MMP.15:4.3 - Derive an expression, using the simplest sufficient route

Begin with a relation that replaces an unobserved intervention quantity by available quantities under an explicit assumption. Then combine it with ordinary conditioning, multiplication and averaging. Every final factor must come from an available law or an earlier justified step. Displayed sums concern discrete variables; for continuous or mixed variables, average against their actual probability laws.

For example, let (L) be measured before action. For a mean effect in the same population as the data, suppose that within each relevant (L=l), action assignment does not select a different mean potential outcome:

[ E[Y(a)\mid L=l]=E[Y(a)\mid A=a,L=l]. ]

This mean exchangeability assumption, consistency and positive probability of each required action in the relevant strata give

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

The first equality averages over the target strata. The second uses exchangeability to compare the same potential outcome, then consistency to replace it by an observed outcome. Average both actions with the same target weights. The different distributions (P(L\mid A=a)) generally answer an association question instead.

A common graphical justification is the back-door criterion: choose pretreatment variables that block every path into (A) capable of linking it to (Y) through other causes. In checking a path, conditioning on an intermediate non-collider blocks it; a collider, where two arrowheads meet, blocks it unless that collider or a descendant is conditioned on. Thus adding every available covariate can open a path rather than remove bias. This criterion is sufficient, not the only way to justify adjustment or identify an effect.

When adjustment of (A)’s effect is unavailable, derive other observable intermediate quantities. In a simple front-door construction, a measured mediator (M) carries every directed path from (A) to (Y); (A) to (M) has no open back-door path; and conditioning on (A) blocks all back-door paths from (M) to (Y). Under these conditions and the required support,

[ P(Y=y\mid do(A=a)) =\sum_m P(M=m\mid A=a) \sum_{a’} P(Y=y\mid M=m,A=a’)P(A=a’). ]

The outer factor identifies the effect of action on the mediator. The inner adjustment identifies the outcome law under intervention on the mediator. The pathway restrictions license combining them for the action’s total effect. Merely including a post-action measurement in a regression does not perform this construction. These classical conditions are sufficient; their failure does not establish that this functional or another identifying expression is impossible. Front-door criteria and their extension, §§2.1.5–3.

For a more involved graph or several input laws, derive a sequence of intermediate distributions using the rules of do-calculus and probability. Each exchange between observation and intervention needs the corresponding separation condition in the modified graph. A suitable identification implementation can carry out that search: provide the graph, the target and the actual input laws, then recover the returned derivation and check its required factors. The historical ID algorithm covers a specified acyclic model class with latent common causes and an observed joint law; generalized search can use several incomplete or experimental laws. Do not replace those inputs by a joint law that the records never supplied. ID algorithm, Figure 3; generalized search, §§2–3.