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 continuous density mode need not distinguish exact zero–one actions

Example

Let the posterior density on $(0, 1)$ be $p (θ) = 6 θ (1 - θ)$ , the $Beta (2, 2)$ density. Here a density mode means a maximizer of this displayed density. It has a unique density mode at $θ = 1 / 2$ , but under exact zero–one loss every point action has posterior risk one.

Facts & Assumptions

Given: $Π$ has Lebesgue density $6 θ (1 - θ)$ on $(0, 1)$ ; actions are points $a \in (0, 1)$ ; the loss is $L (θ, a) = 1_{{θ \neq a}}$ .

[F1]

Posterior risk is the posterior integral of the loss when finite. Finite posterior risk and Bayes action rule

Verification

Given: The hypotheses in the Example and Facts & Assumptions.

1.1givenalgebra

The density is normalized because $\int_{0}^{1} 6 θ (1 - θ) d θ = 6 (1 / 2 - 1 / 3) = 1$ . Also $θ (1 - θ) = 1 / 4 - (θ - 1 / 2)^{2}$ , so the density has its unique maximum at $1 / 2$ .

2.1givenF1algebra∎

Absolute continuity gives $Π ({a}) = 0$ for every point action $a$ . Hence $ρ (a) = \int 1_{{θ \neq a}} Π (d θ) = 1 - Π ({a}) = 1$ for every $a$ . All actions tie under this loss, even though the density has a unique mode.

Depends on

Finite posterior risk and Bayes action rule

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Andrew Gelman et al., Bayesian Data Analysis, 3rd ed., §9.1 (standard reference, not scraped)
Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.3.1, PDF pp. 104–105 (standard reference, not scraped)