CounterexampleConstruction: AI-generatedVerification: 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 high observed weight ESS can miss target mass

Statement refuted

If a finite importance sample has weight-based effective sample size $n$ , then the sample has visited every region to which the target assigns positive probability.

Facts & Assumptions

Given: The claim in “Statement refuted.”

[F1]

Weight-based effective sample size: For nonnegative weights with positive sum, ${ESS}_{w} = (\sum_{i} w_{i})^{2} / \sum_{i} w_{i}^{2}$ .

[F2]

Self-normalized importance sampling: The input $X_{1}, \dots, X_{n}$ is IID from the proposal law $Q$ , and the unnormalized importance weight at a draw is $W = \tilde{p} (X) / q (X)$ when $q (X) > 0$ .

Counterexample

1.1givenF2algebra

Equip $X = {a, b}$ with counting measure $μ$ , set $\tilde{p} (a) = \tilde{p} (b) = 1$ , and let $q (a) = 99 / 100$ and $q (b) = 1 / 100$ . Then $Z_{p} = 2$ and $π (b) = 1 / 2$ , and the proposal law $Q (d x) = q (x) μ (d x)$ has full target support. For any $n \geq 1$ , take the IID $Q$ -sample $X_{1}, \dots, X_{n}$ from [F2]. The event $E_{n} = {X_{1} = \dots = X_{n} = a}$ has probability $(99 / 100)^{n} > 0$ .

2.1F1F2step 1.1algebra∎

On $E_{n}$ , all observed weights equal $W (a) = 100 / 99 > 0$ . By [F1], ${ESS}_{w} = (n W (a))^{2} / (n W (a)^{2}) = n$ , while no sample is at $b$ and the target assigns $b$ probability $1 / 2$ . Thus the claimed certification fails on an event of positive probability.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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