MMP.15:4.1 - Specify the causal quantity and what counts as the same intervention

For an action level (a), let (Y(a)) denote the outcome under the specified intervention setting the action to (a). For a target population (T), a common request is

[ \mu_a=E_T[Y(a)], \qquad \Delta=\mu_1-\mu_0. ]

Define the action, comparator, outcome, population and time horizon far enough to distinguish the actual question. A contrast of average outcomes differs from the chance that a particular unit would benefit. Identifying both marginal laws of (Y(1)) and (Y(0)) does not ordinarily identify their joint law or the distribution of individual differences.

Establish consistency: for a unit actually receiving the specified version of action (a), its observed outcome agrees with (Y(a)). Different doses, implementations or accompanying actions may give the same label different meanings. If one unit’s outcome depends on other units’ actions, represent the relevant joint assignment or policy; the notation (Y(a)) must not silently discard that dependence.

For static interventions, (P(Y\mid do(A=a))) denotes the corresponding intervention law. C.28.MR defines the change of mechanism that gives this notation its meaning. Identification below seeks that law or its required function without assuming all the changed model’s numerical mechanisms are already known.