MMP.15:4.4 - Carry selection, transport and support through the expression

Identify which population supplies every average. For example, an experiment in (S=1) may supply action-specific outcome means by pretreatment (L), while an existing inventory supplies the target distribution (P_T(L)). If the intervention has the same meaning, assignment in the experiment is exchangeable, and the conditional potential-outcome means agree between the experimental and target populations, then

[ \mu_a=\sum_l E[Y\mid A=a,L=l,S=1]P_T(L=l). ]

This is a transport assumption followed by trial identification and target averaging. It requires trial coverage and both action levels wherever the target gives relevant weight. Equal variable names, similar marginal distributions, or randomization within the experiment do not establish the transport assumption. Selection after action or outcome requires its own recording and causal argument; it cannot automatically use this pretreatment bridge. Generalizing trial results, §§3–4.

Inspect the support needed by the chosen expression. Positivity means that a required conditional law is defined on the relevant target support; in the discrete adjustment example, (P(A=a\mid L=l)>0) whenever the target weights that stratum. For continuous variables, use the corresponding support condition. A fixed intervention value in a continuous action space may also need a continuity or other structural restriction: the observed law determines conditional responses only almost everywhere. An empty cell in a small sample does not prove a population probability is zero. A known structural exclusion is different: the missing conditional response cannot be learned there from that source.

Failure of support invalidates that expression at the excluded values. Another source, another identifying argument or an explicit structural restriction may still answer the question. Extrapolation through a response model can produce an assumption-dependent answer, but fitting or regularizing that model does not create observations at the missing support.