Magnus & Callender (2004) argue the no-miracles argument and the pessimistic induction both commit a base-rate fallacy — the vault's denominator problem in formal Bayesian dress, generalized to all of science

philosophy-of-sciencesurvivorship-biasselection-effectsbase-rate-fallacyscientific-realismpessimistic-inductionno-miracles-argumentargument-from-silence

P. D. Magnus and Craig Callender, in "Realist Ennui and the Base Rate Fallacy" (Philosophy of Science, 2004), argue that the two leading "wholesale" arguments about scientific realism — the no-miracles argument (successful theories are probably true, since otherwise their success would be a miracle) and the pessimistic induction (many past successful theories turned out false, so present ones probably will too) — both rest on a base-rate fallacy. They extend a diagnosis Colin Howson (2000) had made of the no-miracles argument and Peter Lewis (2001) of the pessimistic induction.

Formalizing the no-miracles argument in Bayesian notation, they show its load-bearing premise — that an arbitrary theory drawn from the "pool" of candidate theories is antecedently likely to be successful — cannot be checked, because the pool cannot be enumerated: "It is impossible to count up or even fairly sample all the theories that were considered for our mature sciences, and so it is impossible to evaluate whether (5) obtains." The obvious repair — restrict the pool to the theories mature science actually adopted — only swaps one error for another: "The realist would thus avoid base rate neglect, but at the cost of sample selection bias."

That second move is the formal, probabilistic twin of the reasoning the vault's own circuit-theory note makes informally, claim-circuit-theory-vindication-lacks-denominator-survivorship-not-ruled-out: a sample built from the retold, vindicated cases cannot license a claim about the base rate of vindication, because membership in the sample already presupposes the success you are trying to measure. Where the vault reaches that conclusion for systems-neuroscience circuit theories, Magnus and Callender reach it for the entire history of science, which is why this claim sits beside the moc-argument-from-silence cluster — a search that returns only successes is not evidence that failures are rare — rather than under the neuroscience-specific question-do-neural-circuit-theories-systematically-precede-their-anatomy, whose scope it does not close. The base-rate framing also neighbours the population-vs-proportion diagnosis in claim-reference-class-problem-dissolves-with-proportions-not-population-counts: both locate the trouble in how a denominator is drawn.