{"path":"research/reasons-and-cases.md","content":"# Reasons and cases: one arithmetic, two kinds of dependence\n\n*2026-10-23. His question, from a screenshot of three running sessions (link strength; rules and exceptions;\nflywheels and microtasks): \"Are these two first sessions dealing with the same thing from different\ndirections? This is a Just a curious reflection I had right now. What if independent reasons adding with\nlog-odds are LIKE different cases moving the strength of a rule? Or are they just different things\nentirely? Research this extensively online. Ultrathink\"*\n\n*Status: research and proposals. The reasons half is ruled (ledger Q38, Q40, Q46); the cases half is ruled\n(Arc AY, 2026-10-23); the two proposals here are Arc BA, ruled the same evening (*\"Both blessed! But I'm not sure I FULLY grasp all this yet.\"*). Arithmetic: [reasons-and-cases/cases.py](reasons-and-cases/cases.py),\nprinted in [what-cases-printed.txt](reasons-and-cases/what-cases-printed.txt). Reports, verbatim:\n[real-cases-dependent-instances.md](reasons-and-cases/real-cases-dependent-instances.md),\n[literature-weight-of-evidence-and-exchangeability.md](reasons-and-cases/literature-weight-of-evidence-and-exchangeability.md).*\n\n## 0. Answer first\n\n**The same operation, seen from two sides. What differs is what the claim and its rival leave open.**\n\nOne example of each, as it would sit in the graph:\n\n- A claim and two reasons: *It rained last night* ← *The lawn is wet* (support link) and ← *The rain gauge\n  holds 4 mm* (support link).\n- A rule and two cases: *About a third of red-sky mornings bring rain* ← *Red sky on 3 March; it rained*\n  and ← *Red sky on 9 March; it stayed dry* (cases filed under the rule, AY2).\n\n1. **What is the same.** Both weigh evidence by how much likelier it is if the claim is true than if it is\n   false (the Bayes factor), and both add those weights in log-odds: I. J. Good's *weight of evidence*,\n   measured in decibans as Turing did at Bletchley. The sum always holds, by the chain rule of probability,\n   but each term is an item's weight *given the items before it*. Only when the claim and its rival are each\n   one definite possibility (a coin is fair, or lands heads 80 % of the time) does each term equal the\n   item's own weight; then cases add exactly as independent reasons do, ten flips' weights summing to the\n   joint weight digit for digit (§ 5.1). That is Wald's sequential test, and Turing's Banburismus.\n2. **What differs on the cases' side: the rate stays open.** *About a third* is a band, and so is its rival\n   (*anything else*). Neither fixes the rate, so every case still teaches where it lies and the next case\n   weighs less than it would alone. The count computes that conditional sum exactly (AY3). Summing the\n   cases' stand-alone weights instead drifts, in either direction and further as cases accumulate: 40 wet\n   mornings of 120 read 1.000 where the count gives 0.955; 20 of 30 under a wide band read 0.002 where the\n   count gives 0.106 (§ 5.2). The sum is the exact answer to a different claim, one in which every morning\n   has its own rate.\n3. **The reasons' side has the same structure, on the rival.** *It did not rain* is several stories (dew, a\n   sprinkler), so a wet lawn and a wet car roof are independent only once that story is fixed. The ruled\n   *best until independence is read, then add* (ledger Q38, best until read, then add) takes the two ends;\n   the exact answer lies between them: the lawn alone weighs 1.14 nats, lawn and roof exactly 1.31, the two\n   summed 2.27 (§ 5.9). The middle is computable once the shared story is named and drawn as a claim between\n   the reasons and the conclusion (ledger Q42, a shared doubt becomes a claim between, open).\n4. **So there are two kinds of dependence, and both channels meet both.** Evidence can depend on other\n   evidence *through the thing the claim is about* (the rain; the rate), which is what makes it evidence at\n   all; and *through a route that bypasses it* (dew wetting lawn and roof; a repeated test repeating its own\n   error; a publication filter on seven small trials; a confounder shared by thirty observational studies).\n   The first kind is harmless for a yes-or-no claim and computed by the count for a band. The second kind\n   inflates both channels the same way, and the count does not see it.