.8 Comments 506 16.8.1 Communication difficulties 507 17 Principles and pathology of orthodox statistics 509 17.1 Information loss 510 17.2 Unbiased estimators 511 17.3 Pathology of an unbiased estimate 516 17.4 The fundamental inequality of the sampling variance 518 17.5 Periodicity: the weather in Central Park 520 17.5.1 The folly of pre-filtering data 521 17.6 A Bayesian analysis 527 17.7 The folly of randomization 531 17.8 Fisher: common sense at Rothamsted 532 17.8.1 The Bayesian safety device 532 17.9 Missing data 533 17.10 Trend and seasonality in time series 534 17.10.1 Orthodox methods 535 17.10.2 The Bayesian method 536 17.10.3 Comparison of Bayesian and orthodox estimates 540 17.10.4 An improved orthodox estimate 541 17.10.5 The orthodox criterion of performance 544 17.11 The general case 545 17.12 Comments 550 18 The Ap distribution and rule of succession 553 18.1 Memory storage for old robots 553 18.2 Relevance 555 18.3 A surprising consequence 557 18.4 Outer and inner robots 559 Contents 18.5 An application 561 18.6 Laplace s rule of succession 563 18.7 Jeffreys objection 566 18.8 Bass or carp? 567 18.9 So where does this leave the rule? 568 18.10 Generalization 568 18.11 Confirmation and weight of evidence 571 18.11.1 Is indifference based on knowledge or ignorance? 573 18.12 Carnap s inductive methods 574 18.13 Probability and frequency in exchangeable sequences 576 18.14 Prediction of frequencies 576 18.15 One-dimensional neutron multiplication 579 18.15.1 The frequentist solution 579 18.15.2 The Laplace solution 581 18.16 The de Finetti theorem 586 18.17 Comments 588 19 Physical measurements 589 19.1 Reduction of equations of condition 589 19.2 Reformulation as a decision problem 592 19.2.1 Sermon on Gaussian error distributions 592 19.3 The underdetermined case: К is singular 594 19.4 The overdetermined case: К can be made nonsingular 595 19.5 Numerical evaluation of the result 596 19.6 Accuracy of the estimates 597 19.7 Comments 599 19.7.1 A paradox 599 20 Model comparison 601 20.1 Formulation of the problem 602 20.2 The fair judge and the cruel realist 603 20.2.1 Parameters known in advance 604 20.2.2 Parameters unknown 604 20.3 But where is the idea of simplicity? 605 20.4 An example: linear response models 607 20.4.1 Digression: the old sermon still another time 608 20.5 Comments 613 20.5.1 Final causes 614 21 Outliers and robustness 615 21.1 The experimenter s dilemma 615 21.2 Robustness 617 21.3 The two-model model 619 21.4 Exchangeable selection 620 21.5 The general Bayesian solution 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.I 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 С Convolutions and cumulants С. 1 Relation of cumulants and moments C.2 Examples References Bibliography Author index Subject index 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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title_full	Probability theory the logic of science E. T. Jaynes. Ed. by G. Larry Bretthorst
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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
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