CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.
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Posterior RCD exists for a standard-Borel parameter

Statement

Assume AC (The Axiom of Choice). Let a proper Bayesian model have joint prior-predictive law $J$ , and let $(Ω, F, Q)$ be any probability space carrying parameter and data random elements $X, Y$ whose joint law is $J$ (Random elements and real random variables, Law or distribution of a random element). If $(Θ, T)$ is standard Borel (Standard Borel spaces), then a posterior RCD of $X$ given $σ (Y)$ exists. The data space $(Y, Y)$ need only be measurable. In particular, the claim applies to the canonical product realization supplied by A prior and model kernel define a joint law.

Facts & Assumptions

Given: A proper Bayesian model with prior-predictive joint law $J$ , a probability space $(Ω, F, Q)$ , and random elements $X, Y$ on it with joint law $J$ . Assume AC and that $(Θ, T)$ is standard Borel.

[A1]

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

[F1]

A proper prior probability and a parameter-measurable sampling probability kernel define a joint probability $J$ on the product, with $J (A \times B) = \int_{A} K (θ, B) Π (d θ)$ . A prior and model kernel define a joint law

[F2]

A random element is a measurable map and its law is the pushforward probability on its target. Random elements and real random variables, Law or distribution of a random element

[F3]

A random element into a standard-Borel target has a regular conditional distribution given any sub-sigma-algebra, under AC. Existence of regular conditional distributions for standard borel targets

[F4]

On any probability space carrying $X, Y$ with joint law $J$ , a posterior RCD is an RCD of $X$ given $σ (Y)$ . Bayesian posterior as a regular conditional law

Proof

Given: The model, measurable spaces, and hypotheses in the statement.

1.1givenF1F2

By [F1], the model's prior and sampling kernel define the joint probability $J$ on the canonical product space. The given random elements $X, Y$ have this same joint law, and [F2] supplies the random-element and pushforward-law meanings used for this realization.

2.1givenA1F3F4step 1.1choose∎

The target $(Θ, T)$ is standard Borel and AC is assumed. Apply [F3] to $(Ω, F, Q)$ , the given random element $X$ , and the sub-sigma-algebra $G = σ (Y)$ . It gives an RCD of $X$ given $σ (Y)$ ; by [F4] this is the stated posterior RCD. The RCD theorem places no standard-Borel condition on $Y$ .

Depends on

Used by

Standard-Borel Bayesian posterior kernel exists Theorem

Dependency tree · two levels

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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)