\n5. **Every disaster on record is one of the two errors** (§ 5): adding reasons that share a route (a stool\n   test repeated on one patient carried a factor of 16.5 where multiplying the two tests gave 84.8; the\n   *Collins* traits; forecasters summed as if they held separate evidence), or reading a unit's cases as\n   draws from a known rate (Sally Clark's two cot deaths, 1 in 73 million squared from a population rate\n   where a family-level rate gives 1 in 7 to 13 million; Lucia de Berk's shifts, 1 in 3 million where\n   nurse-to-nurse spread gives 1 in 46), or both at once (thirteen BCG trials counted as one big sample\n   claim about 19 times the information they hold).\n6. **And the two channels feed each other.** A reason's weight is a rate that some rule's cases taught (a\n   test validated on 20 sick and 20 well people carries a likelihood ratio of 9 read off its counts, 6.3 for\n   the next patient), and sibling rules' counts give each rule its start (18 batters after 45 at-bats,\n   pooled with each other, predicted the rest of the season 3.5 times better than each batter's own count).\n\nNo published source found says that reasons for a claim and instances of a rule are one structure. Keynes\n(1921) and Peirce (1878) come closest, Good and Jaynes state the formal rule that covers both, and the\nhierarchical models of meta-analysis carry the cases' side; the unification is this note's (§ 3).\n\n**Two proposals** (§ 6, ledger Arc BA): ask the rival-story question between the bundles filed under a rule,\nnever between cases inside one bundle (BA1); and read several cases from one unit (a family, a nurse, a\nlab, one patient's repeated test) as a count under that unit's own rate, with the spread between units,\nnever as independent draws from a population rate (BA2).\n\n## 1. What he asked, and what the question is\n\nThe two sessions are *how independent reasons add* ([how-independent-reasons-add.md](how-independent-reasons-add.md),\nledger Q46: across arguments read as independent, a claim's strength is logit(start) plus the sum of each\nargument's shift in log-odds) and *what a case counts for* ([what-a-case-counts-for.md](what-a-case-counts-for.md),\nArc AY: a rule's cases are pooled as a count in a channel of their own and enter the rule's reasons as one\nargument). The second note already said the reasons channel cannot take a rule's cases one by one\n(§ 8 there). His question asks whether that is a difference in kind or in bookkeeping.\n\n## 2. What the corpus already holds\n\n| Angle | Where | What it says |\n|---|---|---|\n| reasons add in log-odds once independent | `how-independent-reasons-add.md` § 4; ledger Q46 | logit(strength) = logit(start) + Σ shifts; the multiplicative pool, naive Bayes and Potyka's logistic semantics in one line |\n| what independence means for reasons | `retrofitting-the-reasons-channel.md` § 9k; ledger Q40 | *suppose the claim is false: is each reason's most likely explanation the same one?* |\n| the overlap dial | same § 9k point 4; ledger Q43; Clemen and Winkler | *n* / (1 + (*n* − 1)ρ) independent reasons' worth, ceiling 1/ρ |\n| cases are exchangeable, not independent | `what-a-case-counts-for.md` § 8; `what-a-case-counts-for/literature-as-read.md` § 4 (de Finetti) | cases are independent given the rate and dependent without it; counts are sufficient |\n| the count enters the rule's reasons as one argument | ledger AY2 | its shift is the log of the count's Bayes factor |\n| a study is a bundle; spread calls for a narrower rule | ledger AY5; BCG case in `what-a-case-counts-for.md` § 7 | heterogeneity weighs through the computation |\n| two reports beat one only through a third argument | `pollock-against-accrual-and-the-overlap-check.md` § 1c | Pollock: the lift is a statistical syllogism on the pair's joint reliability |\n| enumerative induction is the one place a whole exceeds its parts | same note § 1a | Pollock: the conclusion depends on the *number* of instances, a property of that scheme, never a rule of aggregation |\n| weight of evidence, decibans | `strength-calculus-literature/bayesian-and-informal-logic.md`; `what-counts-as-evidence.md` | Good's log Bayes factor as the unit |\n| a shared origin counts once: sources as nodes, inputs sharing an origin form one channel | `what-counts-as-evidence.md` § 2c; ledger Q4 (independence as structure, open); his words 2026-10-14, *\"Overlapping roots of evidence obviously shall not be added up.