{"path":"research/what-a-case-counts-for/literature-as-read.md","content":"# The literature as read: source check for `what-a-case-counts-for.md`\n\n**What this is**: the verbatim final report of a read-only research helper (Sonnet, Explore type), started\n2026-10-23 at about 15:34 and finished at about 15:54, briefed to check the outside sources the note names\nagainst their texts and to tag each finding with how much of the source was read. Kept unedited, so later\nsessions can see what was checked and what was not. What the note took from it, and the three corrections it\nforced (Reichenbach's wording, Schurz's *high* rather than *nearly always*, and the estimator behind the BCG\nfigures), are in the note's § 3 and § 10.\n\n---\n\nVERIFICATION REPORT (read-only; no files written). Tags: [full text] / [part] / [abstract only] / [secondary source]. \"Unverified\" means I could not check it.\n\n## 1. Laplace's rule of succession\n- Rule (k+1)/(n+2) [full text of the English translation]. Laplace, Essai philosophique sur les probabilités (1814), in Truscott & Emory's translation (Project Gutenberg #58881), Chapter III, the paragraph after the \"Seventh Principle\". Quote: the probability is \"this number increased by unity divided by the same number, increased by two units.\"\n- Page: the rule is on p. 19 [secondary: SEP \"Interpretations of Probability\" cites 1951 Dover p. 19; Wagenmakers et al. 2024 cite \"1814/1902, p. 19\"]. The Gutenberg text has no page numbers.\n- Sunrise example [full text]. Laplace puts history at \"five thousand years ago, or at 1826213 days\" and gives \"a bet of 1826214 to one.\" He then adds the probability is \"incomparably greater\" for someone who knows the mechanism. Wikipedia's \"Rule of succession\" article never gives the 1,826,213 figure. The Gutenberg text does.\n- Uniform prior [full text]. The preceding paragraph says an unknown single-event probability may be taken as \"any value from zero to unity.\" The equal weighting is implicit in his Sixth Principle.\n- Beta(1,1) [full text]. Wagenmakers, Zabell & Gronau, \"J. B. S. Haldane's Rule of Succession\", Statistical Science 39(2):346–354 (2024), arXiv 2307.09489. They credit Laplace (1774/1986, the Mémoire) with a uniform beta(1,1) prior. After n successes in n trials the posterior is beta(n+1,1) and the predictive probability is (n+1)/(n+2).\n- General s-of-n form [full text of entry]: SEP \"The Problem of Induction\" gives p = (n_w+1)/(N+2), citing Laplace 1814. Wikipedia describes it as one success and one failure of pseudocounts [secondary].\n- Priority: the Wikipedia sunrise article says Price (1763) introduced the sunrise problem in his coverage of Bayes [secondary]. Cambridge's blurb for Zabell's chapter says \"Bayes, Price, and Laplace\" [abstract only].\n- Unverified: that the rule appears in Théorie analytique (1812). No source I read says so. Zabell, \"The Rule of Succession\", Erkenntnis 31:283–321 (1989): the full-text link (ece.uvic.ca) is dead, so I know it only by citation.\n- Caveat for your note: De Morgan (1847), quoted in Wagenmakers et al., says the rule gives only \"merely minima\" that background knowledge can raise.\n\n## 2. Carnap's λ-continuum and the zero result\n- Continuum [part]. Carnap, The Continuum of Inductive Methods (Chicago 1952), CMU Carnap project PDF; I read the preface, introduction and §12.\n  - For a binary property the form is c = (s_M + λ/2)/(s+λ), eq. 12-8.\n  - λ=0 is the \"straight rule.\"\n  - λ=∞ gives equal prior weight to every constitution, and is dismissed as unacceptable.\n  - Laplace's rule, after modification, \"leads to λ = 2.\"\n  - SEP \"Problem of Induction\" gives the same family as (N_i+k)/(N+kt), Laplace at t=2, k=1 [full text of passage].\n  - Unverified: the λ=κ ↔ c* identification. I did not see it stated.\n- Zero result [full text, OCR]. Carnap, \"On Inductive Logic\", Philosophy of Science 12(2):72–97 (1945). Page numbers below are from the OCR page headers.\n  - For c* \"for an infinite universe it becomes 0\" (p. 89).\n  - \"c* is again 0\" after counter-instances (p. 90).