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Inverse probability weighting identifies a causal mean

Statement

Assume AC for the conditional-expectation existence convention. Let $P$ be a full-data law with a discrete or atomic treatment $A$ , covariate random element $L$ , observed outcome $Y$ , and potential outcomes $(Y^{b})_{b \in A}$ with the countably discrete convention given by Potential outcomes and consistency. Fix an atom $a$ with $P (A = a) > 0$ and $Y^{a} \in L^{1} (P)$ . Assume consistency $Y = Y^{A}$ , conditional exchangeability $(Y^{b})_{b \in A} ⊥ ⁣ ⁣ ⁣ ⊥ A ∣ σ (L)$ , and positivity $e_{a} (L) : = P (A = a ∣ σ (L)) > 0$ almost surely. For $I_{a} = 1_{{A = a}}$ and the observed law $Q$ of $(A, L, Y)$ , the inverse-probability-weighted functional of Inverse probability weighting for a causal mean is well-defined and satisfies $E_{Q} ⁣ [\frac{I_{a} Y}{e_{a} (L)}] = E_{P} [Y^{a}] .$ Consequently, it identifies the population mean of $Y^{a}$ in the sense of Identifiability of an estimand. No uniform lower bound on $e_{a}$ is assumed. This is a population identity for the true propensity, not a finite-sample stability or estimated-weight result.

For countably discrete treatments, the supplied potential outcomes and global consistency use Potential outcomes and consistency; conditional exchangeability is the full-family sigma-algebra condition of Conditional exchangeability of potential outcomes. More generally, for a treatment in any measurable space with measurable singleton ${a}$ , the same conclusion holds whenever the supplied integrable $Y^{a}$ is conditionally independent of $A$ given $σ (L)$ and $1_{{A = a}} Y = 1_{{A = a}} Y^{a}$ almost surely, with the same positive propensity. This atom-consistency formulation does not require a global measurable selection $Y^{A}$ outside the target atom.

Facts & Assumptions

Given: AC, a probability space under $P$ , a fixed treatment atom $a$ , $Y^{a} \in L^{1} (P)$ , consistency $Y = Y^{A}$ , conditional exchangeability $(Y^{b})_{b \in A} ⊥ ⁣ ⁣ ⁣ ⊥ A ∣ σ (L)$ , and $e_{a} (L) > 0$ almost surely.

[A1]

The point-treatment setup uses potential outcomes and consistency as in Potential outcomes and consistency. For the full-family formulation, assume $Y = Y^{A}$ , integrability of $Y^{a}$ , and conditional independence of the potential-outcome family from $A$ given $σ (L)$ . For the general fixed-atom formulation, assume the atom-consistency and coordinate independence specified in the Statement; those are exactly the two consequences used below.

[A2]

The target is the causal mean relative to the causal model and observed law described by Identifiability of an estimand.

[F1]

The IPW functional is $E_{Q} [I_{a} Y / e_{a} (L)]$ , where $e_{a} = E_{Q} [I_{a} ∣ σ (L)]$ and the weighted variable is required to be integrable. (Inverse probability weighting for a causal mean)

[F2]

For integrable $Z ⊥ ⁣ ⁣ ⁣ ⊥ A ∣ G$ , $E [1_{{A \in C}} Z ∣ G] = E [1_{{A \in C}} ∣ G] E [Z ∣ G]$ . (Conditional treatment-indicator factorization)

[F3]

A bounded $G$ -measurable factor can be taken outside conditional expectation. (Taking out what is known)

[F4]

Integrals of a nondecreasing sequence of nonnegative measurable functions converge to the integral of their pointwise limit. (Monotone convergence for the integral)

[F5]

Dominated convergence gives $L^{1}$ convergence under an integrable majorant and convergence of the corresponding integrals. (Dominated convergence)

[F6]

Conditional expectation preserves the expectation when conditioning down to the trivial sigma-algebra. (Tower property of conditional expectation)

[F7]

An estimand is a specified function on a statistical model. (Estimands on a statistical model)

[F8]

Equal observed sampling laws must give equal values of an identifiable estimand. (Identifiability of an estimand)

[F9]

Identifiability is equivalent to a unique factorization through the statistical model map. (Identifiability is constancy on observational classes)

[F10]

Conditional expectations are almost-sure classes of integrable measurable versions under AC. (Conditional expectation as an ae class)

[F11]

The expectation of an event indicator is its probability: $E [1_{B}] = P (B)$ . (The expectation of an indicator is the probability of the event)

[F12]

Conditional independence is preserved by measurable maps of either variable. Thus full-family exchangeability implies independence of $Y^{a}$ and $A$ given $G$ , and independence of $Y^{a}$ implies that of $∣ Y^{a} ∣$ . (Conditional-independence equivalences and preservation)

[F13]

For a measurable function of a random element, expectation equals its integral against that element's law; integrability of the composite implies integrability under the law. (Change of variables for expectation)

[F14]

Conditional-expectation versions of the same integrable variable agree almost surely. (Conditional expectation is unique almost surely)

