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Independent errors imply mediation cross-world independence

Statement

Assume the Axiom of Choice for the conditional-expectation conventions. In a simple Markovian mediation SCM, for every supplied treatment levels $a, a^{'}$ and mediator value $m$ , $M (a^{'}) ⊥ ⁣ ⁣ ⁣ ⊥ Y (a, m) ∣ σ (L),$ $A ⊥ ⁣ ⁣ ⁣ ⊥ M (a^{'}) ∣ σ (L), A ⊥ ⁣ ⁣ ⁣ ⊥ Y (a, m) ∣ σ (L) .$ Thus independent structural errors supply the cross-world restriction needed by the natural-effect formula, in addition to the single-world treatment-outcome, treatment-mediator, and mediator-outcome independences.

Facts & Assumptions

Given: The simple Markovian mediation SCM and the Axiom of Choice.

[A1]

The Axiom of Choice is used for the conditional-expectation almost-sure classes that define conditional independence. The Axiom of Choice

[F1]

In the simple model, $L = f_{L} (U_{L})$ , $A = f_{A} (L, U_{A})$ , $M (a^{'}) = f_{M} (a^{'}, L, U_{M})$ , and $Y (a, m) = f_{Y} (a, m, L, U_{Y})$ , with all four exogenous variables mutually independent. A simple Markovian mediation model

[F2]

Independent random elements have independent generated sigma-algebras. Independent random elements

[F3]

Conditional independence given $σ (L)$ is the bounded measurable conditional-product identity, interpreted as equality of almost-sure classes. Conditional independence given a sigma-algebra, Conditional expectation given a sigma algebra, Conditional expectation as an ae class

[F4]

The joint law of the four independent exogenous random elements is their finite product law, and bounded integrands may be integrated in either order. Independent random elements have product joint law, Fubini's theorem for L^1 functions on a sigma-finite product

[F5]

Integrating a jointly measurable function against a fixed probability kernel gives a measurable function of the remaining variables. Measurability of integration against a kernel

Proof

Proof technique: test the conditional identities against bounded functions and bounded functions of $L$ ; the structural responses use distinct independent exogenous coordinates.

Given: The hypotheses in the statement.

1.1A1F1F2F3F4F5construct

Let $g_{A}, g_{M}, g_{Y}$ be bounded measurable real functions and let $r$ be a bounded measurable function of $L$ . By [F1], each response is a measurable function of $L$ and its own exogenous coordinate. The exogenous family is independent in the sense of [F2], so its joint law is the product law [F4]; Fubini integrates $U_{A}, U_{M}, U_{Y}$ separately, giving $E [r (L) g_{A} (A) g_{M} (M (a^{'})) g_{Y} (Y (a, m))] = E [r (L) α_{A} (L) α_{M} (L) α_{Y} (L)]$ , where each $α$ is the corresponding integral over one error law and is measurable by [F5]. The same event-testing calculation with two $g$ 's set to $1$ identifies each $α$ as the appropriate conditional-expectation version; testing against every $r$ gives the triple conditional-product identity.

2.1A1F3step 1.1∎

Setting $g_{A}$ , $g_{Y}$ , and $g_{M}$ , respectively, equal to $1$ in the triple conditional-product identity of step 1.1 gives $M (a^{'}) ⊥ ⁣ ⁣ ⁣ ⊥ Y (a, m) ∣ σ (L)$ , $A ⊥ ⁣ ⁣ ⁣ ⊥ M (a^{'}) ∣ σ (L)$ , and $A ⊥ ⁣ ⁣ ⁣ ⊥ Y (a, m) ∣ σ (L)$ by the bounded-function definition [F3]. These are exactly the three claims.

Depends on

Used by

Assumptions for identifying natural effects Definition
The natural-effect mediation formula under independent errors Theorem

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Sources

Miguel A. Hernán and James M. Robins, Causal Inference: What If, Technical Point 23.1 and §23.2 (2020; author revision 19 August 2026) (standard reference, not scraped)
Judea Pearl, Causal inference in statistics: An overview, §§3.1 and 5.1.2 (Statistics Surveys 3, 2009) (standard reference, not scraped)