Preface
1. Plan of the Book
1.1. Outline of the Contents
1.2. Terminology and Notation
1.3. Biographies
PART I DIRECT PROBABILITY, 1750-1805
2. Some Results and Tools in Probability Theory by Bernoulli, de Moivre, and Laplace
2.1. The Discrete Equiprobability Model
2.2. The Theorems of James and Nicholas Bernoulli, 1713
2.3. The Normal Distribution as Approximation to the Binomial. De Moivre’s Theorem, 1733, and Its Modifications by Lagrange, 1776, and Laplace, 1812
2.4. Laplace’s Analytical Probability Theory
3. The Distribution of the Arithmetic Mean, 1756-1781
3.1. The Measurement Error Model
3.2. The Distribution of the Sum of the Number of Points by n Throws of a Die by Montmort and de Moivre
3.3. The Mean of Triangularly Distributed Errors. Simpson, 1756-1757
3.4. The Mean of Multinomially and Continuously Distributed Errors, and the Asymptotic Normality of the Multinomial. Lagrange, 1776
3.5. The Mean of Continuous Rectangularly Distributed Observations. Laplace, 1776
3.6. Laplace’s Convolution Formula for the Distribution of a Sum, 1781
4. Chance or Design. Tests of Significance
4.1. Moral Impossibility and Statistical Significance
4.2. Daniel Bernoulli’s Test for the Random Distribution of the Inclinations of the Planetary Orbits, 1735
4.3. John Michell’s Test for the Random Distribution of the Positions of the Fixed Stars, 1767
4.4. Laplace’s Test of Significance for the Mean Inclination, 1776 and 1812
5. Theory of Errors and Methods of Estimation
5.1. Theory of Errors and the Method of Maximum Likelihood by Lambert, 1760 and 1765
5.2. Theory of Errors and the Method of Maximum Likelihood by Daniel Bernoulli, 1778
5.3. Methods of Estimation by Laplace before 1805
6. Fitting of Equations to Data, 1750-1805
6.1. The Multiparameter Measurement Error Model
6.2. The Method of Averages by Tobias Mayer, 1750
6.3. The Method of Least Absolute Deviations by Boscovich, 1757 and 1760
6.4. Numerical and Graphical Curve Fitting by Lambert, 1765 and 1772
6.5. Laplace’s Generalization of Mayer’s Method, 1787
6.6. Minimizing the Largest Absolute Residual. Laplace, 1786, 1793, and 1799
6.7. Laplace’s Modification of Boscovich’s Method, 1799
6.8. Laplace’s Determination of the Standard Meter, 1799
6.9. Legendre’s Method of Least Squares, 1805
PART II INVERSE PROBABILITY BY BAYES AND LAPLACE, WITH COMMENTS ON LATER DEVELOPMENTS
7. Induction and Probability: The Philosophical Background
7.1. Newton’s Inductive-Deductive Method
7.2. Hume’s Ideas on Induction and Probability, 1739
7.3. Hartley on Direct and Inverse Probability, 1749
8. Bayes, Price, and the Essay, 1764-1765
8.1. Lives of Bayes and Price
8.2. Bayes’s Probability Theory
8.3. The Posterior Distribution of the Probability of Success
8.4. Bayes’s Scholium and His Conclusion
8.5. Price’s Commentary
8.6. Evaluations of the Beta Probability Integral by Bayes and Price
9. Equiprobability, Equipossibility, and Inverse Probability
9.1. Bernoulli’s Concepts of Probability, 1713
9.2. Laplace’s Definitions of Equiprobability and Equipossibility, 1774 and 1776
9.3. Laplace’s Principle of Inverse Probability, 1774
9.4. Laplace’s Proofs of Bayes’s Theorem, 1781 and 1786
10. Laplace’s Applications of the Principle of Inverse Probability in 1774
10.1. Introduction
10.2. Testing a Simple Hypothesis against a Simple Alternative
10.3. Estimation and Prediction from a Binomial Sample
10.4. A Principle of Estimation and Its Application to Estimate the Location Parameter in the Measurement Error Model
10.5. Laplace’s Two Error Distributions
10.6. The Posterior Median Equals the Arithmetic Mean for a Uniform Error Distribution, 1781
10.7. The Posterior Median for Multinomially Distributed Errors and the Rule of Succession, 1781
11. Laplace’s General Theory of Inverse Probability
11.1. The Memoirs from 1781 and 1786
11.2. The Discrete Version of Laplace’s Theory
11.3. The Continuous Version of Laplace’s Theory
12. The Equiprobability Model and the Inverse Probability Model for Games of Chance
12.1. Theoretical and Empirical Analyses of Games of Chance
12.2. The Binomial Case Illustrated by Coin Tossings
12.3. A Solution of the Problem of Points for Unknown Probability of Success
12.4. The Multinomial Case Illustrated by Dice Throwing
12.5. Poisson’s Analysis of Buffon’s Coin-Tossing Data
12.6. Pearson and Fisher’s Analyses of Weldon’s Dice-Throwing Data
12.7. Some Modem Uses of the Equiprobability Model
