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{\Large\textbf{\textit{Theory of Probability}---Sir Harold Jeffreys}}

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\multicolumn{3}{l}{\textbf{I. Fundamental Notions}} \\
\ \\
1.0   & [Induction and its relation to deduction]                    &   1 \\
1.1   & [Principles of inductive reasoning]                          &   8 \\
1.2   & [Axioms for conditional probability]                         &  15 \\
1.21  & [Fallacious applications of the product rule]                &  26 \\
1.22  & [Principles of inverse probability (Bayes)]                  &  26 \\
1.23  & [Arbitrariness of numerical representation]                  &  29 \\
1.3   & [Expected values; ideas of Bayes and Ramsey]                 &  30 \\
1.4   & [The principle of insufficient reason]                       &  33 \\
1.5   & [Consistency of posterior probabilities]                     &  34 \\
1.51  & The infinite regress argument                                &  38 \\
1.52  & The theory of types                                          &  40 \\
1.6   & [Inductive inference approaching certainty]                  &  43 \\
1.61  & [Indistinguishable consequences]                             &  45 \\
1.62  & [Complexity of differential equations]                       &  45 \\
1.7   & [Suppression of an irrelevant premise; chances']            &  50 \\
1.8   & [Expectations of functions]                                  &  53 \\
\ \\
\multicolumn{3}{l}{\textbf{II. Direct Probabilities}} \\
\ \\
2.0   & Likelihood                                                   &  57 \\
2.1   & Sampling [and the hypergeometric law]                        &  59 \\
2.11  & [Sampling with replacement; the binomial law]                &  60 \\
2.12  & [The normal approximation to the binomial]                   &  61 \\
2.13  & [The law of large numbers]                                   &  62 \\
2.14  & [Normal approximation to the hypergeometric]                 &  66 \\
2.15  & Multiple sampling and the multinomial law                    &  67 \\
2.16  & The Poisson law                                              &  68 \\
2.2   & The normal law of error                                      &  70 \\
2.3   & The Pearson laws                                             &  74 \\
2.4   & The negative binomial law                                    &  75 \\
2.5   & Correlation                                                  &  81 \\
2.6   & Distribution functions                                       &  83 \\
2.601 & [Convergence of distribution functions]                      &  83 \\
2.602 & [Lemma needed for the inversion theorem]                     &  84 \\
2.61  & Characteristic functions                                     &  85 \\
2.62  & [Moments; m.g.f.s; semi-invariants]                          &  86 \\
2.63  & [Moments of (negative) binomial and Poisson]                 &  87 \\
2.64  & [M.g.f.\ of the Cauchy distribution]                         &  89 \\
2.65  & The inversion theorem                                        &  90 \\
2.66  & Theorems on limits                                          &  93 \\
2.661 & [Convergence of d.f.s implies that of ch.f.s]                &  93 \\
2.662 & \textit{The smoothed distribution function}                  &  93 \\
2.663 & [Convergence of ch.f.s implies that of d.f.s]                &  94 \\
2.664 & \textit{The central limit theorem}                           &  95 \\
2.67  & [Central limits for Cauchy and Type VII]                     &  97 \\
2.68  & [Case of a finite fourth moment]                             & 100 \\
2.69  & [Symmetric laws over a finite range]                         & 101 \\
2.7   & The $\chi^2$ distribution                                    & 103 \\
2.71  & [Effect of adjustable parameters on $\chi^2$]                & 105 \\
2.72  & [Effect of linear constraints on $\chi^2$                    & 105 \\
2.73  & [$\chi^2$ related to the Poisson law]                        & 106 \\
2.74  & [$\chi^2$ related to the multinomial law]                    & 106 \\
2.75  & [$\chi^2$ related to contingency tables]                     & 107 \\
2.76  & [The interpretation of $\chi^2$; grouping]                   & 107 \\
2.8   & The $t$ and $z$ distributions [$s$ and $s'$]                 & 108 \\
2.81  & [The variance ratio and the $z$ distribution]                & 111 \\
2.82  & [Generalization of $\chi^2$ if variances unknown]            & 112 \\
2.9   & The specification of random noise                            & 114 \\
\ \\
\multicolumn{3}{l}{\textbf{III. Estimation Problems}} \\
\ \\
3.0   & [Introduction]                                               & 117 \\
3.1   & [Conventional priors; the $dv/v$ rule]                       & 117 \\
3.2   & Sampling [and the hypergeometric law]                        & 125 \\
3.21  & [More on the law of succession]                              & 128 \\
3.22  & [The Dirichlet integral; volume of a sphere]                 & 132 \\
3.23  & Multiple sampling                                            & 133 \\
3.3   & The Poisson distribution [$S$ and $\Sigma$]                  & 135 \\
3.4   & The normal law of error                                      & 137 \\
