Contents
Preface XV
1. Plan of the Book 1
1.1. Outline of the Contents
1 1.2. Terminology and Notation 7
1.3. Biographies 8
PART I DIRECT PROBABILITY, 1750-1805
2. Some Results and Tools in Probability Theory by Bernoulli, de Moivre, and Laplace 11
2.1. The Discrete Equiprobability Model 11
2.2. The Theorems of James and Nicholas Bernoulli, 1713 13
2.3. The Normal Distribution as Approximation to the Binomial. De Moivre's Theorem, 1733, and Its Modifications by Lagrange, 1776, and Laplace, 1812 17
2.4. Laplace's Analytical Probability Theory 25
3. The Distribution of the Arithmetic Mean, 1756-1781 33
3.1. The Measurement Error Model 33
3.2. The Distribution of the Sum of the Number of Points by n Throws of a Die by Montmort and de Moivre 34
3.3. The Mean of Triangularly Distributed Errors. Simpson, 1756-1757 35
3.4. The Mean of Multinomially and Continuously Distributed Errors, and the Asymptotic Normality of the Multinomial. Lagrange, 1776 40
3.5. The Mean of Continuous Rectangularly Distributed Observations. Laplace, 1776 50
3.6. Laplace's Convolution Formula for the Distribution of a Sum, 1781 55
4. Chance or Design. Tests of Significance 65
4.1. Moral Impossibility and Statistical Significance 65
4.2. Daniel Bernoulli's Test for the Random Distribution of the Inclinations of the Planetary Orbits, 1735 68
4.3. John Michell's Test for the Random Distribution of the Positions of the Fixed Stars, 1767 70
4.4. Laplace's Test of Significance for the Mean Inclination, 1776 and 1812 74
5. Theory of Errors and Methods of Estimation 79
5.1. Theory of Errors and the Method of Maximum Likelihood by Lambert, 1760 and 1765 79
5.2. Theory of Errors and the Method of Maximum Likelihood by Daniel Bernoulli, 1778 83
5.3. Methods of Estimation by Laplace before 1805 87
6. Fitting of Equations to Data, 1750-1805 91
6.1. The Multiparameter Measurement Error Model 91
6.2. The Method of Averages by Tobias Mayer, 1750 94
6.3. The Method of Least Absolute Deviations by Boscovich, 1757 and 1760 97
6.4. Numerical and Graphical Curve Fitting by Lambert, 1765 and 1772 103
6.5. Laplace's Generalization of Mayer's Method, 1787 107
6.6. Minimizing the Largest Absolute Residual. Laplace, 1786, 1793, and 1799 108
6.7. Laplace's Modification of Boscovich's Method, 1799 112
6.8. Laplace's Determination of the Standard Meter, 1799 116
6.9. Legendre's Method of Least Squares, 1805 118
PART II INVERSE PROBABILITY BY BAYES AND LAPLACE, WITH COMMENTS ON LATER DEVELOPMENTS
7. Induction and Probability: The Philosophical Background 125
7.1. Newton's Inductive-Deductive Method 125
7.2. Hume's Ideas on Induction and Probability, 1739 126
7.3. Hartley on Direct and Inverse Probability, 1749 129
8. Bayes, Price, and the Essay, 1764-1765 133
8.1. Lives of Bayes and Price 133
8.2. Bayes's Probability Theory 136
8.3. The Posterior Distribution of the Probability of Success 138
8.4. Bayes's Scholium and His Conclusion 142
8.5. Price's Commentary 145
8.6. Evaluations of the Beta Probability Integral by
Bayes and Price 147
9. Equiprobability, Equipossibility, and Inverse Probability 155
9.1. Bernoulli's Concepts of Probability, 1713 155
9.2. Laplace's Definitions of Equiprobability and Equipossibility, 1774 and 1776 157
9.3. Laplace's Principle of Inverse Probability, 1774 159
9.4. Laplace's Proofs of Bayes's Theorem, 1781 and 1786 164
10. Laplace's Applications of the Principle of Inverse
Probability in 1774 167
10.1. Introduction 167
10.2. Testing a Simple Hypothesis against a Simple Alternative 167
10.3. Estimation and Prediction from a Binomial Sample 169
