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 posterior median need not be unique

Example

Let $Π = \frac{1}{2} δ_{0} + \frac{1}{2} δ_{1}$ . Every $m \in [0, 1]$ is a posterior median and has the same absolute posterior risk, $1 / 2$ .

Facts & Assumptions

Given: $Π ({0}) = Π ({1}) = 1 / 2$ and there is no other posterior mass.

[F1]

A posterior median has at least half the mass on each closed side. Posterior median

[F2]

Every posterior median minimizes absolute posterior risk when the first absolute moment is finite. Every posterior median minimizes absolute risk when the first absolute moment is finite

Verification

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

1.1givenF1

For $m \in [0, 1]$ , $Π ((- \infty, m]) \geq 1 / 2$ and $Π ([m, \infty)) \geq 1 / 2$ , so [F1] makes every such $m$ a median.

2.1givenF2algebra∎

For $m \in [0, 1]$ , $ρ (m) = \frac{1}{2} ∣ m ∣ + \frac{1}{2} ∣ 1 - m ∣ = \frac{1}{2}$ . For $a < 0$ , $ρ (a) = \frac{1}{2} (- a) + \frac{1}{2} (1 - a) = \frac{1}{2} - a > \frac{1}{2}$ ; for $a > 1$ , $ρ (a) = \frac{1}{2} a + \frac{1}{2} (a - 1) = a - \frac{1}{2} > \frac{1}{2}$ . Thus precisely the interval $[0, 1]$ minimizes risk, in agreement with [F2].

Depends on

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

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.3.1, PDF pp. 104–105 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 8 §8.1 (standard reference, not scraped)