A simple Markovian mediation model

Let $L$ be an observed baseline random element in a standard-Borel space Standard Borel spaces, $A$ a treatment random element Random elements and real random variables taking values in a nonempty countable set Finite, countably infinite, countable, uncountable equipped with the discrete sigma-algebra (the full power set), following the treatment-version convention of Potential outcomes and consistency. Let $M$ be an observed mediator random element in a standard-Borel space Standard Borel spaces, and let $Y$ be an observed real random variable Random elements and real random variables. A simple Markovian mediation SCM is a recursive structural causal model in the sense of Recursive structural causal models on finite DAGs whose endogenous variables are exactly $(L,A,M,Y)$ and whose equations have the form $L=f_L(U_L),A=f_A(L,U_A),M=f_M(A,L,U_M),Y=f_Y(A,M,L,U_Y)\mathrm{.}$ The functions are jointly measurable in the sense of A measurable function between measurable spaces, and the exogenous random elements $U_L,U_A,U_M,U_Y$ are mutually independent Independent random elements, as required by the Markovian condition in that SCM definition. The graph therefore has no latent common cause outside $L$ and no additional post-treatment variable that causes both $M$ and $Y$. This model is a special case of an NPSEM-IE (a nonparametric structural equation model with independent errors); it is a substantive model restriction, not a consequence of the observed distribution. It is a single-unit model; if units interfere, the treatment version must include the relevant joint assignment or exposure, as required by Potential outcomes and consistency.

The mediator and outcome responses are the recursive responses under the interventions from Controlled and natural mediation effects: $M(a)=f_M(a,L,U_M),Y(a,m)=f_Y(a,m,L,U_Y)\mathrm{.}$ In particular, recursion gives $Y^a=Y(a,M(a))$. The model's absence of an additional post-treatment common cause is part of the hypothesis; if such a variable is present, this definition does not cover that causal system.