TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Standard-Borel Bayesian posterior kernel exists

Statement

Assume AC (The Axiom of Choice). In a proper Bayesian model, suppose both parameter and data spaces $(Θ, T)$ and $(Y, Y)$ are standard Borel (Standard Borel spaces). Let $J$ be the joint prior-predictive law and let $J_{Y}$ be its data marginal. There is a data-indexed posterior probability kernel $y \mapsto Π_{y}$ such that, for all $A \in T$ and $B \in Y$ , $J (A \times B) = \int_{B} Π_{y} (A) J_{Y} (d y), J_{Y} (B) = \int_{Θ} K (θ, B) Π (d θ) .$ The composition $Π_{Y (ω)}$ is a posterior RCD. Any two data-indexed posterior kernels agree as probability measures outside one measurable $J_{Y}$ -null set. On any measurable $J_{Y}$ -null set, the conditional-law identities do not select a pointwise value: replacing the kernel there by the prior $Π$ preserves the identities.

Facts & Assumptions

Given: A proper Bayesian model, its prior-predictive joint law $J$ , the parameter and data coordinate maps $X, Y$ , AC, and standard-Borel parameter and data spaces.

[A1]

The Axiom of Choice (AC). The Axiom of Choice

[F1]

The prior and parameter-measurable sampling kernel define the joint probability $J$ and its rectangle law. A prior and model kernel define a joint law

[F2]

Coordinate projections have measurable inverse images because they are measurable rectangles; a measurable map from a probability space is a random element. The product sigma-algebra and its finite iterates, A measurable function between measurable spaces, Random elements and real random variables

[F3]

A posterior RCD exists when the parameter space is standard Borel, with no standard-Borel requirement on the data space. Posterior RCD exists for a standard-Borel parameter

[F4]

If both random-element targets are standard Borel and AC holds, an RCD of $X$ given $σ (Y)$ factors through $Y$ as a probability kernel on the data space. Regular conditional kernels factor through a standard borel conditioning variable

[F5]

A data-indexed probability kernel is a conditional law precisely when its composition with $Y$ satisfies the conditional event identities. Conditional law given a random element

[F6]

A joint probability on a product of standard-Borel spaces admits a disintegration kernel, unique outside one null set of its second marginal. Disintegration of a joint law on standard borel spaces

Proof

Given: The model and hypotheses in the statement.

1.1givenF1F2construct

By [F1], $J$ is a probability measure on $(Θ \times Y, T \otimes Y)$ and $J_{Y} (B) = J (Θ \times B) = \int_{Θ} K (θ, B) Π (d θ)$ . Its coordinate maps $X$ and $Y$ are measurable by [F2].

2.1givenA1F3step 1.1choose

Since $Θ$ is standard Borel and [A1] holds, [F3] supplies a posterior RCD $L$ of $X$ given $σ (Y)$ on the canonical space of step 1.1.

3.1givenA1F4F5step 2.1

Both $Θ$ and $Y$ are standard Borel and [A1] holds, so [F4] factors $L$ through $Y$ as a probability kernel $Π_{y}$ ; hence its evaluations are $Y$ -measurable and every section is a probability. By [F5], the composition $Π_{Y}$ is a posterior RCD, with rectangle identity $J (A \times B) = \int_{B} Π_{y} (A) J_{Y} (d y)$ .

4.1F1F5F6step 1.1step 3.1∎

Apply [F6] to the same joint probability $J$ . It gives uniqueness of the disintegration kernel as a probability measure outside one $J_{Y}$ -null set. Replacing a version on any measurable $J_{Y}$ -null set by the prior $Π$ preserves every rectangle identity, as [F5] records. Thus the conditional identities alone do not choose a pointwise value there; each section must still be a probability measure.

Depends on

Used by

Dependency tree · two levels

45 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, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, Bayesian Data Analysis, 3rd ed., Chapter 1 §§1.1–1.3 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 2 (standard reference, not scraped)