ExampleConstruction: 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.

Ordinary and self-normalized importance sampling on a two-point space

Example

Let $X = {a, b}$ with counting measure, unnormalized target density $\tilde{p} (a) = \tilde{p} (b) = 1$ , proposal probabilities $Q (a) = 3 / 4$ and $Q (b) = 1 / 4$ , and integrand $f = 1_{{b}}$ . Then $Z_{p} = 2$ , the target probabilities are $π (a) = π (b) = 1 / 2$ , and $θ = \int f d π = 1 / 2$ . The proposal has full target support.

Facts & Assumptions

Given: The finite space, target, proposal, integrand, and normalizer in the Example.

[F1]

Ordinary importance sampling: For each proposal draw, ordinary importance sampling uses $Y_{i} = f (X_{i}) \tilde{p} (X_{i}) / (Z_{p} q (X_{i}))$ and returns their sample mean.

[F2]

Self-normalized importance sampling: When the sum of the nonnegative weights $W_{i} = \tilde{p} (X_{i}) / q (X_{i})$ is positive, the self-normalized estimate is $\sum_{i} W_{i} f (X_{i}) / \sum_{i} W_{i}$ .

[F3]

Ordinary importance sampling is unbiased: Under the stated support and integrability assumptions, the ordinary importance sampling estimate has expectation $\int f d π$ .

Verification

Given: Use one proposal draw $X \sim Q$ and the definitions in [F1] and [F2].

1.1givenF3algebra

The target normalizer is $Z_{p} = \tilde{p} (a) + \tilde{p} (b) = 2$ , so $π (a) = π (b) = 1 / 2$ and $θ = E_{π} f = 1 / 2$ . Since both proposal probabilities are positive and $X$ is finite, the support and integrability assumptions of [F3] hold.

2.1F1F3step 1.1algebra

The ordinary weighted value is $0$ at $a$ and $2$ at $b$ . Its expectation is $(3 / 4) 0 + (1 / 4) 2 = 1 / 2 = θ$ , as also guaranteed by [F3].

3.1F2step 1.1algebra∎

The self-normalized weights are $4 / 3$ at $a$ and $4$ at $b$ , so the single-draw ratio in [F2] equals $f (X)$ . Its expectation is $Q (X = b) = 1 / 4$ , and its bias for $θ$ is $1 / 4 - 1 / 2 = - 1 / 4$ . This verifies that self-normalization need not be unbiased at finite sample size.

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