3.41  & [Normal law of unknown variance]                             & 138 \\
3.42  & [Prediction from normal observations]                        & 142 \\
3.43  & [Relation to one-way analysis of variance]                   & 143 \\
3.5   & The method of least squares]                                 & 147 \\
3.51  & [Examples on least squares                                   & 152 \\
3.52  & Equations of unknown weights; grouping                       & 152 \\
3.53  & Least squares equations; successive approximation            & 154 \\
3.54  & [Example of this method]                                     & 157 \\
3.55  & [Positive parameters; prior $d\alpha$ for $\alpha>0$\,]      & 160 \\
3.6   & The rectangular distribution                                 & 161 \\
3.61  & Re-scaling of a law of chance                                & 164 \\
3.62  & Reading of a scale                                           & 164 \\
3.7   & Sufficient [and ancillary] statistics & 165 \\
3.71  & The Pitman-Koopman theorem [and the exponential family]      & 165 \\
3.8   & The posterior probabilities that the true value, or the third \\
& \,observation, will lie between the first two observations   & 170 \\
3.9   & Correlation                                                  & 174 \\
3.10  & Invariance theory $I_m$ and $J$]                             & 179 \\
\ \\
\multicolumn{3}{l}{\textbf{IV. Approximate Methods and Simplifications}} \\
\ \\
4.0   & Maximum likelihood                                           & 193 \\
4.01  & Relation of maximum likelihood to invariance theory          & 195 \\
4.1   & An approach to maximum likelihood [via minimum $\chi^2$]     & 196 \\
4.2   & Combination of estimates with different estimated uncertainties
& 198 \\
4.3   & The use of expectations                                      & 200 \\
4.31  & Orthogonal parameters                                        & 207 \\
4.4   & [Approaches based on the median; outliers]                   & 211 \\
4.41  & [Approximate normality with an example]                      & 214 \\
4.42  & [Linear relations with both variables subject to error]      & 216 \\
4.43  & Grouping                                                     & 217 \\
4.44  & Effects of grouping; Sheppard's correction                   & 220 \\
4.45  & [Case of one known component of variance]                    & 221 \\
4.5   & Smoothing of observed data                                   & 223 \\
4.6   & Correction of a correlation coefficient                      & 227 \\
4.7   & Rank correlation [and Spearman's $\rho_0$]                   & 229 \\
4.71  & Grades and contingency [and examples of $\rho_0$]            & 235 \\
4.8   & The estimation of an unknown and unrestricted integer        \\
& [The tramcar problem]                                        & 238 \\
4.9   & Artificial randomization                                     & 239 \\
\ \\
\multicolumn{3}{l}{\textbf{V. Significance Tests: One new parameter}} \\
\ \\
5.0   & General discussion [and the Bayes factor $K$]                & 245 \\
5.01  & Treatment of old parameters                                  & 249 \\
5.02  & Required properties of $f(\alpha)$                           & 251 \\
5.03  & Comparison of two sets of observations                       & 252 \\
5.04  & Selection of alternative hypotheses                          & 253 \\
5.1   & Test of whether a suggested value of a chance is correct     \\
& [binomial with a uniform prior]                              & 256 \\
5.11  & Simple contingency [$2\times2$ tables]                       & 259 \\
5.12  & Comparison of samples [one margin fixed]                     & 261 \\
5.13  & [A special case]                                             & 263 \\
5.14  & [More general priors; several examples]                      & 263 \\
5.15  & Test for consistency of two Poisson parameters               & 267 \\
5.2   & Test of whether the true value in the normal law is zero;    \\
& standard error originally unknown                            & 268 \\
5.21  & Test of whether a true value is zero; $\sigma$ taken as known
& 274 \\
5.3   & Generalization by invariance theory [and choice of priors]   & 275 \\
5.31  & General approximate form                                     & 277 \\
5.4   & Other tests related to the normal law                        & 278 \\
5.41  & Test of whether two values are equal;                        \\
& standard errors supposed the same                            & 278 \\
5.42  & Test of whether two location parameters are the same,        \\
& standard  errors not supposed equal                          & 280 \\
5.43  & Test of whether a standard error has a suggested value $\sigma_0$
& 281 \\
5.44  & Test of agreement of two estimated standard errors          & 283 \\
5.45  & Test of both the standard error and the location parameter   & 285 \\
5.46  & [Example on the tensile strength of tires]                   & 285 \\
5.47  & The discovery of argon                                       & 287 \\
5.5   & Comparison of a correlation coefficient with a suggested value
& 289 \\
5.51  & Comparison of correlations                                   & 293 \\