13. Laplace’s Methods of Asymptotic Expansion, 1781 and 1785
13.1. Motivation and Some General Remarks
13.2. Laplace’s Expansions of the Normal Probability Integral
13.3. The Tail Probability Expansion
13.4. The Expansion about the Mode
13.5. Two Related Expansions from the 1960s
13.6. Expansions of Multiple Integrals
13.7. Asymptotic Expansion of the Tail Probability of a Discrete Distribution
13.8. Laplace Transforms
14. Laplace’s Analysis of Binomially Distributed Observations
14.1. Notation
14.2. Background for the Problem and the Data
14.3. A Test for the Hypothesis θ≤r Against θ > r Based on the Tail Probability Expansion, 1781
14.4. A Test for the Hypothesis θ≤ r Against θ> r Based on the Normal Probability Expansion, 1786
14.5. Tests for the Hypothesis 02≤^ι Against Θ2 > 0ι, 1781, 1786, and 1812
14.6. Looking for Assignable Causes
14.7. The Posterior Distribution of θ Based on Compound Events, 1812
14.8. Commentaries
15. Laplace’s Theory of Statistical Prediction
15.1. The Prediction Formula
15.2. Predicting the Outcome of a Second Binomial Sample from the Outcome of the First
15.3. Laplace’s Rule of Succession
15.4. Theory of Prediction for a Finite Population. Prevost and Lhuilier, 1799
15.5. Laplace’s Asymptotic Theory of Statistical Prediction, 1786
15.6. Notes on the History of the Indifference Principle and the Rule of Succession from Laplace to Jeffreys (1948)
16. Laplace’s Sample Survey of the Population of France and the Distribution of the Ratio Estimator
16.1. The Ratio Estimator
16.2. Distribution of the Ratio Estimator, 1786
16.3. Sample Survey of the French Population in 1802
16.4. From Laplace to Bowley (1926), Pearson (1928), and Neyman (1934)
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
17.1. The Characteristic Function and the Inversion Formula for a Discrete Distribution by Laplace, 1785
17.2. Laplace’s Central Limit Theorem, 1810 and 1812
17.3. Poisson’s Proofs, 1824, 1829, and 1837
17.4. Bessel’s Proof, 1838
17.5. Cauchy’s Proofs, 1853
17.6. Ellis’s Proof, 1844
17.7. Notes on Later Developments
17.8. Laplace’s Diffusion Model, 1811
17.9. Gram-Charlier and Edgeworth Expansions
18. Derivations of the Normal Distribution as a Law of Error
18.1. Gauss’s Derivation of the Normal Distribution and the Method of Least Squares, 1809
18.2. Laplace’s Large-Sample Justification of the Method of Least Squares and His Criticism of Gauss, 1810
18.3. Bessel’s Comparison of Empirical Error Distributions with the Normal Distribution, 1818
18.4. The Hypothesis of Elementary Errors by Hagen, 1837, and Bessel, 1838
18.5. Derivations by Adrain, 1808, Herschel, 1850, and Maxwell, 1860
18.6. Generalizations of Gauss’s Proof: The Exponential Family of Distributions
18.7. Notes and References
19. Gauss’s Linear Normal Model and the Method of Least Squares, 1809 and 1811
19.1. The Linear Normal Model
19.2. Gauss’s Method of Solving the Normal Equations
19.3. The Posterior Distribution of the Parameters
19.4. Gauss’s Remarks on Other Methods of Estimation
19.5. The Priority Dispute between Legendre and Gauss
20. Laplace’s Large-Sample Theory of Linear Estimation, 1811-1827
20.1. Main Ideas in Laplace’s Theory of Linear Estimation, 1811-1812
20.2. Notation
20.3. The Best Linear Asymptotically Normal Estimate for One Parameter, 1811
20.4. Asymptotic Normality of Sums of Powers of the Absolute Errors, 1812
20.5. The Multivariate Normal as the Limiting Distribution of Linear Forms of Errors, 1811
20.6. The Best Linear Asymptotically Normal Estimates for Two Parameters, 1811
20.7. Laplace’s Orthogonalization of the Equations of Condition and the Asymptotic Distribution of the Best Linear Estimates in the Multiparameter Model, 1816
20.8. The Posterior Distribution of the Mean and the Squared Precision for Normally Distributed Observations, 1818 and 1820
20.9. Application in Geodesy and the Propagation of Error, 1818 and 1820
20.10. Linear Estimation with Several Independent Sources of Error, 1820
20.11. Tides of the Sea and the Atmosphere, 1797-1827
20.12. Asymptotic Efficiency of Some Methods of Estimation, 1818
20.13. Asymptotic Equivalence of Statistical Inference by Direct and Inverse Probability
21. Gauss’s Theory of Linear Unbiased Minimum Variance Estimation, 1823-1828
21.1. Asymptotic Relative Efficiency of Some Estimates of the Standard Deviation in the Normal Distribution, 1816
21.2. Expectation, Variance, and Covariance of Functions of Random Variables, 1823
