Theory of Probability Sir Harold Jeffreys Table of Contents

Size: px

Start display at page:

Download "Theory of Probability Sir Harold Jeffreys Table of Contents"

Sydney Hill
5 years ago
Views:

1 Theory of Probability Sir Harold Jeffreys Table of Contents I. Fundamental Notions 1.0 [Induction and its relation to deduction] [Principles of inductive reasoning] [Axioms for conditional probability] [Fallacious applications of the product rule] [Principles of inverse probability (Bayes)] [Arbitrariness of numerical representation] [Expected values; ideas of Bayes and Ramsey] [The principle of insufficient reason] [Consistency of posterior probabilities] The infinite regress argument The theory of types [Inductive inference approaching certainty] [Indistinguishable consequences] [Complexity of differential equations] [Suppression of an irrelevant premise; chances ] [Expectations of functions] 53 II. Direct Probabilities 2.0 Likelihood Sampling [and the hypergeometric law] [Sampling with replacement; the binomial law] [The normal approximation to the binomial] [The law of large numbers] [Normal approximation to the hypergeometric] Multiple sampling and the multinomial law The Poisson law The normal law of error The Pearson laws The negative binomial law Correlation Distribution functions [Convergence of distribution functions] [Lemma needed for the inversion theorem] Characteristic functions [Moments; m.g.f.s; semi-invariants] [Moments of (negative) binomial and Poisson] [M.g.f. of the Cauchy distribution] The inversion theorem 90 1

2 2.66 Theorems on limits [Convergence of d.f.s implies that of ch.f.s] The smoothed distribution function [Convergence of ch.f.s implies that of d.f.s] The central limit theorem [Central limits for Cauchy and Type VII] [Case of a finite fourth moment] [Symmetric laws over a finite range] The distribution [Effect of adjustable parameters on ] [Effect of linear constraints on [ related to the Poisson law] [ related to the multinomial law] [ related to contingency tables] [The interpretation of ; grouping] The and distributions [ and ] [The variance ratio and the distribution] [Generalization of if variances unknown] The specification of random noise 114 III. Estimation Problems 3.0 [Introduction] [Conventional priors; the rule] Sampling [and the hypergeometric law] [More on the law of succession] [The Dirichlet integral; volume of a sphere] Multiple sampling The Poisson distribution [ and ] The normal law of error [Normal law of unknown variance] [Prediction from normal observations] [Relation to one-way analysis of variance] The method of least squares] [Examples on least squares Equations of unknown weights; grouping Least squares equations; successive approximation [Example of this method] [Positive parameters; prior for ] The rectangular distribution Re-scaling of a law of chance Reading of a scale Sufficient [and ancillary] statistics The Pitman-Koopman theorem [and the exponential family] The posterior probabilities that the true value, or the third observation, will lie between the first two observations 170 2

3 3.9 Correlation Invariance theory and ] 179 IV. Approximate Methods and Simplifications 4.0 Maximum likelihood Relation of maximum likelihood to invariance theory An approach to maximum likelihood [via minimum ] Combination of estimates with different estimated uncertainties The use of expectations Orthogonal parameters [Approaches based on the median; outliers] [Approximate normality with an example] [Linear relations with both variables subject to error] Grouping Effects of grouping; Sheppard s correction [Case of one known component of variance] Smoothing of observed data Correction of a correlation coefficient Rank correlation [and Spearman s ] Grades and contingency [and examples of ] The estimation of an unknown and unrestricted integer [The tramcar problem] Artificial randomization 239 V. Significance Tests: One new parameter ' 5.0 General discussion [and the Bayes factor ] Treatment of old parameters Required properties of! " Comparison of two sets of observations Selection of alternative hypotheses Test of whether a suggested value of a chance is correct [binomial with a uniform prior] Simple contingency [#%$&# tables] Comparison of samples [one margin fixed] [A special case] [More general priors; several examples] Test for consistency of two Poisson parameters Test of whether the true value in the normal law is zero; standard error originally unknown Test of whether a true value is zero; taken as known Generalization by invariance theory [and choice of priors] General approximate form Other tests related to the normal law Test of whether two values are equal; 3

