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 minimumloss 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
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
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: K is singular 594
19.4 The overdetermined case: K 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
21.6 Pure outliers 624
21.7 One receding datum 625
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