The natural-effect mediation formula under independent errors

Write $\mathcal{A}$ for the nonempty countable treatment-value set in [F2]. Its discrete metric $d(a_1,a_2)={\mathbf{1}}_{\{a_1\mathrm{\ne }{a}_2\}}$ is complete, since every Cauchy sequence is eventually constant, and separable, since $\mathcal{A}$ itself is countable and dense. Every singleton is open, so its Borel sigma-algebra is the stipulated full power set. Thus the treatment-value space is Polish and hence standard Borel. By [F4], $\mathcal{A}\times \mathcal{L}$ and $\mathcal{A}\times \mathcal{L}\times \mathcal{M}$ are standard Borel, so the conditioning spaces of $(A,L)$ and $(A,L,M)$ satisfy the hypothesis of [F5]. By [A1] and [F4], these observed random elements admit regular conditional laws for $M$ and $Y$. By [F5], choose corresponding conditional-law kernels $Q$ and $R$ that factor through these conditioning elements. Define $Q_b^{\ast }(l,B)=P(f_M(b,l,U_M)\in B)$ and $R^{\ast }((b,l,m),C)=P(f_Y(b,m,l,U_Y)\in C)$ for Borel $B\subseteq \mathcal{M}$ and $C\subseteq \mathbb{R}$. Their evaluations are measurable by [F7], and each section is a probability measure by pushforward.

For every bounded measurable $h(A,L)$ and Borel $B\subseteq \mathcal{M}$, the product law [F6] and the equations in [F2] give $E[h(A,L){\mathbf{1}}_{\{M\in B\}}]=E[h(A,L)Q_A^{\ast }(L,B)]$, because $U_M$ is independent of $(U_L,U_A)$. For every bounded measurable $g(A,L,M)$ and Borel $C\subseteq \mathbb{R}$, the same calculation gives $E[g(A,L,M){\mathbf{1}}_{\{Y\in C\}}]=E[g(A,L,M)R^{\ast }((A,L,M),C)]$, because $U_Y$ is independent of $(U_L,U_A,U_M)$. Thus $Q^{\ast }$ and $R^{\ast }$ are versions of the observed conditional laws; [F5] makes them equal as measures outside the corresponding conditioning-law null sets. Treatment positivity makes the law of $L$ given $A=b$ equivalent to $P_L$ for each level used.

Put ${\mu }_a^{\ast }(l,m)=\int f_Y(a,m,l,u_Y)P_{U_Y}(d{u}_Y)$ where the absolute integral is finite, and put it equal to zero on the exceptional set. By [F7] this is measurable and is the conditional-mean kernel associated with $R^{\ast }$. The nested response is $Y(a,M(a^{\mathrm{\prime }}))=f_Y(a,f_M(a^{\mathrm{\prime }},L,U_M),L,U_Y)$; integrating first in $U_Y$ and then in $U_M$ using [F6] yields $E[Y(a,M(a^{\mathrm{\prime }}))]={\int }_{\mathcal{L}}{\int }_{\mathcal{M}}{\mu }_a^{\ast }(l,m)Q_{a^{\mathrm{\prime }}}^{\ast }(l,dm)P_L(dl)$. The integrability hypothesis justifies Fubini, and the change-of-variables identity [F8] gives the outer integral over $P_L$. The conditional independence of the nested mediator and outcome responses is also recorded in [F3].

By step 1.2 and [F5], $Q_{a^{\mathrm{\prime }}}^{\ast }=Q_{a^{\mathrm{\prime }}}$ for $P_L$-almost every $l$, and the conditional means from $R^{\ast }$ and $R$ agree for $P_L(dl)Q_a(l,dm)$-almost every $(l,m)$. Since $Q_{a^{\mathrm{\prime }}}(l,\cdot )\ll Q_a(l,\cdot )$ for $P_L$-almost every $l$, they also agree under $P_L(dl)Q_{a^{\mathrm{\prime }}}(l,dm)$. Thus step 2.1 becomes the observed-kernel formula, independently of the chosen RCD versions. The positive propensity and mediator support domination in [F10] ensure both version transfers on the target integration law.

The right-hand side of step 3.1 depends only on the observed law of $(A,L,M,Y)$, so by [F9] it identifies $\theta (a,a^{\mathrm{\prime }})$. Applying the formula to $(a,a^{\mathrm{\prime }})$, $(a^{\mathrm{\prime }},a^{\mathrm{\prime }})$, and $(a,a)$ identifies the two natural-effect contrasts in [F1]. Structural recursion gives $Y(a,M(a))=Y^a$ and $Y(a^{\mathrm{\prime }},M(a^{\mathrm{\prime }}))=Y^{a^{\mathrm{\prime }}}$; adding the contrasts cancels $\theta (a,a^{\mathrm{\prime }})$ and yields $\tau (a,a^{\mathrm{\prime }})$. No regression coefficient was used as a substitute for the target functional.