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 augmented ABC target has the tolerance posterior as its parameter marginal

Statement

Under the hypotheses of Augmented ABC tolerance target with $Z_{ϵ} > 0$ , the $Θ$ -marginal of ${\tilde{Π}}_{ϵ}$ is the tolerance-conditioned law $Π_{ϵ} (\cdot ∣ s_{0})$ defined in Approximate Bayesian computation and the tolerance-conditioned target.

Facts & Assumptions

Given: A prior probability $Π$ , simulator probability kernel $K$ , measurable summary $S$ , observed summary $s_{0}$ , and $ϵ \geq 0$ with $Z_{ϵ} = p_{ϵ} (s_{0}) > 0$ .

[F1]

Augmented ABC tolerance target: $H (C) = \int K (θ, C_{θ}) Π (d θ)$ , $A_{ϵ} = S^{- 1} (B_{ϵ} (s_{0}))$ , and ${\tilde{Π}}_{ϵ} (C) = Z_{ϵ}^{- 1} \int_{C} 1_{A_{ϵ}} (y) d H$ .

[F2]

Approximate Bayesian computation and the tolerance-conditioned target: with $M$ the law of $(θ, S (Y))$ under the prior-simulator joint law, $M (A \times D) = \int_{A} K (θ, S^{- 1} (D)) Π (d θ)$ , $p_{ϵ} (s_{0}) = M (Θ \times B_{ϵ} (s_{0}))$ , and $Π_{ϵ} (A ∣ s_{0}) = M (A \times B_{ϵ} (s_{0})) / p_{ϵ} (s_{0})$ .

[F3]

A prior and model kernel define a joint law: for measurable $A \subseteq Θ$ and $D \subseteq Y$ , $H (A \times D) = \int_{A} K (θ, D) Π (d θ)$ .

Proof

Given: The hypotheses in the Statement.

1.1F1F3given

For each $A \in T$ , [F1] and the rectangle formula [F3] give ${\tilde{Π}}_{ϵ} (A \times Y) = Z_{ϵ}^{- 1} \int_{A} K (θ, A_{ϵ}) Π (d θ)$ .

2.1F1F2step 1.1algebra∎

Since $A_{ϵ} = S^{- 1} (B_{ϵ} (s_{0}))$ , [F2] gives $M (A \times B_{ϵ} (s_{0})) = \int_{A} K (θ, A_{ϵ}) Π (d θ)$ and $p_{ϵ} (s_{0}) = Z_{ϵ}$ ; substitution into [F2] makes step 1.1 exactly $Π_{ϵ} (A ∣ s_{0})$ for every measurable $A$ , so the parameter marginal is the tolerance posterior.

Depends on

Used by

A positive-tolerance ABC-MH stationary law can differ from the exact-data posterior Example

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

Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander, Fundamentals and Recent Developments in Approximate Bayesian Computation, §4 (standard reference, not scraped)
Osvaldo A. Martin, Ravin Kumar, and Junpeng Lao, Bayesian Modeling and Computation in Python, Chapter 8 (standard reference, not scraped)