\"* | the route around a claim is this, named for reasons; BA1 applies it to the bundles under a rule (*added 2026-10-23 on his question whether this had been discussed before: it had, in about twenty notes, and this table first missed it*) |\n\nNot in the corpus before tonight (searched by term across `docs/research`): the chain-rule reading of a\ncount as a sum of conditional weights (the chain rule itself appears, as fact times relevance, in\n`retrofitting-the-reasons-channel.md` § 9j); a marginal likelihood; Wald's sequential test as the point\nwhere cases and reasons coincide (the test itself is in `proving-a-pass-reliable.md`, for auditing a\nmachine pass item by item, which is the same arithmetic on a different claim); Kish's design effect applied to cases (it appears only for reasons); the\ndistinction between dependence through the thing measured and dependence through the route; and the real\ncases in § 5 (Clark, *Collins*, repeated tests, Efron and Morris, de Berk, magnesium, hormone therapy).\n\n## 3. What the literature adds\n\nFull report, verbatim with every citation and read-status: [literature-weight-of-evidence-and-exchangeability.md](reasons-and-cases/literature-weight-of-evidence-and-exchangeability.md).\n\n- **The chain rule is the one always-valid form, and additivity needs two simple hypotheses.** Good (1985,\n  eq. 1, p. 251) writes the weight of two pieces of evidence as the first's weight plus the second's weight\n  *given the first*, and says (p. 252, crediting Herman Rubin) that the simple sum *\"usually requires that\n  both H and H̄ should be simple statistical hypotheses.\"* Jaynes (*Probability Theory*, ch. 4) puts it as\n  *\"we always have additivity, but not independent additivity\"*, and his exercise on more than two\n  hypotheses shows the independence conditions cannot both hold once the rival is a disjunction of several.\n- **A band against its complement is computed as a ratio of integrals.** Shafer, Shen, Vereshchagin and Vovk\n  (2011, § 8.1, § 8.5) give the point-against-point case as a product of per-case ratios and the band case\n  as a ratio of Beta integrals; Jaynes's compound hypotheses come out as an incomplete Beta function, which\n  is AY3's count. Dawid (1984) and Kass and Raftery (1995, § 3.2) read the log Bayes factor as a difference\n  of predictive scores, each case scored given the ones before, with Laplace's rule as the predictive for a\n  flat start.\n- **Multiplying separate Bayes factors is wrong for a common rate, and named as such.** Nikolakopoulos, Edmar\n  and Ntzoufras (2026, § 3.2): a product of study Bayes factors *\"is not appropriate for the, usually\n  assumed, common effect model\"*; the right product is of conditional marginal likelihoods. Rouder and Morey\n  (2011): *\"one should simply multiply odds. Unfortunately, this approach is not valid.\"* The naive product's\n  error has no fixed sign: it can understate (their witnesses), and it overstates a universal law against a\n  mixture (a Jensen bound the helper derived).\n- **A case of a rule need not support it.** Good (1967): with 100 black crows and none other, against 1,000\n  black crows and one white, a black crow is evidence *against* *all crows are black*. Keynes (1921, ch. XX):\n  each instance strengthens a generalisation only *\"so long as the new instance could not have been\n  predicted with certainty from a knowledge of the former instances\"*, which is the conditional term in\n  words.\n- **Argumentation theory has no accrual of examples.** Walton's *argument from example* scheme takes one\n  example; none of its five questions asks how many. Gradual semantics aggregate by sum, product or maximum\n  (Potyka's table), with no dependence; Pollock takes the maximum; Yanal's independent reasons combine by\n  noisy-or. No computational treatment found that handles several examples differently from several reasons.\n- **The nearest outside account of a shared route.** Pilditch, Hahn and Lagnado (2025) treat repeated\n  measurements by one device as the error of reading them like coin tosses, and note that concordant results\n  raise the device's reliability, the same latent-variable shape; De Pretis, Landes and Osimani (2019) model\n  sponsorship bias as a shared variable over a group of studies, *\"what Bayesian statisticians call a\n  hierarchical model\"*.