MMP.15:5.2 - Use a mediator, then withdraw a pathway exclusion

A service’s command (A) can activate a retry mechanism (M), which affects successful completion (Y). Unrecorded load can affect both command selection and completion. The initial causal account permits (A\to M\to Y) and an unobserved common cause of (A,Y), but no other arrows: in particular, no direct effect of (A) on (Y) and no hidden common cause of (A,M) or (M,Y).

All three variables are binary. The available joint law has (P(A=1)=0.5), (P(M=1\mid A=0)=0.25), (P(M=1\mid A=1)=0.75), and:

There is no observed pretreatment adjustment variable for the shared load. The front-door conditions nevertheless hold. First adjust the mediator’s effect over (A): the inner means are (h(0)=0.5(0.10)+0.5(0.30)=0.20) and (h(1)=0.5(0.70)+0.5(0.90)=0.80). Then average over the mediator law induced by each command:

[ \mu_0=0.75(0.20)+0.25(0.80)=0.35,\qquad \mu_1=0.25(0.20)+0.75(0.80)=0.65. ]

Thus (\Delta=0.30). Directly comparing observed commands instead gives (0.75-0.25=0.50).

Changed condition. Inspection of the service reveals that the command can also change completion through a path bypassing retry. The revised account permits a direct (A\to Y) arrow. Retaining the previous formula is no longer justified.

The same full observational law now admits at least two answers. To demonstrate this, let unobserved (U) be a fair binary variable, let observational command selection be (A=U), and generate (M) with probability (0.25+0.50A) using independent random variation. In two alternative models, generate completion with respective probabilities

[ \text{Model I: }0.10+0.60M+0.20U,\qquad \text{Model II: }0.10+0.60M+0.20A, ]

using a further independent random draw. Every probability lies between zero and one. Because (A=U) observationally, both models give every cell of the supplied joint law.

Under (do(A=a)), (U) remains fair. Model I gives (\mu_0=0.35,\mu_1=0.65); Model II gives (\mu_0=0.25,\mu_1=0.75). Their effects are 0.30 and 0.50. Both satisfy the revised account, which permits the listed influences without requiring every permitted influence to be nonzero. Model II was excluded by the original no-direct-effect premise.

This is a proof that the revised assumptions and available law do not identify the average effect. These are ambiguity witnesses, not discovered service mechanisms. More precise estimation of the same law cannot distinguish them. A substantive restriction might; further evidence is chosen by its value for the use.