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.

A proper Gaussian prior gives nonuniform finite-sample coverage on an unbounded truth class

Example

Fix $n \geq 1$ , $σ, τ > 0$ , and $0 < γ < 1$ . Observe independent $Y_{i} \sim N (θ, σ^{2})$ under truth $θ \in R$ and use prior $N (0, τ^{2})$ . Let $v_{n} = (τ^{- 2} + n σ^{- 2})^{- 1}$ , $a_{n} = v_{n} n σ^{- 2}$ , and $z_{γ} = Φ^{- 1} (1 - γ / 2)$ , where $Φ$ is the standard normal cumulative distribution function. The exact posterior credible interval is $[a_{n} \overset{ˉ}{Y} - z_{γ} \sqrt{v_{n}}, a_{n} \overset{ˉ}{Y} + z_{γ} \sqrt{v_{n}}]$ . Since $0 < a_{n} < 1$ , its repeated-sampling coverage at fixed $n$ tends to zero as $∣ θ ∣ \to \infty$ , despite exact posterior computation and exact sampling.

Example references

The calibrated-inversion question and its exact Bayesian baseline. ‡

Facts & Assumptions

Given: The scalar normal observation model and proper prior in the Example, with $n \geq 1$ , $σ, τ > 0$ , and $0 < γ < 1$ .

[A1]

Completing the Gaussian likelihood and prior gives posterior mean $a_{n} \overset{ˉ}{Y}$ , variance $v_{n} = (τ^{- 2} + n σ^{- 2})^{- 1}$ , and central credible radius $r_{n} = z_{γ} \sqrt{v_{n}}$ .

[A2]

The standard normal cumulative distribution has vanishing tails: $Φ (t) \to 0$ as $t \to - \infty$ and $Φ (t) \to 1$ as $t \to + \infty$ .

Verification

1.1givenA1algebra

At every fixed $n$ , $a_{n} = n τ^{2} / (σ^{2} + n τ^{2})$ lies strictly between zero and one, and $r_{n}$ is finite and independent of the observation. Under truth $θ$ , $a_{n} \overset{ˉ}{Y} - θ$ has mean $- (1 - a_{n}) θ$ and fixed positive standard deviation $a_{n} σ / \sqrt{n}$ .

1.2A1algebra

Standardizing the event in the Example gives $c_{n} (θ) = Φ ⁣ (\frac{r_{n} + (1 - a_{n}) θ}{a_{n} σ / \sqrt{n}}) - Φ ⁣ (\frac{- r_{n} + (1 - a_{n}) θ}{a_{n} σ / \sqrt{n}})$ .

2.1A1A2step 1.2algebra∎

As $θ \to + \infty$ , both arguments in step 1.2 tend to $+ \infty$ , so both cumulative probabilities tend to one by [A2] and their difference tends to zero. As $θ \to - \infty$ , both arguments tend to $- \infty$ and their difference again tends to zero. Thus the infimum coverage over $θ \in R$ is zero at each fixed $n$ .

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