TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge 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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ABC Metropolis–Hastings preserves its augmented tolerance target

Statement

Assume AC and the hypotheses of ABC Metropolis–Hastings on simulator-augmented states with $Z_{ϵ} > 0$ . Its output $P_{ϵ}$ is reversible with respect to and leaves invariant the augmented ABC tolerance target ${\tilde{Π}}_{ϵ}$ on $Θ \times Y$ . This is invariance for the fixed tolerance target only; it gives no convergence or full-data posterior conclusion.

Facts & Assumptions

Given: AC, a standard-Borel augmented state space, $ϵ \geq 0$ with $Z_{ϵ} > 0$ , the target ${\tilde{Π}}_{ϵ}$ , the simulator proposal kernel $Q$ , and the ABC-MH procedure.

[A1]

The Axiom of Choice: AC states that every family of nonempty sets has a choice function.

[F1]

Augmented ABC tolerance target: when $Z_{ϵ} > 0$ , ${\tilde{Π}}_{ϵ} = Z_{ϵ}^{- 1} h_{ϵ} H$ is a probability measure on the product of the standard-Borel parameter and data spaces.

[F2]

ABC Metropolis–Hastings on simulator-augmented states: the algorithm applies Metropolis–Hastings kernel with target ${\tilde{Π}}_{ϵ}$ and proposal probability kernel $Q$ , retaining rejected proposals as self-transitions.

[F3]

Metropolis–Hastings flow is reversible and invariant: under AC, for a standard-Borel space, probability target, and probability proposal kernel, the general Metropolis–Hastings output is reversible with respect to and leaves invariant its target.

Proof

Given: The hypotheses in the Statement.

1.1F1F2given

By [F1], ${\tilde{Π}}_{ϵ}$ is a probability target on the product state space; by [F2], the proposal is a probability kernel and the returned kernel is exactly the general Metropolis–Hastings output for this target and proposal.

2.1A1F1F2F3step 1.1∎

The state space is standard Borel and [A1] supplies AC, so [F3] applies to the target and proposal identified in step 1.1; it gives reversibility and invariance of $P_{ϵ}$ with respect to ${\tilde{Π}}_{ϵ}$ , with no irreducibility or convergence assertion.

Depends on

Used by

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

Cited to discharge well-definedness by ABC Metropolis–Hastings on simulator-augmented states.

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Sources

Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander, Fundamentals and Recent Developments in Approximate Bayesian Computation, §5 and Appendix, Algorithm 3 (standard reference, not scraped)
Benjamin M. Turner and Thomas Van Zandt, A tutorial on approximate Bayesian computation, §3.1 (standard reference, not scraped)