5.6   & The intraclass correlation coefficient                       & 295 \\
5.61  & Systematic errors; further discussion                        & 300 \\
5.62  & estimation of intraclass correlation                         & 302 \\
5.63  & Suspiciously close agreement [very small $\chi^2$]           & 307 \\
5.64  & [Eddington's \textit{Fundamental Theory}]                    & 310 \\
5.65  & [The effect of smoothing data]                               & 311 \\
5.7   & Test of the normal law of error                              & 314 \\
5.8   & Test for independence in rare events                         & 319 \\
5.9   & Introduction of new functions                                & 325 \\
5.91  & [Relation to normal distribution theory]                     & 324 \\
5.92  & Allowance for old functions                                  & 325 \\
5.93  & Two sets of observations relevant to the same parameter      & 326 \\
5.94  & Continuous departure from a uniform distribution of chance   \\
& [distribution of angles; the circular normal (von Mises) law]& 328 \\
5.95  & [Independence of the establishment and explanation of laws]  & 331 \\
\ \\
\multicolumn{3}{l}{\textbf{VI. Significance Tests: Various Complications}} \\
\ \\
6.0   & Combination of tests                                         & 332 \\
6.1   & [Tests on several new parameters at once]                    & 340 \\
6.11  & [Simultaneous consideration of a new function               \\
& and of correlation]                                          & 341 \\
6.12  & [Occam's rule (razor)]                                       & 342 \\
6.2   & [Fitting of two new harmonics]                               & 346 \\
6.3   & Partial and serial correlation                               & 356 \\
6.4   & Contingency affecting only diagonal elements                 & 360 \\
6.5   & Deduction as an approximation                                & 365 \\
\ \\
\multicolumn{3}{l}{\textbf{VII. Frequency Definitions and Direct Methods}} \\
\ \\
7.0   & [Introduction]                                               & 369 \\
7.01  & [Alternative definitions of probability]                     & 369 \\
7.02  & [Objections to probability as the ratio of favourable cases  \\
& \,to all cases (Neyman)]                                     & 370 \\
7.03  & [Objections to probability as a limiting frequency           \\
& \,(Venn and von Mises) and to probability in terms of        \\
& \,a hypothetical infinite population (Fisher)]               & 373 \\
7.04  & [Non-equivalence of the above theories]                      & 375 \\
7.05  & [Need for probabilities of hypotheses]                       & 377 \\
7.1   & [Problem of the uncertainty of a mean as treated             \\
& by Student' and Fisher]                                     & 378 \\
7.11  & [Different sets of data with the same hypothesis] & 382      \\
7.2   & [Criticisms of the use of $P$ values in tests]               & 383 \\
7.21  & [Use of $P$ values in estimation]                            & 387 \\
7.22  & [Uselessness of rejection in the absence of an alternative]  & 390 \\
7.23  & [Separation of $\chi^2$ into components]                     & 391 \\
7.3   & [Karl Pearson and the method of moments]                     & 392 \\
7.4   & [Similarities with R.A.\ Fisher's methods]                   & 393 \\
7.5   & [Criticism of the Neyman-Pearson notion of errors of the     \\
& \,second kind]                                               & 395 \\
7.6   & [Statistical mechanics; ergodic theory]                      & 398 \\
\ \\
\multicolumn{3}{l}{\textbf{VIII. General Questions}}
\ \\
8.0   & [Prior probabilities are \textit{not} frequencies            & 401 \\
8.1   & [Necessity of using prior probabilities]                     & 405 \\
8.2   & [Scientific caution']                                       & 409 \\
8.3   & [Parallels with quantum mechanics]                           & 411 \\
8.4   & [Should the rejection of unobservables be accepted?]         & 412 \\
8.5   & [Agreement with observations is not enough]                  & 417 \\
8.6   & [Recapitulation of main principles]                          & 419 \\
8.7   & [Realism versus idealism; religion versus materialism]       & 422 \\
8.8   & [Unprovability of idealism]                                  & 424 \\
\ \\
\multicolumn{3}{l}{\textbf{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 \\
\ \\
\multicolumn{3}{l}{\textbf{Appendix B. Tables of $K$}}               \\
& [Introduction; grades of $K$]                                & 432 \\
I     & [\S6.0, eq.\ (1), p.\,333]                                   & 437 \\
II    & [\S6.2, eq.\ (21), p.\,346;                                  \\
& \,note the formula here is right and eq.\ (21), p.\,346 is wrong]
& 438 \\
III   & [\S5.92, first displayed equation, p.\,325]                  & 439 \\
IIIA  & [\S5.2, eq.\ (33), p.\,274]                                  & 439 \\
IV    & [\S6.21, eq.\ (37), p.\,348]                                 & 440 \\
IVA   & [\S6.21, eq.\ (42), p.\,349]                                 & 440 \\
V     & [\S5.43, eq.\ (11) and eq.\ (14), p.\,282]                   & 441 \\
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