ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

Exact Gaussian credible-ball coverage with a score-error budget of zero

Example

Let $n \geq 1$ , $σ, τ > 0$ , and $0 < γ < 1$ . Given a scalar truth $θ \in R$ , observe independent $Y_{i} \sim N (θ, σ^{2})$ , $i = 1, \dots, n$ , and use the proper prior $θ \sim N (0, τ^{2})$ . Write $\overset{ˉ}{Y} = n^{- 1} \sum_{i} Y_{i}$ , $v_{n} = (τ^{- 2} + n σ^{- 2})^{- 1}$ , $a_{n} = v_{n} n σ^{- 2}$ , and $z_{γ} = Φ^{- 1} (1 - γ / 2)$ , where $Φ$ is the standard normal cumulative distribution function. The exact posterior is $N (a_{n} \overset{ˉ}{Y}, v_{n})$ and its central $(1 - γ)$ credible interval is $C_{n} (\overset{ˉ}{Y}) = [a_{n} \overset{ˉ}{Y} - z_{γ} \sqrt{v_{n}}, a_{n} \overset{ˉ}{Y} + z_{γ} \sqrt{v_{n}}]$ . Its frequentist coverage at truth $θ$ is $P_{θ} {θ \in C_{n} (\overset{ˉ}{Y})} = Φ ⁣ (\frac{z_{γ} \sqrt{v_{n}} + (1 - a_{n}) θ}{a_{n} σ / \sqrt{n}}) - Φ ⁣ (\frac{- z_{γ} \sqrt{v_{n}} + (1 - a_{n}) θ}{a_{n} σ / \sqrt{n}}) .$

Example references

The calibration frontier whose exact-computation baseline is isolated here. ‡

Facts & Assumptions

Given: The normal observation model, the proper $N (0, τ^{2})$ prior, the posterior credible interval, $n \geq 1$ , $σ, τ > 0$ , and $0 < γ < 1$ from the Example.

[A1]

The likelihood is proportional to $\exp (- \frac{1}{2 σ^{2}} \sum_{i = 1}^{n} (Y_{i} - θ)^{2})$ and the prior density is proportional to $\exp (- θ^{2} / (2 τ^{2}))$ .

[A2]

$Φ (t) = P (Z \leq t)$ for $Z \sim N (0, 1)$ , and $z_{γ}$ is its $(1 - γ / 2)$ quantile.

Verification

1.1givenA1A2algebra

Multiplying the likelihood by the prior and completing the square in $θ$ gives posterior precision $τ^{- 2} + n σ^{- 2}$ and posterior mean $(τ^{- 2} + n σ^{- 2})^{- 1} n σ^{- 2} \overset{ˉ}{Y} = a_{n} \overset{ˉ}{Y}$ . Thus the posterior variance is $v_{n}$ and the stated interval has posterior mass $1 - γ$ .

2.1givenA1A2step 1.1algebra∎

Under the fixed truth $θ$ , $a_{n} \overset{ˉ}{Y} - θ$ is normal with mean $- (1 - a_{n}) θ$ and standard deviation $a_{n} σ / \sqrt{n}$ . The event $θ \in C_{n} (\overset{ˉ}{Y})$ is exactly $- z_{γ} \sqrt{v_{n}} \leq a_{n} \overset{ˉ}{Y} - θ \leq z_{γ} \sqrt{v_{n}}$ ; standardizing the two endpoints gives the displayed difference of normal cumulative probabilities.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.