TheoremStatement: Literature-sourcedProof: 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 sufficient summary preserves the Bayesian posterior

Statement

Assume AC. Let $(Θ, T)$ , $(Y, Y)$ , and $(T, C)$ be standard Borel spaces (Standard Borel spaces), let $Π$ be a proper prior, and let $K$ be a sampling probability kernel as in Bayesian model as a prior and sampling kernel. Let $S : Y \to T$ be a measurable statistic (Statistics and estimators) that is sufficient in the common-kernel sense of Sufficient statistics. Write $K_{S} (θ, C) = K (θ, S^{- 1} (C))$ , let $M$ be the joint prior-predictive law of $(θ, S (Y))$ , and let $J_{Y}$ and $J_{S}$ be the data and summary marginals. A posterior kernel $Π_{t}^{S}$ of $θ$ given $S (Y) = t$ exists. Then ${\hat{Π}}_{y} (A) : = Π_{S (y)}^{S} (A)$ is a data-indexed posterior version for the full observation: for every $A \in T$ and $D \in Y$ , $J (Θ \in A, Y \in D) = \int_{D} {\hat{Π}}_{y} (A) J_{Y} (d y) .$ Thus full-data posterior information is preserved by a sufficient summary, up to the usual almost-everywhere version convention.

In a dominated branch, suppose $μ$ and $ν$ are sigma-finite reference measures for the full data and summary, respectively, and the specified jointly measurable densities satisfy $K (θ, d y) = p_{θ} (y) μ (d y), K_{S} (θ, d t) = h_{θ} (t) ν (d t), p_{θ} (y) = g (y) h_{θ} (S (y))$ on the specified common versions, with $g : Y \to [0, \infty]$ independent of $θ$ . For any $y$ where the factorization holds $Π$ -almost everywhere, $0 < g (y) < \infty$ , and $0 < Z_{S} (S (y)) : = \int_{Θ} h_{θ} (S (y)) Π (d θ) < \infty,$ the explicit full-data and summary posterior updates agree at $y$ by cancellation of $g (y)$ . No equality is asserted on zero or infinite evidence fibres.

Facts & Assumptions

Given: The model, proper prior, standard Borel spaces, AC, statistic $S$ , common conditional-data kernel from sufficiency, and, in the dominated branch, the stated reference measures and factorized density versions.

[A1]

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

[F1]

Sufficiency means there is one probability kernel $Q (t, D)$ , independent of $θ$ , satisfying $K (θ, D \cap S^{- 1} (C)) = \int_{C} Q (t, D) K_{S} (θ, d t)$ for all $θ$ , data events $D$ , and summary events $C$ . Sufficient statistics

[F2]

A proper prior and a parameter-measurable sampling kernel define a joint probability law; its rectangle law is prior mass integrated against the sampling kernel. A prior and model kernel define a joint law

[F3]

Under AC, a proper Bayesian model with standard Borel parameter and data spaces has a data-indexed posterior probability kernel satisfying the joint event identity. Standard-Borel Bayesian posterior kernel exists

[F4]

A data-indexed posterior kernel is a probability kernel whose evaluations satisfy the conditional-law event identities. Bayesian posterior as a regular conditional law, Conditional law given a random element

[F5]

Increasing nonnegative simple approximations pass through integrals by monotone convergence. Monotone convergence for the integral

[F6]

On positive finite evidence fibres, the dominated Bayes formula gives the posterior update by normalizing the likelihood against the prior. Evidence normalizer and fiberwise posterior update, Bayes formula for dominated kernels

[F7]

Measurable real functions are closed under the product operation used below. Arithmetic and lattice operations preserve measurability whenever they are defined

Proof

Given: The hypotheses in the Statement.

1.1givenA1F3F4construct

The summary simulator $K_{S} (θ, C) = K (θ, S^{- 1} (C))$ is a probability kernel because $S$ is measurable. Both its parameter space and target are standard Borel, so [A1] and [F3], applied to this summary experiment, give a posterior kernel $Π_{t}^{S}$ satisfying the joint summary-law identity. Define ${\hat{Π}}_{y} (A) = Π_{S (y)}^{S} (A)$ . This is a probability kernel in $y$ by measurability of $S$ and the posterior-kernel property.

