ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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

Example

Let $Θ = {0, 1}$ have the uniform prior and let $Y = {0, 1, 2}$ . The simulator probabilities are $K (0, {0}) = \frac{1}{2}, K (0, {1}) = \frac{1}{2}, K (0, {2}) = 0,$ $K (1, {0}) = \frac{1}{4}, K (1, {1}) = \frac{1}{2}, K (1, {2}) = \frac{1}{4} .$ Take summary $S (0) = S (1) = 0$ , $S (2) = 1$ , observed summary $s_{0} = 0$ , distance $∣ s - s_{0} ∣$ , and positive tolerance $ϵ = \frac{1}{2}$ . Thus the accepted data set is ${0, 1}$ . Use the independence simulator proposal $Q ((θ, y), \cdot) = H (\cdot)$ , where $H = Π K$ is the prior-simulator law.

Verification

Given: The finite prior, simulator, summary, tolerance, and independence proposal specified in the Example.

[F1] Augmented ABC tolerance target: the augmented target is ${\tilde{Π}}_{ϵ} (C) = Z_{ϵ}^{- 1} \int_{C} 1_{A_{ϵ}} (y) H (d θ, d y)$ when $Z_{ϵ} > 0$ .

[F2] The augmented ABC target has the tolerance posterior as its parameter marginal: the parameter marginal of the augmented target is the tolerance-conditioned ABC law.

[F3] ABC Metropolis–Hastings on simulator-augmented states: ABC-MH applies the general Metropolis–Hastings accepted-flow construction to this target and the simulator-based proposal, keeping rejected proposals as self-transitions.

[F4] ABC Metropolis–Hastings preserves its augmented tolerance target: the resulting kernel is reversible with respect to and leaves invariant this fixed-tolerance target.

[F5] Metropolis–Hastings kernel: the general accepted-flow procedure uses the common part of the forward and transposed proposal flows and retains rejected proposals as self-transitions.

1.1F1givenalgebra

The accepted prior-simulator masses are $H (0, 0) = 1 / 4$ , $H (0, 1) = 1 / 4$ , $H (1, 0) = 1 / 8$ , and $H (1, 1) = 1 / 4$ , while $(1, 2)$ has mass $1 / 8$ outside the tolerance; hence $Z_{ϵ} = 7 / 8$ .

2.1F1F2step 1.1algebra

Dividing those four accepted masses by $7 / 8$ gives augmented masses $(2 / 7, 2 / 7, 1 / 7, 2 / 7)$ and, by [F2], parameter marginal $(4 / 7, 3 / 7)$ .

3.1F2step 2.1algebra

Given exact data $Y = 0$ , the prior-weighted likelihood masses are $1 / 4$ at $θ = 0$ and $1 / 8$ at $θ = 1$ , so the exact posterior is $(2 / 3, 1 / 3)$ , different from $(4 / 7, 3 / 7)$ because the tolerance summary merges $Y = 0$ with $Y = 1$ .

4.1F1F3F4F5step 1.1step 2.1algebra∎

For accepted current and proposed states, the forward flow is ${\tilde{Π}}_{ϵ} (d x) H (d y) = \frac{8}{7} H (d x) H (d y)$ , symmetric under swap, so [F5] accepts; for proposal $(1, 2)$ the transposed flow is zero and the proposal is rejected. Thus $P (x, B) = H (B \cap A_{ϵ}) + \frac{1}{8} 1_{B} (x) = \frac{7}{8} {\tilde{Π}}_{ϵ} (B) + \frac{1}{8} 1_{B} (x)$ for accepted $x$ , and integration gives ${\tilde{Π}}_{ϵ} P (B) = {\tilde{Π}}_{ϵ} (B)$ , verifying stationarity consistently with [F4].

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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 (standard reference, not scraped)
Benjamin M. Turner and Thomas Van Zandt, A tutorial on approximate Bayesian computation, §3.1 (standard reference, not scraped)