Separable interventionist mediation effects

Suppose an expanded structural causal model decomposes a treatment into two substantively meaningful components $N$ and $O$ that can in principle be set separately. Write $M(n,o)$ and $Y(n,o)$ for the mediator and outcome under $do(N=n,O=o)$, and $Y(n,o,m)$ for the outcome under the additional intervention $do(M=m)$, using the intervention semantics of Recursive structural causal models on finite DAGs. Here $Y(n,o)$ leaves the mediator equation active. Mixed component values must correspond to meaningful intervention protocols; an arbitrary mathematical relabeling of $A$ is not a component decomposition. For reference component levels $n,n^{\mathrm{\prime }},o,o^{\mathrm{\prime }}$, the separable component contrasts are ${\mathrm{SE}}_N(n,n^{\mathrm{\prime }};o)=E[Y(n,o)-Y(n^{\mathrm{\prime }},o)],$ ${\mathrm{SE}}_O(o,o^{\mathrm{\prime }};n)=E[Y(n,o)-Y(n,o^{\mathrm{\prime }})],$ whenever the contrasts are integrable Integrable real and complex functions, and their integrals. They refer to feasible component interventions, not to the nested individual mediator value $M(a^{\mathrm{\prime }})$ in a natural effect.

For the simple single-mediator structural system, the relevant dismissible-component restrictions are the structural exclusions $M(n,o)=M(n),Y(n,o,m)=Y(n^{\mathrm{\prime }},o,m)\text{for all }n,n^{\mathrm{\prime }},o,m\mathrm{.}$ The first says that the outcome-only component $O$ has no effect on the mediator. The second says that the mediator-only component $N$ has no direct effect on the outcome when the mediator is fixed. In other causal systems the formal dismissible-component conditions are model-specific. Meaningful component interventions, consistency, positivity, and appropriate exchangeability conditions are required before these contrasts can be identified. No identification theorem for separable effects is asserted here; they define a different target from the natural direct and indirect effects in Controlled and natural mediation effects.