\n  - In an infinite domain \"the degree of confirmation is 0 for either law\" (p. 92).\n  - Carnap's own workaround is \"instance confirmation\" and \"qualified-instance confirmation\" (§14, pp. 90–93).\n- Carnap 1952 [part]: the introduction notes that for \"All ravens are black\" in an infinite universe, the straight rule gives c=1 and c* gives 0. So the zero belongs to the finite-λ Dirichlet-type methods with the limit-of-finite-N convention. It is not true of every Carnapian rule.\n- Critics [secondary]. SEP \"Rudolf Carnap\", Supplement C (Leitgeb & Carus) calls it \"one of the better-known problems.\" It cites Popper 1935 [1959: §80], Earman 1992 and Carnap, LFP 1950, p. 571. It names the repairs as Hintikka 1966, Kuipers 1978 and Hintikka & Niiniluoto 1980.\n- Zabell: Dirichlet priors give zero to the vertices of the simplex, \"a number of critics pointed to this property\" [full text, Wagenmakers et al.].\n- I could not read Popper §80 or LFP.\n\n## 3. The point-mass repair\n- Broad 1918 (Mind 27:389–404) [secondary: Etz & Wagenmakers 2017, Statistical Science 32(2):313–329, arXiv 1511.08180, full text]. With the Laplace uniform prior on a finite population of n, after m positive instances the probability that all n have the property is (m+1)/(n+1).\n- Jeffreys, recalling Broad: \"The result is preposterous\" (1980, p. 452) [secondary, via Etz & Wagenmakers].\n- Wrinch & Jeffreys 1919 (Phil. Mag. 38:715–731), 1921 (Phil. Mag. 42:369–390) and 1923. The fix is \"a finite initial probability, independent of the population's size,\" given to the general law [secondary, full text of Etz & Wagenmakers].\n- Haldane 1932 (Proc. Camb. Phil. Soc. 28:55–61) first used a mixture prior of point mass plus continuous density. Jeffreys 1933 (Proc. Camb. Phil. Soc. 29:83–87) gave essentially the same rule for a finite population [secondary, Wagenmakers et al., full text].\n- Worked rule [full text, Wagenmakers et al.]:\n  - Take P(UG) = ½ and θ|¬UG ~ Beta(1,1).\n  - The Bayes factor is n+1 and P(UG|n) = (n+1)/(n+2).\n  - The predictive probability is 1 − 1/(n+2)².\n  - They note that in Dirichlet extensions, with T=1 observed type the universal generalization gets positive mass. At T=2 it gets zero, but \"only these two types\" keeps positive mass.\n- Jeffreys, Theory of Probability (1939; 2nd ed. 1948; 3rd ed. 1961) [TOC only]. The York-hosted TOC lists §3.2 \"Sampling\" (p. 125) and §3.21 \"More on the law of succession\" (p. 128). I did not read the text. The sources point to 1961 pp. 127–128 and 1939 pp. 106–109. Jeffreys, Scientific Inference (1973, p. 53): \"quite inadequate to account for the high probability that we often attach to a general law\" [secondary, via Wagenmakers et al.].\n- Hintikka 1966 [secondary only]. \"A Two-Dimensional Continuum of Inductive Methods\", in Hintikka & Suppes (eds.), Aspects of Inductive Logic (North-Holland), pp. 113–132. SEP \"Probability\" lists it as 1965. Per Britannica (via search snippet, page not fetchable) and Kuipers' REP summary [abstract only], the repair gives nonzero prior weight to constituents. I read no Hintikka text.\n- Zabell, \"Confirming Universal Generalizations\", Erkenntnis 45:267–283 (1996); also in Costantini & Galavotti (eds.), Probability, Dynamics and Causality (1997) [abstract only]. The zero is \"an immediate consequence of the sufficientness postulate.\" Modifying that postulate minimally gives a continuum that confirms universal generalizations. Zabell 2011, \"Carnap and the Logic of Inductive Inference\", Handbook of the History of Logic vol. 10, pp. 265–309 [citation only].\n\n## 4. de Finetti's representation theorem\n- Statement [full text]. SEP \"Problem of Induction\": for an infinite exchangeable sequence the joint law is \"a mixture of distribution functions\" under which the data behave as independent draws. Wagenmakers et al. give the form P(S_n=k) = ∫ C(n,k) θ^k(1−θ)^(n−k) dμ(θ), with S_n/n converging a.s. Counts are sufficient. With a Beta μ this gives the Beta-binomial (my gloss).\n- Originals [citations only]: de Finetti 1931 (via SEP Carnap supplement). 