10.4. A Principle of Estimation and Its Application to Estimate the Location Parameter in the Measurement Error Model 171
10.5. Laplace's Two Error Distributions 176
10.6. The Posterior Median Equals the Arithmetic Mean for a Uniform Error Distribution, 1781 180
10.7. The Posterior Median for Multinomially Distributed Errors and the Rule of Succession, 1781 181
11. Laplace's General Theory of Inverse Probability 185
11.1. The Memoirs from 1781 and 1786 185
11.2. The Discrete Version of Laplace's Theory 185
11.3. The Continuous Version of Laplace's Theory 188
12. The Equiprobability Model and the Inverse Probability Model for Games of Chance 191
12.1. Theoretical and Empirical Analyses of Games of Chance 191
12.2. The Binomial Case Illustrated by Coin Tossings 192
12.3. A Solution of the Problem of Points for Unknown Probability of Success 196
12.4. The Multinomial Case Illustrated by Dice Throwing 197
12.5. Poisson's Analysis of Buffon's Coin-Tossing Data 198
12.6. Pearson and Fisher's Analyses of Weldon's Dice-Throwing Data 200
12.7. Some Modem Uses of the Equiprobability Model 201
13. Laplace's Methods of Asymptotic Expansion, 1781 and 1785 203
13.1. Motivation and Some General Remarks 203
13.2. Laplace's Expansions of the Normal Probability Integral 206
13.3. The Tail Probability Expansion 210
13.4. The Expansion about the Mode 212
13.5. Two Related Expansions from the 1960s 216
13.6. Expansions of Multiple Integrals 218
13.7. Asymptotic Expansion of the Tail Probability of a Discrete Distribution 220
13.8. Laplace Transforms 222
14. Laplace's Analysis of Binomially Distributed Observations 229
14.1. Notation 229
14.2. Background for the Problem and the Data 230
14.3. A Test for the Hypothesis 6 _< r Against 0 > r Based on the Tail Probability Expansion, 1781 232
14.4. A Test for the Hypothesis 0:< r Against 0 > r Based on the Normal Probability Expansion, 1786 234
14.5. Tests for the Hypothesis 02< 01 Against 02 > 01, 1781, 1786, and 1812 235
14.6. Looking for Assignable Causes 240
14.7. The Posterior Distribution of 0 Based on Compound Events, 1812 242
14.8. Commentaries 245
15. Laplace's Theory of Statistical Prediction 249
15.1. The Prediction Formula 249
15.2. Predicting the Outcome of a Second Binomial Sample from the Outcome of the First 249
15.3. Laplace's Rule of Succession 256
15.4. Theory of Prediction for a Finite Population. Prevost and Lhuilier, 1799 262
15.5. Laplace's Asymptotic Theory of Statistical Prediction, 1786 264
15.6. Notes on the History of the Indifference Principle and the Rule of Succession from Laplace to Jeffreys (1948) 268
16. Laplace's Sample Survey of the Population of France and the Distribution of the Ratio Estimator 283
16.1. The Ratio Estimator 283 16.2. Distribution of the Ratio Estimator, 1786 284
16.3. Sample Survey of the French Population in 1802 286
16.4. From Laplace to Bowley (1926), Pearson (1928), and Neyman (1934) 289
PART III THE NORMAL DISTRIBUTION, THE METHOD OF LEAST SQUARES, AND THE CENTRAL LIMIT THEOREM. GAUSS AND LAPLACE, 1809-1828
17. Early History of the Central Limit Theorem, 1810-1853 303
17.1. The Characteristic Function and the Inversion Formula for a Discrete Distribution by Laplace, 1785 303
17.2. Laplace's Central Limit Theorem, 1810 and 1812 307
17.3. Poisson's Proofs, 1824, 1829, and 1837 317
17.4. Bessel's Proof, 1838 327
17.5. Cauchy's Proofs, 1853 329
17.6. Ellis's Proof, 1844 333
17.7. Notes on Later Developments 335
17.8. Laplace's Diffusion Model, 1811 337
17.9. Gram-Charlier and Edgeworth Expansions 344
