LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

The lower posterior median is a measurable rule

Statement

Let $Π$ be a posterior probability kernel on $R$ . The rule $m (y) = q_{1 / 2} (y)$ is measurable and is a posterior median on every data fiber.

Facts & Assumptions

Given: The posterior kernel and the median definition in the Statement.

[F1]

A posterior median satisfies $Π_{y} ((- \infty, m]) \geq 1 / 2$ and $Π_{y} ([m, \infty)) \geq 1 / 2$ . Posterior median

[F2]

The lower quantile is measurable and satisfies $Π_{y} ((- \infty, q_{1 / 2} (y))) \leq 1 / 2 \leq Π_{y} ((- \infty, q_{1 / 2} (y)])$ . Posterior quantiles are measurable and bracket their probability level

Proof

Given: The posterior probability kernel in the Statement.

1.1givenF2

By [F2], $m (y) = q_{1 / 2} (y)$ is measurable and $Π_{y} ((- \infty, m (y)]) \geq 1 / 2$ .

2.1F1F2step 1.1algebra∎

Since $Π_{y}$ is a probability measure, $Π_{y} ([m (y), \infty)) = 1 - Π_{y} ((- \infty, m (y))) \geq 1 / 2$ by [F2]. Thus both inequalities in [F1] hold, so $m (y)$ is a median for every $y$ .

Depends on

Used by

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

Dependency tree · two levels

18 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)