ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Extreme IPW weights under poor overlap

Example

For each $ε \in (0, 1 / 2)$ , let $P (L = 0) = P (L = 1) = 1 / 2$ , let $P (A = 1 ∣ L = 0) = ε$ and $P (A = 1 ∣ L = 1) = 1 / 2$ , and set $Y^{1} = 1_{{L = 0}}$ , $Y^{0} = 0$ , and $Y = Y^{A}$ . The potential outcomes are deterministic given $L$ , so conditional exchangeability holds. For every $ε > 0$ , positivity holds; as $ε$ decreases, the treated weight in stratum $L = 0$ grows.

Facts & Assumptions

Given: The finite-stratum family in the Example, consistency, and AC for the conditional-expectation conventions.

[A1]

$Y^{1}$ is integrable in the point-treatment setup of Potential outcomes and consistency, and the true treatment probability is $e_{1} (L) = P (A = 1 ∣ L)$ .

[F1]

The IPW functional uses the population weight $I_{1} / e_{1} (L)$ and its expectation, when integrable. (Inverse probability weighting for a causal mean)

[F2]

Under the stated causal assumptions and positive propensity, the population IPW mean equals $E [Y^{1}]$ . (Inverse probability weighting identifies a causal mean)

Verification

1.1givenA1F1F2algebra

The event ${L = 0, A = 1}$ has probability $ε / 2$ and there the IPW summand $W = I_{1} Y / e_{1} (L)$ equals $1 / ε$ . Everywhere else $W = 0$ . Hence $E [W] = (ε / 2) (1 / ε) = 1 / 2 = E [Y^{1}]$ , consistently with [F2].

2.1step 1.1algebra

The second moment and variance are $E [W^{2}] = \frac{ε}{2} \frac{1}{ε^{2}} = \frac{1}{2 ε}$ and $Var (W) = \frac{1}{2 ε} - \frac{1}{4}$ . As $ε ↓ 0$ , the realized treated weight $1 / ε$ and this population variance diverge, although the exact population identity remains valid at each $ε > 0$ .

3.1givenstep 2.1∎

Thus strict positivity at each model does not impose a uniform weight or variance bound across models; the example makes no claim about finite-sample estimator consistency.

Depends on

Used by

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