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.

A diffuse normal prior changes the point-null Bayes factor

Example

Compare $M_{0} : θ = 0$ with $M_{1} : θ \sim N (0, τ^{2})$ and $Y ∣ θ \sim N (θ, σ^{2})$ , where $τ^{2}, σ^{2} > 0$ . At the fixed observation $y = 0$ , the Bayes factor for the alternative against the point null depends on the spread $τ^{2}$ of the proper alternative prior and tends to zero as that prior becomes diffuse.

Facts & Assumptions

Given: The two-model normal comparison in the Example and the observation $y = 0$ .

[A1]

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

[F1]

A proper model evidence is the integral of the specified likelihood density against its proper parameter prior; a Bayes factor is the ratio of two finite positive evidences. Marginal likelihoods, proper model evidence, and Bayes factors

[F2]

For $m \in R$ and $s > 0$ , $N (m, s^{2})$ has Lebesgue density $(2 π s^{2})^{- 1 / 2} \exp (- (x - m)^{2} / (2 s^{2}))$ . A positive-variance normal law has its location-scale density

Verification

Given: The proper priors, normal likelihoods, and observation in the Example.

1.1A1F1F2givenalgebra

By [F2], $m_{0} (0) = (2 π σ^{2})^{- 1 / 2}$ . The alternative evidence is $m_{1} (0) = \int_{R} \frac{e^{- θ^{2} / (2 τ^{2})}}{\sqrt{2 π τ^{2}}} \frac{e^{- θ^{2} / (2 σ^{2})}}{\sqrt{2 π σ^{2}}} d θ$ . The exponent coefficient is $(τ^{2} + σ^{2}) / (2 τ^{2} σ^{2})$ ; substituting $z = θ \sqrt{(τ^{2} + σ^{2}) / (τ^{2} σ^{2})}$ and using the standard normal integral gives $m_{1} (0) = (2 π (τ^{2} + σ^{2}))^{- 1 / 2}$ . Both evidences are finite and positive.

2.1F1step 1.1algebra∎

By [F1], the alternative-to-null Bayes factor is $B F_{10} (0) = m_{1} (0) / m_{0} (0) = \sqrt{σ^{2} / (σ^{2} + τ^{2})}$ . As $τ^{2} \to \infty$ with $σ^{2}$ fixed, this tends to zero. Therefore the evidence ratio changes with the proper alternative prior's spread even though the observed datum and point-null model remain fixed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Andrew Gelman et al., Bayesian Data Analysis, 3rd ed., §7.4 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 8, §8.2.2 (standard reference, not scraped)