18. Derivations of the Normal Distribution as a Law of Error 351
18.1. Gauss's Derivation of the Normal Distribution and the Method of Least Squares, 1809 351
18.2. Laplace's Large-Sample Justification of the Method of Least Squares and His Criticism of Gauss, 1810 357
18.3. Bessel's Comparison of Empirical Error Distributions with the Normal Distribution, 1818 360
18.4. The Hypothesis of Elementary Errors by Hagen, 1837, and Bessel, 1838 365
18.5. Derivations by Adrain, 1808, Herschel, 1850, and Maxwell, 1860 368
18.6. Generalizations of Gauss's Proof: The Exponential Family of Distributions 373
18.7. Notes and References 380
19. Gauss's Linear Normal Model and the Method of Least Squares, 1809 and 1811 381
19.1. The Linear Normal Model 381
19.2. Gauss's Method of Solving the Normal Equations 383
19.3. The Posterior Distribution of the Parameters 386
19.4. Gauss's Remarks on Other Methods of Estimation 393
19.5. The Priority Dispute between Legendre and Gauss 394
20. Laplace's Large-Sample Theory of Linear Estimation, 1811-1827  397
20.1. Main Ideas in Laplace's Theory of Linear Estimation, 1811-1812  397
20.2. Notation  398
20.3. The Best Linear Asymptotically Normal Estimate for One Parameter, 1811  399
20.4. Asymptotic Normality of Sums of Powers of the Absolute Errors, 1812 401
20.5. The Multivariate Normal as the Limiting Distribution of Linear Forms of Errors, 1811 402
20.6. The Best Linear Asymptotically Normal Estimates for Two Parameters, 1811 405
20.7. Laplace's Orthogonalization of the Equations of Condition and the Asymptotic Distribution of the Best Linear Estimates in the Multiparameter Model, 1816 410
20.8. The Posterior Distribution of the Mean and the Squared Precision for Normally Distributed Observations, 1818 and 1820 418
20.9. Application in Geodesy and the Propagation of Error, 1818 and 1820 424
20.10. Linear Estimation with Several Independent Sources of Error, 1820 430
20.11. Tides of the Sea and the Atmosphere, 1797-1827 431 20.12. Asymptotic Efficiency of Some Methods of Estimation, 1818 444
20.13. Asymptotic Equivalence of Statistical Inference byDirect and Inverse Probability 452
21. Gauss's Theory of Linear Unbiased Minimum Variance Estimation, 1823-1828 455
21.1. Asymptotic Relative Efficiency of Some Estimates of the Standard Deviation in the Normal Distribution, 1816 455
21.2. Expectation, Variance, and Covariance of Functions of Random Variables, 1823 459
21.3. Gauss's Lower Bound for the Concentration of the Probability Mass in a Unimodal Distribution, 1823 462
21.4. Gauss's Theory of Linear Minimum Variance Estimation, 1821 and 1823 465
21.5. The Theorem on the Linear Unbiased Minimum Variance Estimate, 1823 467
21.6. The Best Estimate of a Linear Function of the Parameters, 1823 476
21.7. The Unbiased Estimate of a2 and Its Variance, 1823 477
21.8. Recursive Updating of the Estimates by an Additional Observation, 1823 480
21.9. Estimation under Linear Constraints, 1828 484
21.10. A Review 488
PART IV SELECTED TOPICS IN ESTIMATION THEORY, 1830-1930
22. On Error and Estimation Theory, 1830-1890 493
22.1. Bibliographies on the Method of Least Squares 493
22.2. State of Estimation Theory around 1830 494
22.3. Discussions on the Method of Least Squares and Some Alternatives 496
23. Bienaymé's Proof of the Multivariate Central Limit Theorem and His Defense of Laplace's Theory of Linear Estimation, 1852 and 1853 501
23.1. The Multivariate Central Limit Theorem, 1852 501
23.2. Bravais's Confidence Ellipsoids, 1846 504