1937, \"La prévision…\", Ann. Inst. H. Poincaré 7:1–68 (via SEP \"Interpretations of Probability\"). Hewitt & Savage 1955, Trans. AMS 80(2):470–501, is the general version (Crossref citation only).\n- Independent vs exchangeable [full text, Wagenmakers et al.]. On Johnson's theorem: \"either the sequence is independent (so that the future does not depend on the past)\" or it is a Dirichlet mixture. So exchangeable cases are conditionally independent given θ but marginally dependent. Kemp et al. (item 5) show the cost of fixing θ.\n- Finite caveat [secondary, via Wagenmakers et al.]. Diaconis & Freedman, \"Finite exchangeable sequences\", Ann. Probab. 8(4):745–764 (1980). If a binary sequence extends to length n, the distance to the nearest i.i.d. mixture is ≤ 4k/n for the first k; for t categories it is ≤ 2tk/n. I did not retrieve the abstract. Sampling without replacement is exchangeable but not extendable.\n\n## 5. Kemp, Perfors & Tenenbaum (2007)\nDevelopmental Science 10(3):307–321, doi 10.1111/j.1467-7687.2007.00585.x. [full text, UCI course-page PDF]\n- Marbles (p. 308). Goodman's example: bags uniform in colour, a single black draw from a new bag n. Hypothesis H is \"All marbles in bag n are black\"; overhypothesis O is \"Each bag in stack S contains marbles that are uniform in color.\"\n- Model (pp. 310–312). θ_i ~ Dirichlet(αβ). α is within-bag variability, and small α means near-uniform bags. β is the colour mix across bags. There is a uniform prior on β and an exponential prior on α (mean λ=1).\n- Result (Fig. 3, pp. 311–313):\n  - After 10 all-white and 10 all-black bags, the posterior on θ_new is \"very close to 1.\"\n  - The text puts the mean at 0.99, against 0.75 for a model with α and β fixed at prior means.\n  - Fig. 3b caption: for the fixed model \"observing 20 previous bags provides no information about a new bag.\"\n  - Barratos example (Nisbett et al. 1983): one draw is \"almost as informative as observing 20 members of that tribe\" for skin colour, but not for obesity (α high).\n  - Fig. 6 (pp. 314–315): with a fixed total of observations, two draws from each of many bags teach homogeneity best.\n- Mill: the paper does not cite or mention Mill. The Mill link is yours, not theirs.\n- Mill's question [full text, Gutenberg #27942]. System of Logic, Book III, ch. III, §3: \"Why is a single instance, in some cases, sufficient for a complete induction, while in others, myriads of concurring instances…\" It then calls the answer the solution to \"the problem of induction.\"\n- Goodman, Fact, Fiction, and Forecast (1955): I did not read it. Chapter and page for \"overhypothesis\" are unverified.\n- Note: there is no point mass here, so the answer is 0.99, not 1.\n\n## 6. Reichenbach and Kyburg\n- Reichenbach, The Theory of Probability (1949), p. 374 [secondary; original not read]. Hájek (Synthese 156, 2007) quotes: \"An individual thing or event may be incorporated in many reference classes, from which different probabilities will result.\"\n- Rule [secondary]. Bacchus, Grove, Halpern & Koller (arXiv cs/0307056) quote: \"the narrowest [smallest] reference class for which suitable statistics can be compiled.\" Your phrase \"reliable statistics\" is not what the sources show. The rule is conditional on adequate statistics.\n- Kyburg [secondary]. His definitions first appear in 1961, with the full presentation in the 1974 book. The refined formulation is \"The Reference Class\", Philosophy of Science 50(3):374–397 (1983) (Crossref). Per Loui, \"Computing Reference Classes\" (UAI 1986, arXiv 1304.3099 [part]):\n  - Kyburg uses interval-valued statistics.\n  - Narrower classes are better \"unless its interval contains the interval associated with the other class.\"\n  - Nested intervals signal agreement.\n  - Disagreement is resolved by the most specific class.\n  - A narrower class overrides only when its interval disagrees. This bears on \"explanation proposes a narrower class.\"\n- I read no Kyburg original.\n\n## 7. Schurz (2001)\nPhilosophy of Science 68(4):476–497, doi 10.1086/392938.