ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 candidate outside posterior support is not admissible

Example

On $Ω = {0, 1}$ with counting measure, take $γ = (1, 0)$ . Then $Z = 1$ and $π = (1, 0)$ . The probability density $q = (0, 1)$ is supported entirely outside the target support, so it is not admissible and has no finite ELBO.

Facts & Assumptions

Given: The two-point counting space, target density $γ = (1, 0)$ , and candidate $q = (0, 1)$ .

[F1]

Variational family and evidence lower bound: admissibility requires $q = 0$ on ${π = 0}$ and $\int q (∣ \log γ ∣ + ∣ \log q ∣) < \infty$ .

Verification

Given: The data in the Example.

1.1F1algebra

Counting measure gives $Z = 1$ and $π = (1, 0)$ . At the point $1$ , $q (1) = 1$ but $π (1) = 0$ , so the support condition in [F1] fails.

2.1F1step 1.1algebra∎

Also $γ (1) = 0$ and $q (1) = 1$ , so $q (1) ∣ \log γ (1) ∣ = + \infty$ . The absolute-integrability condition in [F1] fails as well; the finite-ELBO identity is not applicable to this candidate.

Depends on

Variational family and evidence lower bound

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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