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.

Exact discrete variational family attains log evidence

Example

Let $Ω = {0, 1}$ with counting measure, $γ (0) = 2$ , and $γ (1) = 1$ . Then $Z = 3$ and $π = (2 / 3, 1 / 3)$ . Taking $q = π$ attains $ELBO (q) = \log 3$ . The distinct admissible density $q = (1 / 2, 1 / 2)$ gives $ELBO (q) = \frac{3}{2} \log 2 < \log 3$ .

Facts & Assumptions

Given: The two-point counting space, target weights $γ = (2, 1)$ , and ELBO convention.

[F1]

Variational family and evidence lower bound: $Z = \int γ d μ$ , $π = γ / Z$ , and $ELBO (q) = \sum_{x} q (x) (\log γ (x) - \log q (x))$ for admissible $q$ .

[F2]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is strictly increasing, $\log 1 = 0$ , and it satisfies the product and quotient laws on positive reals.

Verification

Given: The data in the Example.

1.1F1algebra

Counting measure gives $Z = 2 + 1 = 3$ and $π = (2 / 3, 1 / 3)$ . For $q = π$ , each positive coordinate has $\log q (x) = \log γ (x) - \log 3$ , so the defining sum is $ELBO (π) = \log 3$ . Both terms are finite and $q$ is supported on $π$ .

2.1F1F2step 1.1algebra∎

For $q = (1 / 2, 1 / 2)$ , the defining sum is $\frac{1}{2} \log 2 + \log 2 = \frac{3}{2} \log 2$ . Since $(2^{3 / 2})^{2} = 8 < 9 = 3^{2}$ and both bases are positive, $2^{3 / 2} < 3$ . The log laws give $\log 3 - \frac{3}{2} \log 2 = \frac{1}{2} \log (9 / 8) > 0$ , because $9 / 8 > 1$ and [F2] makes $\log$ strictly increasing with $\log 1 = 0$ . This density is admissible because both target weights are positive and the space is finite.

Depends on

Used by

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