ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A prior-predictive null fiber has no quotient posterior

Example

Let $Θ = {a, b}$ have its full power-set sigma-algebra and prior mass $1 / 2$ at each point. Let the data space be $Y = {0, 1, 2}$ with its full power-set sigma-algebra and counting measure. Conditional on $a$ , the observation is $0$ with probability one; conditional on $b$ , it is $1$ with probability one. The sampling kernel has density $ℓ (a, y) = 1_{{0}} (y)$ and $ℓ (b, y) = 1_{{1}} (y)$ ; it is jointly measurable on these finite power-set spaces. Then $Z (2) = 0$ , so the quotient formula is undefined at $2$ ; the prior-predictive probability of that observation is zero.

Facts & Assumptions

[F1]

Evidence normalizer and fiberwise posterior update: at a fixed data value, the evidence is the integral of the nonnegative likelihood section against the prior measure, and a quotient update is defined only for finite positive evidence.

[F2]

Fiberwise posterior propriety is equivalent to finite positive evidence: zero evidence cannot be made into a probability by finite positive scaling.

[F3]

Bayes formula for dominated kernels: for a proper prior and a jointly measurable normalized sampling-kernel density, the evidence $m$ is the data marginal density; the kernel uses the quotient where $0 < m < \infty$ and the prior as the stated filler where $m = 0$ or $m = \infty$ . The filled fibers have zero prior-predictive marginal mass.

Given: The finite prior, data space, conditional laws, and likelihood density specified in the Example.

Verification

1.1givenF1algebra

The likelihood sections are normalized: $L_{0} (a) = 1$ , $L_{1} (b) = 1$ , and all other parameter-data pairs have likelihood zero. Therefore $Z (0) = 1 / 2$ , $Z (1) = 1 / 2$ , and $Z (2) = 0$ .

2.1step 1.1F2F3algebra∎

By [F3], the prior-predictive mass at $2$ is $Z (2)$ times its counting measure, hence zero. At this fiber every numerator $\int_{A} L_{2} d Π$ is also zero, so the displayed quotient is $0 / 0$ and is not defined; the theorem's chosen filler is $K (2, A) = Π (A)$ . This filler is a version value on a prior-predictive null event, not an update inferred from the observation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Gelman et al., Bayesian Data Analysis, 3rd ed., §1.3 (standard reference, not scraped)
Johnson, Ott & Dogucu, Bayes Rules!, §§2.1.3–2.1.4, 2.3.4–2.3.6 (standard reference, not scraped)