\n- **Not read:** Wald's own texts, Good's weighted-average theorem, Fitelson (2001) beyond its slides, Sober\n  (1989), Walton's book, Pollock's statistical induction (listed in the report's last section).\n\n## 4. The arithmetic, made plain\n\n**One operation.** For any evidence *E₁ … Eₙ* about a claim *H* against its rival *¬H*,\n\nlogit P(H | all) = logit P(H) + log [P(E₁ … Eₙ | H) / P(E₁ … Eₙ | ¬H)],\n\nand the joint factor always splits by the chain rule:\n\nlog factor(all) = Σᵢ log [P(Eᵢ | E₁ … Eᵢ₋₁, H) / P(Eᵢ | E₁ … Eᵢ₋₁, ¬H)].\n\nSo every pile of evidence is a sum of weights. The only question is whether each weight may be taken *on\nits own*, or must be taken *given the ones before it*.\n\n**Three bookkeepings of that one sum.**\n\n| The evidence | Each term is | What Deliberus does |\n|---|---|---|\n| reasons whose routes are separate, for a yes/no claim | the reason's own weight, since given *H* (and given *¬H*) the others tell nothing more | add (Q46) |\n| reasons that share a route (dew; one source; a repeated test's error) | smaller than its own weight once the earlier reasons are known, and larger than nothing | best one until independence is read (Q38); the middle once the shared story is a claim between (Q42, open) |\n| cases of a rule stated as a band | the case's weight given the earlier cases, because the band does not fix the rate | the count computes the whole sum at once (AY3) |\n\nThe reasons rule is the special case in which the claim and its rival fix everything the evidence depends\non. The cases rule is the same sum with the dependence known in advance (every case is a draw from one\nunknown rate), which is why it can be computed instead of read. Reasons whose shared story is unnamed sit\nbetween: the dependence is real and its shape is unknown, so the ruled default takes the safe end (the best\nreason) until the rival question is asked.\n\n**Why summing a rule's cases' stand-alone weights goes wrong.** A case's stand-alone weight averages the\nrate over the band, and over everything outside it. Summing thirty such weights treats each case as if it\ncame from the band's average rate and the rival's average rate: two point hypotheses. That is close while\nthe band is narrow and the case count small (10 of 30: 0.702 against 0.691), and wrong once the cases have\ntaught the rate something the averages do not hold: with 40 of 120 the cases have pinned the rate near a\nthird, inside the band, and the sum overshoots to 1.000; with 20 of 30 under a wide band the cases point to\ntwo thirds, a rate the rival contains and explains well, which the stand-alone weights never learn, so the\nsum calls the rule refuted (0.002) where the count leaves it 0.106.\n\n**Two kinds of dependence, named.** *Dependence through the quantity* runs through what the claim is about:\nboth reasons depend on whether it rained, every case on the rate. It is the reason the items are evidence.\n*Dependence through the route* runs around the claim: under *¬H* one alternative produces several items\n(dew; a lab's contamination; a publication filter; a family's predisposition when the claim is about a\npopulation). It is the reason items overstate. Q40's question detects the second kind; the count handles\nthe first kind for cases; nothing handles the second kind for cases today.\n\n## 5. Tests on real cases\n\nAll recomputed in `cases.py` from the published counts; the helper's figures and these agree.\n\n**5.1 Exact, by construction: a point claim.** A coin is fair or lands heads 80 % of the time, start 0.5.\nAfter 8 heads in 10 flips the sum of the flips' log factors is 1.927448, the log of the joint factor\n1.927448: each head adds 2.04 decibans, each tail takes away 3.98, and *biased* reads 0.873. Cases of a\npoint hypothesis add exactly like independent reasons.\n\n**5.2 Exact, by construction: a band claim.** *About a third of red-sky mornings bring rain* (0.25 to 0.42),\nflat start 0.170; the chain of conditional weights equals the joint weight to four decimals in every row.