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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A compact scale hyperprior removes the tail term and permits an explicit bandwidth balance

Example

Take $G$ to be uniform on $[1, 2]$ , and suppose $p_{0}$ satisfies the hypotheses of Dirichlet normal-mixture posterior contraction. Let $σ_{n} = n^{- 1 / 5} (\log n)^{2 / 5} .$ For $γ = 2$ , use $a_{1} = 1 / 4$ and $a_{2} = 1 / 6$ .

Facts & Assumptions

Given: $G = Unif [1, 2]$ , $γ = 2$ , $σ_{n} = n^{- 1 / 5} (\log n)^{2 / 5}$ , and a truth satisfying the main theorem hypotheses.

[F1]

Dirichlet normal-mixture posterior contraction: when $G$ is compactly supported in $(0, \infty)$ , the middle tail term is absent and the rate is $\max {(n σ_{n})^{- 1 / 2} \log n, σ_{n}^{2} \log n}$ .

Verification

1.1givenalgebra

For large $n$ , $n^{- 1 / 4} \leq σ_{n} \leq n^{- 1 / 6}$ because $n^{1 / 20} (\log n)^{2 / 5} \to \infty$ and $n^{- 1 / 30} (\log n)^{2 / 5} \to 0$ . Thus $0 < a_{2} < a_{1} < 1$ and $a_{2} = (4 + γ)^{- 1}$ . The compactly supported $G$ satisfies the lower scale-tail bound since $G (s) = 0$ for $s < 1$ , and the upper tail is zero for $s \geq 2$ .

2.1F1givenstep 1.1algebra∎

Direct calculation gives $(n σ_{n})^{- 1 / 2} \log n = n^{- 2 / 5} (\log n)^{4 / 5}$ and $σ_{n}^{2} \log n = n^{- 2 / 5} (\log n)^{4 / 5}$ . Hence [F1] yields this Hellinger contraction upper bound in $P_{0}^{n}$ -probability. It is not a minimax claim.

Depends on

Dirichlet normal-mixture posterior contraction

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Subhashis Ghosal and Aad W. van der Vaart, Posterior convergence rates of Dirichlet mixtures at smooth densities, Theorem 1 (standard reference, not scraped)