ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

An improper flat-prior scale makes a formal evidence arbitrary

Example

Consider one observation $Y$ from the normal location model $Y ∣ θ \sim N (θ, σ^{2})$ with known $σ^{2} > 0$ . For any $c > 0$ , the measure $Π_{c} (d θ) = c λ (d θ)$ is an improper flat prior, where $λ$ is Lebesgue measure (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ). The likelihood integral against it is finite but equals $c$ . If two competing models use this same normal-location likelihood, with the same observed $y$ and known variance, but use independent arbitrary constants $c_{1}$ and $c_{2}$ , their formal evidence ratio is $c_{2} / c_{1}$ , so it has no value determined by the models.

Facts & Assumptions

Given: A normal location likelihood with $σ^{2} > 0$ , an observation $y \in R$ , and the sigma-finite prior measure $c d θ$ for $c > 0$ .

[A1]

AC is assumed for the library normal-law convention. The Axiom of Choice

[F1]

A proper-model evidence in this pair is defined by integrating a specified likelihood density against a probability prior. Marginal likelihoods, proper model evidence, and Bayes factors

[F2]

The BCS-02 fiberwise normalizer is the likelihood integral against the supplied sigma-finite prior measure; an improper prior has infinite total mass. Evidence normalizer and fiberwise posterior update

[F3]

AC implies Countable Choice. AC implies DC implies countable choice

[F4]

Under Countable Choice, Lebesgue measure is sigma-finite and finite on bounded sets. Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure

[F5]

Lebesgue measure of a real interval is its length, including a degenerate interval of measure zero. A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included

[F6]

For $σ > 0$ , the normal location likelihood density is $(2 π σ^{2})^{- 1 / 2} \exp (- (y - θ)^{2} / (2 σ^{2}))$ . A positive-variance normal law has its location-scale density

Verification

Given: The model and prior measure in the Example.

1.1A1F1F2F3F4F5F6givenalgebra

By [A1, F3, F4], $λ$ is sigma-finite, so $Π_{c} = c λ$ is sigma-finite. By [F5], $Π_{c} ([- n, n]) = 2 c n$ for every positive integer $n$ , hence $Π_{c} (R) = + \infty$ and it is improper in the sense of [F2]. By [F6], its formal likelihood integral is $\int_{R} \frac{c}{\sqrt{2 π σ^{2}}} \exp ⁣ (- \frac{(y - θ)^{2}}{2 σ^{2}}) d θ$ . The substitution $z = (y - θ) / σ$ has absolute Jacobian $σ$ , so this equals $c \int_{R} (2 π)^{- 1 / 2} e^{- z^{2} / 2} d z = c$ . This value is positive and finite for every fixed $c > 0$ , but $Π_{c} (R) = + \infty$ , so it is not an evidence under [F1].

2.1F1F2step 1.1algebra∎

For the two copies of the same normal-location likelihood specified above, with model-specific constants $c_{1}, c_{2} > 0$ , the formal ratio is $c_{2} / c_{1}$ by step 1.1. Rescaling either improper prior changes this ratio arbitrarily while leaving its unnormalized density proportional to Lebesgue measure; hence no generic Bayes factor is defined by these improper priors.

Depends on

Used by

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Sources

Andrew Gelman et al., Bayesian Data Analysis, 3rd ed., §7.4 (standard reference, not scraped)