\n\n| Mornings | From the count | Summing stand-alone weights | Best single case |\n|---|---|---|---|\n| 10 wet of 30 | 0.691 | 0.702 | 0.226 |\n| 40 wet of 120 | 0.955 | 1.000 | |\n| 100 wet of 300 | 0.998 | 1.000 | |\n| 20 wet of 30 | 0.0034 | 0.0006 | |\n| same, wide band 0.15 to 0.55: 20 wet of 30 | 0.106 | 0.002 | |\n\n**5.3 Reasons that share a route: a stool test repeated on the same patients** (Fanshawe et al. 2024;\n10 cancers, 1,299 without). A positive first test multiplies the odds by 9.06, a positive second by 9.36;\nmultiplied as independent reasons, 84.8. The double positive carried 16.5, about five times less, because\namong people without cancer a false positive is followed by another in 0.49 of cases against 0.085 overall:\nthe second test mostly repeats the first one's error. From a start of 1 %, the independent sum says 0.46,\nthe data 0.14. Serial antigen tests for SARS-CoV-2 (Soni et al. 2023) show the same in symptomatic\npatients (two tests 93.4 % sensitive where independence predicts 96.9 %; three 94.3 % against 99.5 %), and\nthe opposite on the first day, when the second test gains from the infection growing: the shared route can\nalso be a shared trend.\n\n**5.4 Reasons summed across people: forecasters.** In the Good Judgment Project an average of about 235\nforecasters per question was best pushed outwards by a factor near 2 in log-odds (99 % interval 1.16 to\n3.92; Satopää et al. 2014), where summing as independent evidence would mean a factor of 235. Their\ninformation overlapped almost entirely; the fitted factor reads as about two independent forecasters'\nworth (with each forecaster's own caution mixed in).\n\n**5.5 A unit's cases read as draws from a known rate.** Sally Clark: two cot deaths in one family, squared\nfrom 1 in 8,543 to 1 in 73 million. A second death in a family is 5 to 10 times likelier after a first\n(Hill 2004 from the national study's counts; 5.7 from a separate programme; other estimates 1.9 to 6), so\nthe figure is 1 in 7 to 13 million on the same base rate, and 1 in 169,000 to 296,000 on the population\nrate. A recurrence factor of 10 is what families whose rates vary with a coefficient of variation of about\n3 produce: the cases are dependent because they share the family's unknown rate. Lucia de Berk: 14 of a\nward's 27 incidents in her 201 of its 1,704 shifts, every shift an independent draw, is 1 in 3.1 million;\nallowing nurses' rates to differ, as they do, it is 1 in 46 (Gill, Groeneboom and de Jong). She was\nacquitted. *People v. Collins*: six traits multiplied to 1 in 12 million, where the traits overlap (a man\nwith a beard often has a moustache); the court and later statisticians showed the product valid only as a\nchain of conditional probabilities, which nobody had.\n\n**5.6 Bundles that share a route outside the rule.** Seven small magnesium trials gave an odds ratio of 0.45\nfor death after heart attack; the 58,000-patient ISIS-4 trial gave 1.06, and all 22 trials together 0.99.\nOnly 2 of 15 magnesium trials described concealed allocation. Thirty-one observational studies of hormone\ntherapy pooled to a relative risk of 0.56 for heart disease; the Women's Health Initiative trial found 1.29,\nabout 16 standard errors away; adjusting for socioeconomic status alone moved the observational figure to\nabout 1.0. Ego depletion (198 tests, d = 0.62, against a 23-lab replication at 0.04) and Bem's precognition\nexperiments (a Bayes factor of 5 billion against a replication 72 times likelier under chance) are the same\nshape. In each, the bundles shared a route around the rule (a selection filter, a confounder, a common\nsmall-study quality) and the count, which assumes exchangeable bundles, could not see it.\n\n**5.7 The BCG trials both ways.** Every one of the 357,347 people counted as an independent case: risk ratio\n0.65, z = −10.6. Cases sharing their trial's setting (random effects): 0.49, z = −4.0. The shared setting\nmultiplies the variance by 19.5, so the data hold about a twentieth of the information the independent\nreading claims. This is the cases' version of the reasons channel's overlap dial (Kish's design effect for\nclustered samples, the formula the corpus already cites for reasons, `retrofitting-the-reasons-channel.md`\n§ 9k point 4), with the spread between trials in the place of the reasons' overlap.\n\n**5.8 The channels feeding each other.