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 likelihood tilt can be a finite measure of mass below one

Example

On $X = {0, 1}$ with counting measure, let $p_{0} (0) = 1 / 10$ , $p_{0} (1) = 9 / 10$ , and model densities $p_{a} (0) = 9 / 10$ , $p_{a} (1) = 1 / 10$ , $p_{b} (0) = 1$ , $p_{b} (1) = 0$ . Give $a$ and $b$ prior probability $1 / 2$ each. The density $p_{a}$ is the unique KL minimizer, while the likelihood tilt at $b$ has total mass $1 / 9$ .

Facts & Assumptions

Given: The finite model, prior, and IID law in the Example.

[F1]

KL projection sets and relative KL neighborhoods: the projection minimizes the directional contrast $K L (P_{0} ∥ P_{θ})$ .

[A1]

Assume AC through the imported KL convention. No further choice is used in this finite model. The Axiom of Choice

[F2]

Likelihood-tilted measures at a KL projection: the total likelihood-tilt mass is $E_{P_{0}} (p_{b} / p_{a})$ and is not required to equal one.

[F3]

Likelihood tilts are sigma-finite measures: the tilted set function is a measure; a finite total mass is allowed to be below one.

Verification

1.1A1F1algebra

The KL loss at $a$ is $K (a) = \frac{1}{10} \log \frac{1}{9} + \frac{9}{10} \log 9 = \frac{4}{5} \log 9 < \infty .$ At $b$ , $P_{0}$ is not absolutely continuous with respect to $P_{b}$ , so $K (b) = + \infty$ . Thus $a$ is the unique minimizer and every KL neighborhood has prior mass at least $1 / 2$ .

2.1F2F3algebra∎

The likelihood tilt at $b$ assigns mass $Q_{b} ({0}) = \frac{1}{10} \frac{1}{9 / 10} = \frac{1}{9}, Q_{b} ({1}) = \frac{9}{10} \frac{0}{1 / 10} = 0.$ Hence $Q_{b} (X) = 1 / 9 < 1$ , so $Q_{b}$ is a finite nonprobability measure.

Depends on

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

B. J. K. Kleijn and A. W. van der Vaart, The Bernstein-Von-Mises Theorem under Misspecification, §3.2, pp. 368–370 (standard reference, not scraped)