[proofplan]
We separate two kinds of information. The front-door formula identifies total-effect interventional distributions using single-world intervention quantities, while natural mediation effects require a joint coupling between counterfactuals from incompatible intervention worlds. We construct two binary counterfactual couplings with the same single-intervention marginal laws and the same total-effect laws under both treatment levels, but with different values of the nested counterfactual $\mathbb E[Y_{1,M_0}]$. This proves that the front-door-identified information alone cannot determine the natural mediation component.
[/proofplan]
[step:Fix the single-intervention marginal information shared by both couplings]
Let
\begin{align*}
\Omega:=\{0,1\}^3
\end{align*}
and let $\mathcal F:=2^\Omega$. Define $\mathbb P:\mathcal F\to[0,1]$ to be the uniform probability measure on $\Omega$. Let
\begin{align*}
W,V,R:(\Omega,\mathcal F)\to(\{0,1\},2^{\{0,1\}})
\end{align*}
be the coordinate maps. Then $W,V,R$ are independent binary random variables satisfying
\begin{align*}
\mathbb P(W=1)=\mathbb P(V=1)=\mathbb P(R=1)=\frac12.
\end{align*}
In both counterfactual couplings define mediator potential outcomes
\begin{align*}
M_0:=W
\end{align*}
and
\begin{align*}
M_1:=V.
\end{align*}
Thus
\begin{align*}
\mathbb P(M_0=1)=\mathbb P(M_1=1)=\frac12.
\end{align*}
We now define two joint laws for the outcome counterfactuals $Y_{a,m}$.
For the first coupling, define
\begin{align*}
Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=R,\quad Y_{1,1}:=R.
\end{align*}
For the second coupling, define
\begin{align*}
Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=1-W,\quad Y_{1,1}:=W.
\end{align*}
In both couplings, for every $a,m\in\{0,1\}$,
\begin{align*}
\mathbb P(Y_{a,m}=1)=\frac12.
\end{align*}
Indeed, this is immediate for the variables equal to $R$, and in the second coupling it also holds for $Y_{1,0}=1-W$ and $Y_{1,1}=W$ because $W$ has the Bernoulli law with parameter $1/2$. Therefore the two couplings agree on the marginal laws of every $M_a$ and every $Y_{a,m}$.
[guided]
The point of the construction is to keep fixed the quantities that single-intervention front-door calculations can see, while changing how variables from different intervention worlds are coupled. We use the finite [probability space](/page/Probability%20Space)
\begin{align*}
\Omega:=\{0,1\}^3
\end{align*}
with $\mathcal F:=2^\Omega$ and uniform probability measure $\mathbb P$. The coordinate maps
\begin{align*}
W,V,R:(\Omega,\mathcal F)\to(\{0,1\},2^{\{0,1\}})
\end{align*}
are independent binary random variables, each taking the value $1$ with probability $1/2$ and the value $0$ with probability $1/2$.
In both couplings we define the mediator potential outcomes by
\begin{align*}
M_0:=W
\end{align*}
and
\begin{align*}
M_1:=V.
\end{align*}
Thus the marginal mediator laws are fixed:
\begin{align*}
\mathbb P(M_0=1)=\mathbb P(W=1)=\frac12
\end{align*}
and
\begin{align*}
\mathbb P(M_1=1)=\mathbb P(V=1)=\frac12.
\end{align*}
For the first coupling, every outcome-response variable is the same Bernoulli variable $R$:
\begin{align*}
Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=R,\quad Y_{1,1}:=R.
\end{align*}
For the second coupling, the untreated-outcome response variables remain equal to $R$, but the treated-outcome response variables are tied to $W$:
\begin{align*}
Y_{0,0}:=R,\quad Y_{0,1}:=R,\quad Y_{1,0}:=1-W,\quad Y_{1,1}:=W.
\end{align*}
Every one of these outcome-response variables takes the value $1$ with probability $1/2$. For $R$ this follows from $\mathbb P(R=1)=1/2$. For $W$ it follows from $\mathbb P(W=1)=1/2$. For $1-W$, we compute
\begin{align*}
\mathbb P(1-W=1)=\mathbb P(W=0)=\frac12.
