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 iid samples estimate two-mode posterior weights

Example

Let $b > 0$ , $0 < w < 1$ , and let the two-mode posterior be $π_{w} = w δ_{- b} + (1 - w) δ_{b}$ on $R$ . From $M \geq 1$ independent exact posterior draws $X_{1}, \dots, X_{M}$ , define $\hat{w} = M^{- 1} \sum_{i} 1 {X_{i} = - b}$ and ${\hat{π}}_{M} = \hat{w} δ_{- b} + (1 - \hat{w}) δ_{b}$ . The one-Wasserstein error and the probability that at least one posterior mode is absent from the sample are $W_{1} ({\hat{π}}_{M}, π_{w}) = 2 b ∣ \hat{w} - w ∣, P {\hat{w} \in {0, 1}} = (1 - w)^{M} + w^{M},$ and $E W_{1} ({\hat{π}}_{M}, π_{w}) \leq 2 b \sqrt{w (1 - w) / M}$ .

Example references

The finite-compute posterior-fidelity question, with score and time-discretization errors set to zero.

Facts & Assumptions

Given: $b > 0$ , $0 < w < 1$ , $M \geq 1$ , posterior $π_{w} = w δ_{- b} + (1 - w) δ_{b}$ , and independent draws $X_{i} \sim π_{w}$ .

[A1]

$W_{1}$ is the infimum of $E ∣ X - X^{'} ∣$ over couplings of the two probability measures.

Verification

1.1givenA1construct

Put $\hat{w} = M^{- 1} \sum_{i} 1 {X_{i} = - b}$ . In any coupling of ${\hat{π}}_{M}$ and $π_{w}$ , mass $\min {\hat{w}, w}$ at $- b$ and $\min {1 - \hat{w}, 1 - w}$ at $b$ can be matched at zero cost; the remaining mass is $∣ \hat{w} - w ∣$ and must move between the atoms, a distance $2 b$ . This coupling is feasible and no coupling can move less unmatched mass, so it is optimal and gives $W_{1} = 2 b ∣ \hat{w} - w ∣$ .

2.1givenstep 1.1algebra∎

The count $M \hat{w}$ is binomial $(M, w)$ , so $E \hat{w} = w$ and $Var (\hat{w}) = w (1 - w) / M$ . Cauchy–Schwarz gives $E ∣ \hat{w} - w ∣ \leq \sqrt{w (1 - w) / M}$ ; step 1.1 yields the expected Wasserstein bound. The event of missing $- b$ has probability $(1 - w)^{M}$ , and the event of missing $b$ has probability $w^{M}$ ; for $M \geq 1$ these events are disjoint, giving the displayed miss probability.

Used by

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Dependency tree · 0 levels

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