TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Inverse response-probability weighting identifies the mean under MAR

Statement

Assume AC. Let $(Ω, F, P)$ be a probability space, $X$ a fully observed random element, $G = σ (X)$ , $R \in {0, 1}$ the response indicator ( $R = 1$ exactly when the real full-data outcome $Y$ is observed, under the one-outcome convention of Full and observed data under a missingness pattern), and $Y \in L^{1} (P)$ . Suppose $R$ and $Y$ are conditionally independent given $G$ , the MAR condition in the assigned one-outcome, fully observed- $X$ formulation. Let $π$ be a chosen version of $E [R ∣ G]$ and assume $π > 0$ almost surely on the target law. Define $R Y / π$ to be zero on the null set where $π = 0$ . Then $R Y / π \in L^{1} (P)$ and $E ⁣ [\frac{R Y}{π}] = E [Y] .$

Here $π$ is the true response probability, not a fitted estimate. This is a population identity; by itself it gives no finite-sample stability or estimated-weight inference result.

Facts & Assumptions

Given: AC, the probability space and random elements above, $Y \in L^{1} (P)$ , $R \in {0, 1}$ , $R ⊥ ⁣ ⁣ ⁣ ⊥ Y ∣ G$ , $π = E [R ∣ G]$ , and $π > 0$ almost surely.

[L1]

Under conditional independence, $E [R φ (Y) ∣ G] = E [R ∣ G] E [φ (Y) ∣ G]$ for every Borel $φ$ with $φ (Y)$ integrable. (Conditional-independence factorization for integrable outcomes)

[F1]

Conditional-expectation versions are measurable and integrable and have the defining event integrals. (Conditional expectation given a sigma algebra)

[F2]

Conditional expectation is positive, preserves order and constants, and preserves the unconditional expectation. (Basic algebra and order properties of conditional expectation)

[F3]

Assume AC. For $H \subseteq G$ and integrable $U$ , $E [E [U ∣ G] ∣ H] = E [U ∣ H]$ almost surely. (Tower property of conditional expectation)

[F4]

Increasing nonnegative measurable functions pass to the limit under the integral. (Monotone convergence for the integral)

[F5]

The expectation of an integrable real random variable is its Lebesgue integral. (Expectation of a nonnegative or integrable random variable)

[A1]

AC is assumed for conditional independence, conditional-expectation existence, and the tower property used here. (The Axiom of Choice, Conditional independence given a sigma-algebra, Basic algebra and order properties of conditional expectation, Tower property of conditional expectation)

Proof

1.1givenA1F1F2F4L1construct

Let $G = σ (X)$ and choose the stated version $π = E [R ∣ G]$ . By [F2], $0 \leq π \leq 1$ almost surely, so replacing it by $\min (1, \max (0, π))$ gives a $G$ -measurable version with $0 \leq π \leq 1$ everywhere. For each nonnegative $S \in {∣ Y ∣, Y^{+}, Y^{-}}$ , choose a $G$ -measurable version $U_{S}$ of $E [S ∣ G]$ , replace it by $\max (U_{S}, 0)$ , and set it equal to zero on ${π = 0}$ . Positivity [F2] and $π > 0$ almost surely show these changes are on $G$ -measurable null sets, so each $U_{S}$ remains a version by [F1]; it is pointwise nonnegative and vanishes on ${π = 0}$ . On ${π > 0}$ set $q_{m} = \min (m, 1 / π)$ and set $q_{m} = 0$ on ${π = 0}$ . For fixed $m$ , the bounded $G$ -measurable functions $q_{m, n} = 2^{- n} ⌊ 2^{n} q_{m} ⌋$ are simple, nonnegative, and increase to $q_{m}$ . For any such $S$ , [F1] gives $E [q_{m, n} R S] = E [q_{m, n} E [R S ∣ G]]$ by summing the event-integral identities over the level sets of $q_{m, n}$ . Applying [L1] with $φ (y) = ∣ y ∣$ , $y^{+}$ , or $y^{-}$ , then using [F4] on the increasing nonnegative products as $n \to \infty$ , yields $E [q_{m} R S] = E [q_{m} π U_{S}]$ .

2.1givenF1F2F3F4F5step 1.1

As $m \to \infty$ , $q_{m} R S ↑ R S / π$ under the stated zero convention. On ${π > 0}$ , $q_{m} π U_{S} = \min (m π, 1) U_{S} ↑ U_{S}$ ; on ${π = 0}$ both sides are zero by the representative chosen in step 1.1. Thus $q_{m} π U_{S} ↑ U_{S}$ pointwise. By [F4], the equalities in step 1.1 pass to these limits. The tower property [F3] with $H = {\emptyset, Ω}$ gives $E [U_{S} ∣ H] = E [S ∣ H]$ ; by the defining event-integral property [F1], conditioning on this trivial sigma-algebra yields the constant unconditional expectation. Thus $E [U_{S}] = E [S]$ , with [F5] identifying these finite expectations with their integrals. This holds for each of $S = ∣ Y ∣, Y^{+}, Y^{-}$ . In particular $E ∣ R Y / π ∣ = E ∣ Y ∣ < \infty$ .

3.1F5step 2.1algebra∎

Subtracting the finite identities for $Y^{+}$ and $Y^{-}$ from step 2.1 is legitimate by the absolute-integrability conclusion there. Since $Y = Y^{+} - Y^{-}$ , this gives $E [R Y / π] = E [Y]$ .

Remarks

The proof uses the true conditional response probability and the explicit conditional-independence hypothesis. If $π$ is estimated, or if treatment and observation are weighted jointly, a separate sampling or joint/sequential law and its own positivity and proof are required. Strict positivity alone does not bound the weights; the companion example exhibits this boundary.

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Sources

Miguel A. Hernán and James M. Robins, Causal Inference: What If, updated author-hosted PDF (standard reference, not scraped)
National Research Council, The Prevention and Treatment of Missing Data in Clinical Trials, Chapter 4 (standard reference, not scraped)
L. Mao, Missing Data: Foundations and Methods, Chapter 1 (standard reference, not scraped)