\end{align*}
Hence, for every $a,m\in\{0,1\}$, both couplings satisfy
\begin{align*}
\mathbb P(Y_{a,m}=1)=\frac12.
\end{align*}
So the single-intervention marginal information is identical in the two constructions. The only thing we have changed is the cross-world dependence structure.
[/guided]
[/step]
[step:Verify that the total-effect laws are unchanged]
For treatment $0$, both couplings satisfy $Y_{0,0}=Y_{0,1}=R$. Since $M_0$ is $\{0,1\}$-valued, both couplings have
\begin{align*}
Y_{0,M_0}=R.
\end{align*}
Thus the total-effect law of $Y_{0,M_0}$ is the law of $R$ in both couplings.
In the first coupling,
\begin{align*}
Y_{1,M_1}=R
\end{align*}
because $Y_{1,0}=Y_{1,1}=R$. Therefore
\begin{align*}
\mathbb E[Y_{1,M_1}]=\mathbb E[R]=\frac12.
\end{align*}
In the second coupling, since $M_1=V$, $Y_{1,0}=1-W$, and $Y_{1,1}=W$, we have
\begin{align*}
Y_{1,M_1}=(1-V)(1-W)+VW.
\end{align*}
Using independence of $V$ and $W$,
\begin{align*}
\mathbb E[Y_{1,M_1}]=\mathbb E[(1-V)(1-W)]+\mathbb E[VW].
\end{align*}
The two terms are
\begin{align*}
\mathbb E[(1-V)(1-W)]=\mathbb P(V=0,W=0)=\frac14
\end{align*}
and
\begin{align*}
\mathbb E[VW]=\mathbb P(V=1,W=1)=\frac14.
\end{align*}
Hence
\begin{align*}
\mathbb E[Y_{1,M_1}]=\frac12.
\end{align*}
Thus the two couplings agree on the total-effect law of $Y_{1,M_1}$, because $Y_{1,M_1}$ is binary and its law is determined by the common expectation $\mathbb E[Y_{1,M_1}]=\frac12$. Therefore they agree on the total-effect information for both treatment levels.
[/step]
[step:Compute the cross-world nested counterfactual under the mediator from treatment $0$]
In the first coupling, $Y_{1,0}=Y_{1,1}=R$, so
\begin{align*}
Y_{1,M_0}=R.
\end{align*}
Therefore
\begin{align*}
\mathbb E[Y_{1,M_0}]=\frac12.
\end{align*}
In the second coupling, $M_0=W$, $Y_{1,0}=1-W$, and $Y_{1,1}=W$. Hence
\begin{align*}
Y_{1,M_0}=(1-W)Y_{1,0}+WY_{1,1}.
\end{align*}
Substituting the definitions of $Y_{1,0}$ and $Y_{1,1}$ gives
\begin{align*}
Y_{1,M_0}=(1-W)(1-W)+W^2.
\end{align*}
Since $W$ is $\{0,1\}$-valued, $(1-W)^2=1-W$ and $W^2=W$. Thus
\begin{align*}
Y_{1,M_0}=1.
\end{align*}
Consequently
\begin{align*}
\mathbb E[Y_{1,M_0}]=1.
\end{align*}
[/step]
[step:Conclude that front-door identification alone does not identify natural mediation effects]
The two couplings agree on the marginal laws of all single-intervention variables $M_a$ and $Y_{a,m}$, and they agree on the total-effect laws of $Y_{0,M_0}$ and $Y_{1,M_1}$. These are the kinds of single-world quantities determined by a front-door total-effect identification. However, they disagree on the nested cross-world counterfactual:
\begin{align*}
\mathbb E[Y_{1,M_0}]=\frac12
\end{align*}
in the first coupling, while
\begin{align*}
\mathbb E[Y_{1,M_0}]=1
\end{align*}
in the second coupling.
Therefore no functional depending only on the front-door-identified single-intervention information can determine $\mathbb E[Y_{1,M_0}]$. Natural direct and indirect effects require additional assumptions specifying the joint distribution of cross-world potential outcomes, not merely identification of total-effect interventional laws. This proves the claim.
[/step]