Proof

1.1givenA1F1F6F10F11F13F14

Put $G = σ (L)$ , $I_{a} = 1_{{A = a}}$ , $e_{a} = E_{P} [I_{a} ∣ G]$ , $μ_{a} = E_{P} [Y^{a} ∣ G]$ , and $H_{n} = \min (n, 1 / e_{a})$ on ${e_{a} > 0}$ , setting $H_{n} = 0$ on the null complement. Let $λ = P_{L} = Q_{L}$ be the common covariate law. Since $G = σ (L) = {L^{- 1} (B) : B \in S}$ , each real $G$ -measurable version $V$ factors as $v (L)$ : for each rational $r$ choose $B_{r} \in S$ with ${V < r} = L^{- 1} (B_{r})$ , and put $v (x) = \inf {r \in Q : x \in B_{r}}$ , replacing nonfinite values by $0$ . Its rational sublevel sets make $v$ measurable, and $v (L) = V$ . Apply this to versions of the $P$ - and $Q$ -conditional propensities to obtain functions ${\overset{ˉ}{e}}_{a}$ and ${\tilde{e}}_{a}$ . By [F13] both are integrable under $λ$ . For every measurable set $B$ in that space, $\int_{B} {\tilde{e}}_{a} d λ = Q (A = a, L \in B) = E_{P} [I_{a} 1_{{L \in B}}] = \int_{B} {\overset{ˉ}{e}}_{a} d λ .$ The difference is integrable and has zero integral over every $B$ , so it and zero are conditional-expectation versions of zero given the full sigma-algebra on $(L, S, λ)$ . Uniqueness [F14] gives ${\tilde{e}}_{a} = {\overset{ˉ}{e}}_{a}$ $λ$ -almost everywhere. Thus the propensity used by [F1] agrees with the full-data propensity. By [F6] and [F11], $P (A = a) = E_{P} [I_{a}] = E_{P} [e_{a}]$ . Then $H_{n}$ is bounded and $G$ -measurable, $H_{n} ↑ 1 / e_{a}$ almost surely, and $I_{a} Y = I_{a} Y^{a}$ almost surely by global consistency or the stated atom-consistency. The conditional expectations exist by AC.

2.1A1F2F3F4F5F12step 1.1

Full-family exchangeability, or the stated coordinate independence directly, gives $Y^{a} ⊥ ⁣ ⁣ ⁣ ⊥ A ∣ G$ . By [F12], this also holds for $∣ Y^{a} ∣$ , so [F2] gives $E [I_{a} ∣ Y^{a} ∣ ∣ G] = e_{a} E [∣ Y^{a} ∣ ∣ G]$ . For each $n$ , taking out the bounded factor $H_{n}$ yields $E [I_{a} ∣ Y^{a} ∣ H_{n}] = E [e_{a} H_{n} E [∣ Y^{a} ∣ ∣ G]]$ . Monotone convergence [F4] on the left and dominated convergence [F5] on the right, since $0 \leq e_{a} H_{n} ↑ 1$ and $E [∣ Y^{a} ∣ ∣ G] \in L^{1}$ , give $E [I_{a} ∣ Y^{a} ∣ / e_{a}] = E ∣ Y^{a} ∣ < \infty$ . Thus the observed weighted outcome is integrable.

3.1F2F3F5step 2.1

By [F3], [F2] also gives $E [I_{a} Y^{a} ∣ G] = e_{a} μ_{a}$ . For each $n$ , $E [I_{a} Y^{a} H_{n}] = E [e_{a} H_{n} μ_{a}]$ . The left sides converge to $E [I_{a} Y^{a} / e_{a}]$ by [F5] with integrable majorant $I_{a} ∣ Y^{a} ∣ / e_{a}$ from step 2.1; the right sides converge to $E [μ_{a}]$ by [F5] with majorant $∣ μ_{a} ∣$ . Hence $E [I_{a} Y^{a} / e_{a}] = E [μ_{a}]$ .

4.1A1F1F6F13step 1.1step 2.1step 3.1

By step 1.1 choose the common covariate-law representative ${\overset{ˉ}{e}}_{a}$ and set the observed-coordinate ratio to zero on its $λ$ -null nonpositive set. The change-of-variables identity [F13], first for the absolute value and then for the integrable signed function, shows that the Q-integrand in [F1] and its P observed-coordinate version have the same integral. Their common absolute integral is $E_{P} [I_{a} ∣ Y^{a} ∣ / e_{a}]$ , finite by step 2.1 and $I_{a} Y = I_{a} Y^{a}$ almost surely, so [F1] is well-defined. The same identity, atom-consistency, step 3.1, and the tower property [F6] give $E_{Q} [I_{a} Y / e_{a} (L)] = E_{P} [I_{a} Y^{a} / e_{a}] = E_{P} [Y^{a}]$ .

5.1A2F1F7F8F9step 1.1step 4.1construct∎

Let $Θ_{a}$ be the class of full-data laws on the fixed measurable spaces for the variables in the applicable formulation of the Statement that satisfy its integrability, consistency, conditional-exchangeability (or atom-specific independence), and positivity assumptions. Define $ψ_{a} (P) = E_{P} [Y^{a}]$ and the observation map $O (P) = L_{P} (A, L, Y) = Q$ , the pushforward of $P$ under $(A, L, Y)$ . Step 4.1 proves for every $P \in Θ_{a}$ that $ψ_{a} (P) = Ψ_{I P W} (O (P); a)$ . This factor is a function of the observed law alone, with its conditional propensity class determined by that law as in step 1.1; [F7]–[F9] therefore give identifiability on this model.

Remarks

Strict positivity is required at each target covariate stratum but need not be bounded away from zero. Near-positivity can make the integrand very variable, as the examples page shows. No finite-sample stability, estimated-propensity consistency, or validity after weight truncation follows from this identity. Continuous-treatment density weights are outside this theorem's scope.

Depends on

Used by

A two-stratum standardization and IPW calculation Example
Extreme IPW weights under poor overlap Example
Weight truncation changes the IPW target Example

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Sources

Hernán and Robins, Causal Inference: What If, updated 2026 author-hosted edition, Chapter 2, Technical Points 2.2–2.3 and Chapter 3 (standard reference, not scraped)
The Mathematics of Indirect Treatment Comparisons, Chapters 9–10, inverse-probability weighting proof (standard reference, not scraped)