** A diagnostic test validated on 20 sick and 20 well people (18 true\nand 2 false positives) has a likelihood ratio of 9.0 read off its counts and 6.3 predicted for the next\npatient; on 200 and 200 with the same rates, 8.6. From a start of 10 %, a positive reads 0.50 or 0.41: the\nweight a reason carries is a rate learned from cases, and a small count should carry its uncertainty into\nthe reason. And Efron and Morris's 18 batters: each batter's own 45 at-bats predicted the rest of his season\nwith a squared error of 17.6 (on their arcsine scale); pooled with his siblings (weight 0.21 on his own\ncount), 5.0, and closer for 16 of 18 on the averages themselves. Sibling rules' counts give each rule its\nstart, which AY3 already names (*the start comes from sibling rules where they exist*).\n\n**5.9 By construction: two reasons that share a rival.** *It rained last night*, from a wet lawn and a wet\ncar roof. Each is wet with probability 0.95 if it rained; if it did not, dew came on three nights in ten\nand wets each with probability 0.9, and otherwise each is wet one time in twenty. Given rain, given dew, or\ngiven neither, the two are independent; given only *it did not rain*, they are not. The lawn alone weighs\n1.136 nats (from even odds, 0.757); lawn and roof together exactly 1.305 (0.787); the two summed as\nindependent 2.272 (0.907). The roof adds 0.169, not 1.136. The best reason (the ruled default before\nindependence is read) is close to the truth here, the sum far from it, and neither is exact; drawn as Q42\nproposes, *everything left outdoors was wet* supporting *it rained* once with *it was not dew* under that\nlink, the structure carries the middle.\n\n## 6. Proposals (ledger Arc BA, ruled 2026-10-23, unbuilt), in order of dependence\n\n1. **BA1, the rival-story question is asked between bundles, never between the cases inside one.** Q40's\n   question (*suppose the claim is false: is each item's most likely explanation the same one?*) is asked\n   between the bundles filed under a rule (studies, sources, labs), and bundles it reads as sharing a route\n   around the rule (a selection filter, a confounder, one lab, one source of cases) are read as one bundle,\n   or the shared route is filed as an explanation that names a kind (AY4) and proposes the narrower\n   reading; it is never asked between cases inside one bundle, whose dependence through the rate the count\n   already holds. Seven small magnesium trials and thirty-one hormone studies (§ 5.6) are what the count\n   alone cannot catch.\n2. **BA2, a unit's cases are counted under the unit's own rate.** When several cases come from one unit (a\n   family, a nurse, a patient's repeated test, a lab), a claim about that unit is read through a count of\n   that unit's cases with the spread of rates between units, never by multiplying a population rate once\n   per case. Sally Clark (1 in 73 million squared against 1 in 7 to 13 million) and Lucia de Berk (1 in 3.1\n   million against 1 in 46) are this error (§ 5.5); a stool test repeated on one patient is the same error\n   in the reasons channel (§ 5.3).\n\nNo row is proposed for the reasons' side: the shared-rival structure that computes the middle is already\nopen as Q42 (a shared doubt becomes a claim between), and § 5.9 is evidence for it. How Q42 fits the object model and the folds, and\nthe four clauses it was ruled with the same night: `where-a-shared-doubt-sits.md` (2026-10-23).\n\n## 7. What only he decided\n\n*Decided 2026-10-23 with BA1 as written: the bundles keep their counts and the shared route is filed as an explanation.* Whether BA1's bundles found to share a route count as one bundle outright, or keep their count with the\nshared route filed as an explanation whose reading is computed. The first is the reasons channel's default\n(best one until independence is read); the second is what meta-analysis does (bias correction, a narrower\nreading). Proposed: the second, since the cases inside each bundle are real cases.\n\n## Cross-references\n\n[how-independent-reasons-add.md](how-independent-reasons-add.md) · [what-a-case-counts-for.md](what-a-case-counts-for.md) ·\n[retrofitting-the-reasons-channel.md](retrofitting-the-reasons-channel.md) § 9k ·\n[pollock-against-accrual-and-the-overlap-check.md](pollock-against-accrual-and-the-overlap-check.md) ·\n[starting-values-and-the-problem-of-the-prior.md](starting-values-and-the-problem-of-the-prior.md) ·\n[long-form-sources-and-meta-analysis-weighing.md](long-form-sources-and-meta-analysis-weighing.md) ·\nledger Q38, Q40, Q43, Q46, Arc AY, Arc BA ([session26-open-rulings.md](session26-open-rulings.md)).\n"}