2.1F1F2F5F7step 1.1

Let $A \in T$ , $D \in Y$ , and $C \in C$ . By [F1], for each $θ$ , $K (θ, D \cap S^{- 1} (C)) = \int_{C} Q (t, D) K_{S} (θ, d t) .$ Integrating over $A$ under $Π$ gives the joint event probability on the left. To justify the corresponding right-hand integral, the measurable function $h (t) = 1_{C} (t) Q (t, D)$ takes values in $[0, 1]$ by [F1] and [F7]. Its dyadic lower approximations $h_{n} (t) = 2^{- n} ⌊ 2^{n} h (t) ⌋$ are finite-valued simple functions: their level sets are preimages under $h$ of Borel intervals, and every positive level set is contained in $C$ . The nested dyadic grids give $h_{n} ↑ h$ , with $0 \leq h - h_{n} < 2^{- n}$ . For each such simple function, the rectangle identity for $M$ from [F2] gives $\int_{A} \int h_{n} (t) K_{S} (θ, d t) Π (d θ) = \int h_{n} (t) M (A, d t) .$ Applying [F5] on both sides as $n \to \infty$ yields $J (Θ \in A, Y \in D, S (Y) \in C) = \int_{C} Q (t, D) M (A, d t) .$ By the summary posterior identity from step 1.1, $M (A, d t) = Π_{t}^{S} (A) J_{S} (d t)$ .

3.1F1F2step 2.1

Set $C = T$ in step 2.1 to obtain $J (Θ \in A, Y \in D) = \int_{T} Q (t, D) Π_{t}^{S} (A) J_{S} (d t) .$ Taking $A = Θ$ in the same identity gives the prior-predictive conditional-data formula $J_{Y} (D) = \int Q (t, D) J_{S} (d t)$ .

4.1F4F5F7step 2.1step 3.1

Let $h (t) = Π_{t}^{S} (A)$ , a measurable function with values in $[0, 1]$ . For a simple $h = \sum_{j = 1}^{m} c_{j} 1_{C_{j}}$ with disjoint measurable $C_{j}$ , step 2.1 with $A = Θ$ gives $\int_{D} h (S (y)) J_{Y} (d y) = \sum_{j} c_{j} J_{Y} (D \cap S^{- 1} (C_{j})) = \sum_{j} c_{j} \int_{C_{j}} Q (t, D) J_{S} (d t) = \int h (t) Q (t, D) J_{S} (d t) .$ For general $h$ , use the same dyadic construction $h_{n} = 2^{- n} ⌊ 2^{n} h ⌋$ as in step 2.1 and apply [F5] on both sides. Hence $\int_{D} {\hat{Π}}_{y} (A) J_{Y} (d y) = \int Q (t, D) Π_{t}^{S} (A) J_{S} (d t) = J (Θ \in A, Y \in D),$ where the last equality is step 3.1. By [F4], $\hat{Π}$ is a posterior version for the full observation.

5.1givenF6algebra∎

In the dominated branch, on a specified good fibre $y$ let $Z_{S} = \int h_{θ} (S (y)) Π (d θ)$ . The factorization and $0 < g (y) < \infty$ give $Z_{Y} (y) : = \int p_{θ} (y) Π (d θ) = g (y) Z_{S},$ so $Z_{Y} (y)$ is also positive finite. Applying [F6] to both likelihoods gives, for every $A \in T$ , $\frac{\int_{A} p_{θ} (y) Π (d θ)}{Z_{Y} (y)} = \frac{g (y) \int_{A} h_{θ} (S (y)) Π (d θ)}{g (y) Z_{S}} = \frac{\int_{A} h_{θ} (S (y)) Π (d θ)}{Z_{S}} .$ This is the stated cancellation, and no quotient is formed on excluded evidence fibres.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

72 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, §§1.3.2 and 2.3.1 (printed pp. 14–15 and 33–34) (standard reference, not scraped)
Noah Fithian, Stat 210a Reader: Sufficiency (standard reference, not scraped)
Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander, Fundamentals and Recent Developments in Approximate Bayesian Computation (standard reference, not scraped)
Paul Fearnhead and Dennis Prangle, Constructing Summary Statistics for Approximate Bayesian Computation: Semi-automatic ABC (standard reference, not scraped)