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.

Finite ABC-SMC particles can miss target support despite a valid mixture weight

Example

Let $Θ = {0, 1}$ have the uniform prior and $Y = {0, 1, 2}$ . The simulator has $K (0, {0}) = K (0, {1}) = \frac{1}{2}$ and $K (0, {2}) = 0$ , while $K (1, {0}) = \frac{1}{4}$ , $K (1, {1}) = \frac{1}{2}$ , and $K (1, {2}) = \frac{1}{4}$ . Take $S (0) = S (1) = 0$ , $S (2) = 1$ , $s_{0} = 0$ , and $ϵ_{1} = \frac{1}{2}$ . Let the previous two-particle population be $(0, 0)$ with equal weights and use the stage-1 kernel $Q_{1} (0, {0}) = \frac{99}{100}$ , $Q_{1} (0, {1}) = \frac{1}{100}$ , with the same row from $θ = 1$ . The proposal mixture has full prior support, and its Radon–Nikodym weights are finite.

Verification

Given: The stage-1 prior, simulator, tolerance event, proposal mixture, and particle count $N = 2$ specified in the Example.

[F1] Sequential ABC importance-resampling with tolerance populations: a candidate receives weight $1_{A_{ϵ_{1}}} (Y) d Π / d ϕ_{1}^{N} (θ)$ , and a zero-total-weight population is reported as degenerate.

[F2] Sequential ABC Radon–Nikodym weights target each tolerance law: the theorem does not claim that a finite self-normalized particle population represents all positive target mass.

1.1F1givenalgebra

The proposal masses are $ϕ_{1}^{N} ({0}) = 99 / 100$ and $ϕ_{1}^{N} ({1}) = 1 / 100$ , so the uniform prior is dominated and the finite RN weights are $r_{1}^{N} (0) = 50 / 99$ and $r_{1}^{N} (1) = 50$ . Tolerance acceptance is certain at $θ = 0$ and has probability $3 / 4$ at $θ = 1$ , so $Z_{ϵ_{1}} = 1 / 2 + (1 / 2) (3 / 4) = 7 / 8$ and the unnormalized target mass at $θ = 1$ is $3 / 8$ , giving normalized mass $3 / 7$ .

2.1F1step 1.1algebra

Each proposal has parameter $0$ with probability $99 / 100$ , independently conditional on the previous population, so both have parameter $0$ with probability $(99 / 100)^{2} = 9801 / 10000$ ; on this event both weights equal $50 / 99$ and the normalized empirical measure assigns zero mass to $θ = 1$ .

3.1F1F2step 1.1step 2.1algebra∎

Step 1.1 gives true target mass $3 / 7$ at $θ = 1$ , whereas the positive-probability event in step 2.1 has no empirical mass there; any resampled offspring also stay at $θ = 0$ . This illustrates the finite-particle caveat in [F2] despite valid support and weights.

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Sources

Osvaldo A. Martin, Ravin Kumar, and Junpeng Lao, Bayesian Modeling and Computation in Python, Chapter 8 (standard reference, not scraped)
Jarno Lintusaari, Michael U. Gutmann, Ritabrata Dutta, Samuel Kaski, and Jukka Corander, Fundamentals and Recent Developments in Approximate Bayesian Computation, §5.2 (standard reference, not scraped)