4 standard errors supposed the same Test of whether two location parameters are the same, standard errors not supposed equal Test of whether a standard error has a suggested value '( Test of agreement of two estimated standard errors Test of both the standard error and the location parameter [Example on the tensile strength of tires] The discovery of argon Comparison of a correlation coefficient with a suggested value Comparison of correlations The intraclass correlation coefficient Systematic errors; further discussion estimation of intraclass correlation Suspiciously close agreement [very small ] [Eddington s Fundamental Theory] [The effect of smoothing data] Test of the normal law of error Test for independence in rare events Introduction of new functions [Relation to normal distribution theory] Allowance for old functions Two sets of observations relevant to the same parameter Continuous departure from a uniform distribution of chance [distribution of angles; the circular normal (von Mises) law] [Independence of the establishment and explanation of laws] 331 VI. Significance Tests: Various Complications 6.0 Combination of tests [Tests on several new parameters at once] [Simultaneous consideration of a new function and of correlation] [Occam s rule (razor)] [Fitting of two new harmonics] Partial and serial correlation Contingency affecting only diagonal elements Deduction as an approximation 365 VII. Frequency Definitions and Direct Methods 7.0 [Introduction] [Alternative definitions of probability] [Objections to probability as the ratio of favourable cases to all cases (Neyman)] [Objections to probability as a limiting frequency (Venn and von Mises) and to probability in terms of 4

5 ) ) a hypothetical infinite population (Fisher)] [Non-equivalence of the above theories] [Need for probabilities of hypotheses] [Problem of the uncertainty of a mean as treated by Student and Fisher] [Different sets of data with the same hypothesis] [Criticisms of the use of values in tests] [Use of values in estimation] [Uselessness of rejection in the absence of an alternative] [Separation of into components] [Karl Pearson and the method of moments] [Similarities with R.A. Fisher s methods] [Criticism of the Neyman-Pearson notion of errors of the second kind] [Statistical mechanics; ergodic theory] 398 VIII. General Questions 8.0 [Prior probabilities are not frequencies [Necessity of using prior probabilities] [ Scientific caution ] [Parallels with quantum mechanics] [Should the rejection of unobservables be accepted?] [Agreement with observations is not enough] [Recapitulation of main principles] [Realism versus idealism; religion versus materialism] [Unprovability of idealism] 424 Appendix A. Mathematical Theorems A.1 [If the sum of finite subsets of a set of reals is bounded the set is countable] 425 A.2 [A bounded sequence of functions on a countable set has a convergent subsequence] 425 A.21 [The Arzela-Ascoli theorem] 425 A.22 [Weak compactness of the set of d.f.s] 426 A.23 [Uniquensess of limits of d.f.s] 426 A.3 Stieltjes integrals 426 A.31 Inversion of the order of integration 427 A.4 Approximations 428 A.41 Abel s lemma 428 A.42 Watson s lemma 429 Appendix B. Tables of [Introduction; grades of ] 432 I [* 6.0, eq. (1), p. 333] 437 II [* 6.2, eq. (21), p. 346; note the formula here is right and eq. (21), p. 346 is wrong] 438 5

6 III [* 5.92, first displayed equation, p. 325] 439 IIIA [* 5.2, eq. (33), p. 274] 439 IV [* 6.21, eq. (37), p. 348] 440 IVA [* 6.21, eq. (42), p. 349] 440 V [* 5.43, eq. (11) and eq. (14), p. 282] 441 6

PART I INTRODUCTION The meaning of probability Basic definitions for frequentist statistics and Bayesian inference Bayesian inference Combinatorics

PART I INTRODUCTION The meaning of probability Basic definitions for frequentist statistics and Bayesian inference Bayesian inference Combinatorics Table of Preface page xi PART I INTRODUCTION 1 1 The meaning of probability 3 1.1 Classical definition of probability 3 1.2 Statistical definition of probability 9 1.3 Bayesian understanding of probability