21.3. Gauss’s Lower Bound for the Concentration of the Probability Mass in a Unimodal Distribution, 1823
21.4. Gauss’s Theory of Linear Minimum Variance Estimation, 1821 and 1823
21.5. The Theorem on the Linear Unbiased Minimum Variance Estimate, 1823
21.6. The Best Estimate of a Linear Function of the Parameters, 1823
21.7. The Unbiased Estimate of σ2 and Its Variance, 1823
21.8. Recursive Updating of the Estimates by an Additional Observation, 1823
21.9. Estimation under Linear Constraints, 1828
21.10. A Review
PART IV SELECTED TOPICS IN ESTIMATION THEORY, 1830-1930
22. On Error and Estimation Theory, 1830-1890
22.1. Bibliographies on the Method of Least Squares
22.2. State of Estimation Theory around 1830
22.3. Discussions on the Method of Least Squares and Some Alternatives
23. Bienayme’s Proof of the Multivariate Central Limit Theorem and His Defense of Laplace’s Theory of Linear Estimation, 1852 and 1853
23.1. The Multivariate Central Limit Theorem, 1852
23.2. Bravais’s Confidence Ellipsoids, 1846
23.3. Bienayme’s Confidence Ellipsoids and the χ2 Distribution, 1852
23.4. Bienayme’s Criticism of Gauss, 1853
23.5. The Bienayme Inequality, 1853
24. Cauchy’s Method for Determining the Number of Terms To Be Included in the Linear Model and for Estimating the Parameters, 1835-1853
24.1. The Problem
24.2. Solving the Problem by Means of the Instrumental Variable ±1, 1835
24.3. Cauchy’s Two-Factor Multiplicative Model, 1835
24.4. The Cauchy-Bienayme Dispute on the Validity of the Method of Least Squares, 1853
25. Orthogonalization and Polynomial Regression
25.1. Orthogonal Polynomials Derived by Laplacean Orthogonalization
25.2. Chebyshev’s Orthogonal Polynomials, Least Squares, and Continued Fractions, 1855 and 1859
25.3. Chebyshev’s Orthogonal Polynomials for Equidistant Arguments, 1864 and 1875
25.4. Gram’s Derivation of Orthogonal Functions by the Method of Least Squares, 1879, 1883, and 1915
25.5. Thiele’s Free Functions and His Orthogonalization of the Linear Model, 1889, 1897, and 1903
25.6. Schmidt’s Orthogonalization Process, 1907 and 1908
25.7. Notes on the Literature after 1920 on Least Squares Approximation by Orthogonal Polynomials with Equidistant Arguments
26. Statistical Laws in the Social and Biological Sciences. Poisson, Quetelet, and Galton, 1830-1890
26.1. Probability Theory in the Social Sciences by Condorcet and Laplace
26.2. Poisson, Bienayme, and Cournot on the Law of Large Numbers and Its Applications, 1830-1843
26.3. Quetelet on the Average Man, 1835, and on the Variation around the Average, 1846
26.4. Galton on Heredity, Regression, and Correlation, 1869-1890
26.5. Notes on the Early History of Regression and Correlation, 1889-1907
27. Sampling Distributions under Normality
27.1. The Helmert Distribution, 1876, and Its Generalization to the Linear Model by Fisher, 1922
27.2. The Distribution of the Mean Deviation by Helmert, 1876, and by Fisher, 1920
27.3. Thiele’s Method of Estimation and the Canonical Form of the Linear Normal Model, 1889 and 1903
27.4. Karl Pearson’s Chi-Squared Test of Goodnes of Fit, 1900, and Fisher’s Amendment, 1924
27.5. “Student's” t Distribution by Gosset, 1908
27.6. Studentization, the F Distribution, and the Analysis of Variance by Fisher, 1922-1925
27.7. The Distribution of the Correlation Coefficient, 1915, the Partial Correlation Coefficient, 1924, the Multiple Correlation Coefficient, 1928, and the Noncentral χ2 and F Distributions, 1928, by Fisher
28. Fisher’s Theory of Estimation, 1912-1935, and His Immediate Precursors
28.1. Notation
28.2. On the Probable Errors of Frequency Constants by Pearson and Filon, 1898
28.3. On the Probable Errors of Frequency Constants by Edgeworth, 1908 and 1909
28.4. On an Absolute Criterion for Fitting Frequency Curves by Fisher, 1912
28.5. The Parametric Statistical Model, Sufficiency, and the Method of Maximum Likelihood. Fisher, 1922
28.6. Efficiency and Loss of Information. Fisher, 1925
28.7. Sufficiency, the Factorization Criterion, and the Exponential Family. Fisher, 1934
28.8. Loss of Information by Using the Maximum Likelihood Estimate and Recovery of Information by Means of Ancillary Statistics. Fisher, 1925
28.9. Examples of Ancillarity and Conditional Inference. Fisher, 1934 and 1935
28.10. The Discussion of Fisher’s 1935 Paper
28.11. A Note on Fisher and His Books on Statistics
References
Index