\n- [abstract only; paywalled]. Normic laws \"if A, then normally B.\" Their evolutionary origin \"must establish a systematic connection between prototypical and statistical normality.\"\n- Normality-to-probability claim [secondary]. SEP \"Ceteris Paribus Laws\" (Reutlinger, Schurz, Hüttemann; rev. 2024), §8.1:\n  - It calls this the \"statistical consequence thesis.\" Normic laws imply unspecified \"Most A are B\" generalizations and carry \"high conditional statistical probabilities.\"\n  - Self-regulating systems must be in prototypical states \"in the high statistical majority of cases.\"\n  - Exceptions come from dysfunctions.\n  - Challenged by McCarthy 1986 and Millikan 1984.\n- Caveat: the claim is \"high,\" not \"near zero.\" No threshold appears in what I read. Your \"near-zero unexplained-failure rate\" is an interpretation. SEP's reference list misprints the volume as 28.\n- German companion: JGPS 32(1):65–107 (2001), doi 10.1023/A:1011209401546 (metadata only).\n\n## 8. BCG data\n- Dataset exists, 13 trials [full text]. Metadata: wviechtb/metadat, man/dat.bcg.Rd.\n- Columns: trial, author, year, tpos, tneg, cpos, cneg, ablat, alloc. alloc is random, alternate or systematic.\n- Where it lives: NOT in wviechtb/metafor. NEWS.md says \"datasets moved to the `metadat` package\" (version not pinned: the NEWS.md structure is irregular). metafor DESCRIPTION has Depends: metadat. In metafor the data appear only in README.md and tests/testthat/test_misc_metan_vs_rma.{mh,peto,uni}_with_dat.bcg.r.\n- In wviechtb/metadat: data/dat.bcg.rda, data-raw/dat.bcg.txt (13 rows), data-raw/dat.bcg.r, man/dat.bcg.Rd. dat.colditz1994 is the identical data, \"included twice for backwards compatibility.\" The Rd cites Colditz et al. 1994, JAMA 271(9):698–702.\n- Published findings:\n  - Colditz abstract [abstract only, Europe PMC]: \"Geographic latitude of the study site and study validity score explained 66% of the heterogeneity among trials.\" That is two moderators. It reports 14 prospective trials, against 13 in the dataset. I did not find the reason.\n  - metafor README [full text], REML: total τ²=0.3132, I²=92.22%, Q(12)=152.23.\n  - With ablat: residual τ²=0.0764, I²=68.39%, R²=75.62%, QE(11)=30.73 (p=0.0012), slope −0.0291.\n  - Berkey et al. 1995 (Stat Med 14:395–411) [abstract only, plus the metafor-project page]: EB estimator gives τ² 0.268 → 0.157, R²=41.38%.\n- My own check: a pure-Python REML on the 13 rows reproduces τ² 0.3132, I² 92.22%, residual τ² 0.0763, I² 68.39%, R² 75.63% and slope −0.0291.\n- Caveat: \"variance explained\" ranges from 41% (EB) to 76% (REML) with k=13. Always name the estimator.\n\n## 9. DerSimonian & Laird (1986)\nControlled Clinical Trials 7(3):177–188, doi 10.1016/0197-2456(86)90046-2. [abstract only]\n- It proposes a random-effects approach that \"incorporates the heterogeneity of effects.\"\n- The model \"can be extended to include relevant covariates which would reduce the heterogeneity,\" giving \"more specific therapeutic recommendations.\"\n- It also proposes a \"simple noniterative procedure.\"\n- Meta-regression [abstract only]. Thompson & Higgins, Stat Med 21:1559–1573 (2002):\n  - Regression is weighted by within-trial variance plus residual between-trial heterogeneity.\n  - Results are observational.\n  - \"Data dredging is the main pitfall,\" so covariates should be pre-specified.\n- Higgins & Thompson (2002, Stat Med 21:1539–58) define I² as the proportion of total variation due to heterogeneity [abstract only].\n- van Houwelingen et al. (2002, Stat Med 21:589–624) is a BCG-based meta-regression tutorial [abstract only].\n- Caution: a moderator is a candidate narrower rule, but only a pre-specified one is safe.\n\n## Not verified\n- Théorie analytique statement of the rule.\n- Zabell 1989 text.\n- Jeffreys ToP text beyond its TOC.\n- Hintikka 1966 text.\n- Schurz 2001 full text.\n- Popper §80 and Carnap LFP.\n- Reichenbach 1949 and Kyburg originals.\n- Goodman's chapter.\n- The Diaconis & Freedman abstract.\n- Why dat.bcg has 13 trials when Colditz reports 14.\n"}