ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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 Bayesian posterior RCD indexed by the sample point

Example

Let $Θ = Y = {0, 1}$ , give each parameter prior probability $1 / 2$ , and set $K (θ, {y}) = 3 / 4$ when $y = θ$ and $1 / 4$ otherwise. On the canonical joint space $Ω = Θ \times Y$ , the posterior RCD is the random probability measure $L (ω, \cdot) = Π_{Y (ω)}$ , where $Π_{y} (y) = 3 / 4$ and $Π_{y} (1 - y) = 1 / 4$ . The RCD is evaluated at a sample point $ω$ ; the data-indexed version is the separate kernel $y \mapsto Π_{y}$ .

Facts & Assumptions

Given: The finite model and the canonical joint probability specified in the example.

[F1]

A proper Bayesian model supplies a prior probability and a parameter-measurable sampling probability kernel. Bayesian model as a prior and sampling kernel

[F2]

A posterior RCD is indexed by the sample point, while a data-indexed posterior is a kernel on the data space whose composition with $Y$ satisfies the conditional-law identity. Bayesian posterior as a regular conditional law

[F3]

The prior and sampling kernel define the joint probability on the product space. A prior and model kernel define a joint law

Verification

Given: The assumptions and finite model in the example.

1.1givenF1F3algebra

The joint law assigns mass $3 / 8$ to each diagonal point $(0, 0), (1, 1)$ and $1 / 8$ to each off-diagonal point; hence $J_{Y} (0) = J_{Y} (1) = 1 / 2$ .

2.1givenF2step 1.1algebraconstruct

For each $y$ , $Π_{y}$ has masses $3 / 4$ at $y$ and $1 / 4$ at $1 - y$ , so it is a probability, with $Π_{y} ({y}) J_{Y} (y) = 3 / 8 = J (y, y)$ and $Π_{y} ({1 - y}) J_{Y} (y) = 1 / 8 = J (1 - y, y)$ . Define $L (ω, A) = Π_{Y (ω)} (A)$ . Since the spaces are finite, both $y \mapsto Π_{y} (A)$ and $ω \mapsto L (ω, A)$ are measurable for every $A \subseteq Θ$ .

3.1F2step 1.1step 2.1algebra∎

Every event $H \in σ (Y)$ is $Θ \times B$ for some $B \subseteq Y$ . For any $A \subseteq Θ$ , $\int_{H} L (ω, A) J (d ω) = \sum_{y \in B} Π_{y} (A) J_{Y} (y) = \sum_{y \in B} J (A \times {y}) = J (H \cap {X \in A})$ . Thus $L$ is an RCD of $X$ given $σ (Y)$ , and by [F2] it is the posterior RCD.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 2 §§2.1.1–2.1.4 (standard reference, not scraped)
Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, Bayesian Data Analysis, 3rd ed., Chapter 1 §1.3 (standard reference, not scraped)