ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

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At a boundary, the normal posterior limit is truncated

Example

Let $X_{i}$ be IID $N (0, 1)$ , let the working model be $N (θ, 1)$ with $Θ = [0, \infty)$ , and put an exponential prior with density $π (θ) = e^{- θ} 1_{[0, \infty)} (θ)$ . The truth $θ^{*} = 0$ is on the boundary. The exact posterior of $H_{n} = \sqrt{n} θ$ is a normal law truncated to $[0, \infty)$ , so it cannot approach the untruncated Gaussian LAN approximation in total variation.

Facts & Assumptions

Given: AC; IID standard-normal data; the normal-mean likelihood on $[0, \infty)$ ; and the exponential prior in the Example.

[A1]

The Axiom of Choice: AC is used for the standard-normal law and the iid central limit theorem.

[F1]

Standard normal and normal laws, The standard normal density has total mass one: $ϕ (x) = e^{- x^{2} / 2} / \sqrt{2 π}$ is a probability density, and its translates are the normal location laws used here.

[F2]

Identical distribution and IID families: The observations are mutually independent and share the standard-normal law.

[F3]

Lindeberg-Levy iid central limit theorem: IID, mean-zero, variance-one observations have $\sqrt{n} \overset{ˉ}{X} \Rightarrow N (0, 1)$ .

[F4]

Portmanteau theorem: Under weak convergence, probabilities converge on Borel sets whose boundary has zero probability under the limit law.

[F5]

Total variation distance for probability laws: The distance is the supremum of event-probability differences.

Verification

Given: The model and prior in the Example. Write $Δ_{n} = \sqrt{n} \overset{ˉ}{X}$ .

1.1F1F2givenalgebra

The prior is proper because $\int_{0}^{\infty} e^{- θ} d θ = 1$ . The local log-likelihood ratio is exactly $ℓ_{n} (h / \sqrt{n}) - ℓ_{n} (0) = h Δ_{n} - h^{2} / 2$ for $h \geq 0$ , so the local Gaussian candidate is the untruncated $N (Δ_{n}, 1)$ . The center $θ^{*} = 0$ is a boundary point of $Θ = [0, \infty)$ , not an interior point.

1.2F1givenalgebra

Transform the posterior variable by $h = \sqrt{n} θ$ . Its unnormalized density on $h \geq 0$ is proportional to $\exp (h Δ_{n} - h^{2} / 2 - h / \sqrt{n})$ . Completing the square gives $\exp [- \frac{1}{2} (h - (Δ_{n} - n^{- 1 / 2}))^{2}]$ . Thus the posterior is $N (Δ_{n} - n^{- 1 / 2}, 1)$ conditioned on $[0, \infty)$ , and assigns probability zero to $(- \infty, 0)$ .

2.1A1F2F3F4F5step 1.2givenalgebra∎

By [F5], the total-variation distance from $N (Δ_{n}, 1)$ is at least its mass on $(- \infty, 0)$ , namely $Φ (- Δ_{n})$ . On ${Δ_{n} \leq 0}$ this bound is at least $1 / 2$ . The event $(- \infty, 0]$ is a continuity set for $N (0, 1)$ , so [F3] and [F4] give $P_{0} (Δ_{n} \leq 0) \to 1 / 2$ . Therefore the total-variation error is at least $1 / 2$ with limiting probability at least $1 / 2$ , and it does not converge to zero in probability.

Depends on

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Sources

Natalia Bochkina, Bernstein-von Mises theorem and misspecified models: a review, §1.4.1 (standard reference, not scraped)