Controlled and natural mediation effects

Fix a target population, two treatment versions $a,a^{\mathrm{\prime }}$, a mediator random element $M$ in a standard-Borel space Standard Borel spaces, and a real random variable $Y$ Random elements and real random variables. Let $M(a)$ be the mediator response under $do(A=a)$ and let $Y(a,m)$ be the outcome response under the joint intervention $do(A=a,M=m)$, with the intervention semantics of Recursive structural causal models on finite DAGs. These are full-data random variables; consistency alone does not identify their joint law from observed data.

When the displayed differences are integrable Integrable real and complex functions, and their integrals, the controlled direct effect at mediator setting $m$ is ${\mathrm{CDE}}_m(a,a^{\mathrm{\prime }})=E[Y(a,m)-Y(a^{\mathrm{\prime }},m)]\mathrm{.}$ It requires that setting the mediator to $m$ be a meaningful intervention. The natural direct effect and natural indirect effect, with $a^{\mathrm{\prime }}$ the reference treatment level, are $\mathrm{NDE}(a,a^{\mathrm{\prime }})=E[Y(a,M(a^{\mathrm{\prime }}))-Y(a^{\mathrm{\prime }},M(a^{\mathrm{\prime }}))],$ $\mathrm{NIE}(a,a^{\mathrm{\prime }})=E[Y(a,M(a))-Y(a,M(a^{\mathrm{\prime }}))]\mathrm{.}$ The first argument is the active outcome treatment level and the second is the reference level that supplies the mediator response. When $a\mathrm{\ne }{a}^{\mathrm{\prime }}$, these targets involve nested counterfactuals from different intervention regimes. They are defined on a full-data law whether or not they are identified by the observed law. The total mean contrast is $\tau (a,a^{\mathrm{\prime }})=E[Y^a-Y^{a^{\mathrm{\prime }}}],$ using the treatment-only potential outcomes of Potential outcomes and consistency. For $a=1,a^{\mathrm{\prime }}=0$, this agrees with the ATE convention in Average and conditional average treatment effects.