23.3. Bienaymé's Confidence Ellipsoids and the x2 Distribution, 1852 506
23.4. Bienaymé's Criticism of Gauss, 1853 509
23.5. The Bienaymé Inequality, 1853 510
24. Cauchy's Method for Determining the Number of Terms To Be Included in the Linear Model and for Estimating the Parameters, 1835-1853 511
24.1. The Problem 511
24.2. Solving the Problem by Means of the Instrumental Variable +1, 1835 512
24.3. Cauchy's Two-Factor Multiplicative Model, 1835 516
24.4. The Cauchy-Bienaymé Dispute on the Validity of the Method of Least Squares, 1853 520
25. Orthogonalization and Polynomial Regression 523
25.1. Orthogonal Polynomials Derived by Laplacean Orthogonalization 523
25.2. Chebyshev's Orthogonal Polynomials, Least Squares, and Continued Fractions, 1855 and 1859 525
25.3. Chebyshev's Orthogonal Polynomials for Equidistant Arguments, 1864 and 1875 535
25.4. Gram's Derivation of Orthogonal Functions by the Method of Least Squares, 1879, 1883, and 1915 540
25.5. Thiele's Free Functions and His Orthogonalization of the Linear Model, 1889, 1897, and 1903 550
25.6. Schmidt's Orthogonalization Process, 1907 and 1908 556
25.7. Notes on the Literature after 1920 on Least Squares Approximation by Orthogonal Polynomials with Equidistant Arguments 557
26. Statistical Laws in the Social and Biological Sciences. Poisson, Quetelet, and Galton, 1830-1890 567
26.1. Probability Theory in the Social Sciences by Condorcet and Laplace 567
26.2. Poisson, Bienaymé, and Cournot on the Law of Large Numbers and Its Applications, 1830-1843 571
26.3. Quetelet on the Average Man, 1835, and on the Variation around the Average, 1846 586
26.4. Galton on Heredity, Regression, and Correlation, 1869-1890 599
26.5. Notes on the Early History of Regression and Correlation, 1889-1907 616
27. Sampling Distributions under Normality 633
27.1. The Helmert Distribution, 1876, and Its Generalization to the Linear Model by Fisher, 1922 633
27.2. The Distribution of the Mean Deviation by Helmert, 1876, and by Fisher, 1920 641
27.3. Thiele's Method of Estimation and the Canonical Form of the Linear Normal Model, 1889 and 1903 645
27.4. Karl Pearson's Chi-Squared Test of Goodnes of Fit, 1900, and Fisher's Amendment, 1924 648
27.5. "Student's" t Distribution by Gosset, 1908 664
27.6. Studentization, the F Distribution, and the Analysis of Variance by Fisher, 1922-1925 669
27.7. The Distribution of the Correlation Coefficient, 1915, the Partial Correlation Coefficient, 1924, the Multiple Correlation Coefficient, 1928, and the Noncentral x2 and F Distributions, 1928, by Fisher 675
28. Fisher's Theory of Estimation, 1912-1935, and His Immediate Precursors 693
28.1. Notation 693
28.2. On the Probable Errors of Frequency Constants by Pearson and Filon, 1898 695
28.3. On the Probable Errors of Frequency Constants by Edgeworth, 1908 and 1909 697
28.4. On an Absolute Criterion for Fitting Frequency Curves by Fisher, 1912 707
28.5. The Parametric Statistical Model, Sufficiency, and the Method of Maximum Likelihood. Fisher, 1922 713
28.6. Efficiency and Loss of Information. Fisher, 1925 720
28.7. Sufficiency, the Factorization Criterion, and the Exponential Family. Fisher, 1934 727
28.8. Loss of Information by Using the Maximum Likelihood Estimate and Recovery of Information by Means ofAncillary Statistics. Fisher, 1925 729
28.9. Examples of Ancillarity and Conditional Inference. Fisher, 1934 and 1935 732
28.10. The Discussion of Fisher's 1935 Paper 733
28.11. A Note on Fisher and His Books on Statistics 734
References 741
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Index 777