MMP.15:4 - Solution

Define the target, recover the available laws and describe the admitted causal models. Derive a common expression for their intervention consequence, or demonstrate how they disagree. Limit the returned conclusion to what that argument establishes.

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.

MMP.15:4.2 - Recover the available laws and the admissible causal models

Use MMP.7 to recover how records arise. With inclusion indicator (S), selected records supply a law such as (P(A,Y,L\mid S=1)), not automatically (P(A,Y,L)). Keep assignment, actual action, measurement and inclusion distinct when those distinctions affect the target. State which variables were jointly observed, which interventions were performed, and which population each source concerns.

For identification, provisionally treat these population laws as known. This asks what unlimited data of those kinds could determine. MMP.13 handles finite-sample estimation.

Describe the admitted causal relations. An acyclic causal graph is a useful representation: directed arrows allow direct causal influence, and a shared unobserved cause can be represented explicitly or by a bidirected edge. The absence of an arrow excludes a possible influence relative to the represented variables. A good observational fit does not justify that exclusion. Time-indexed variables can express feedback across time; a theorem for acyclic graphs must not be applied unchanged to an equilibrium model with unresolved cycles.

When these causal relations or material rival mechanisms still need construction, C.28.CM develops the model family, its subject meanings and the consequences that could distinguish its members. Bring that result back to the identification question. A sufficient already supplied family needs no additional construction.

The operative identification test is this: whenever two admitted causal models induce the same available laws, must they give the same requested quantity? If the assumptions themselves conflict with the available laws, return that conflict rather than declaring a result identified through an empty model class.

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.

MMP.15:4.3.1 - Use a binary instrument for a specified local effect

Use this branch when a binary assignment or encouragement (Z) changes actual action (A\in{0,1}), and an instrumental-variable argument might identify an effect of (A). The effect of offering an action and the effect of performing it are different targets. If the receiving question requires the population average effect, determine whether the local quantity below answers it before using the result.

Let (A(z)) be the action a unit would take under assignment (z). Start with (Y(z,a)) for its outcome under assignment (z) and action (a). Use the simple argument only under these assumptions:

• Consistency and stable versions: observed (A=A(Z)) and (Y=Y(Z,A)); the specified interventions have fixed meanings and no interference between units.
• Assignment independence and support: (Z) is independent of the potential actions and outcomes, and both assignment levels have positive probability in the population supplying the law.
• Exclusion: (Y(z,a)=Y(a)); assignment has no effect on the outcome except through actual action.
• Monotonicity: (A(1)\geq A(0)) for each unit. Assignment 1 never discourages a unit that would act under assignment 0.
• Relevance and finite means: (\delta_A=E[A\mid Z=1]-E[A\mid Z=0]>0), and the required outcome expectations exist.

These premises can be more demanding than random assignment alone. Randomizing an offer can support assignment independence without establishing exclusion or monotonicity. This is the binary local-effect argument of Angrist, Imbens and Rubin, 1996, §§2–4.

The response pair ((A(0),A(1))) separates always-takers ((1,1)), never-takers ((0,0)), compliers ((0,1)) and defiers ((1,0)). These are potential-response groups; observing one assignment and action generally does not label the individual’s group. Monotonicity excludes defiers.

To derive the result, use exclusion and the binary-action identity (Y(a)=Y(0)+a{Y(1)-Y(0)}). Independence and consistency give

[ \delta_Y=E[Y\mid Z=1]-E[Y\mid Z=0] =E[(A(1)-A(0))(Y(1)-Y(0))]. ]

The same argument gives (\delta_A=E[A(1)-A(0)]). Under monotonicity, that difference is one for compliers and zero for the other admitted groups. Consequently,

[ \frac{\delta_Y}{\delta_A} =E[Y(1)-Y(0)\mid A(1)>A(0)]. ]

This Wald ratio identifies the local average treatment effect for compliers under the named instrument and population. It is neither automatically the effect for everyone who took the action nor the population average effect. Changing the encouragement can change the complier group.

If (\delta_A=0), the ratio is undefined. A small nonzero population first stage can still identify the local quantity under the assumptions, while finite-sample uncertainty and sensitivity to premise violations can be large; MMP.13 must supply suitable inference for the receiving use. If defiers are possible, the numerator and denominator instead mix oppositely signed response-group contributions. A direct (Z)-to-(Y) route, assignment dependence or interference also requires a different argument. Conditional instrument validity, continuous action and transport to another population need their own derivations; the simple unadjusted ratio does not supply them.

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.

MMP.15:4.5 - Distinguish a proof of ambiguity from an unfinished search

To demonstrate nonidentifiability, construct two causal models satisfying the stated assumptions, reproducing all the available laws, and giving different values of the requested target. C.28.MR can calculate each model’s intervention consequence. Show both the observational agreement and the difference after intervention, as in :5.2.

A complete identification algorithm can also return an obstruction; check its class and query. A time limit or exhausted search without an applicable completeness result leaves the derivation unresolved. Failure to identify a full intervention distribution also does not by itself prove failure to identify a particular mean or contrast. ID completeness; scope of search completeness, §3.4.

When a point is not identified, use the assumptions to bound the requested quantity. Optimize it over compatible models, or derive an inequality valid for all of them. One compatible example establishes possibility; it does not establish a bound. If a bound is claimed sharp, show that compatible models attain or approach its endpoints. A bound that already settles the receiving question needs no complete reconstruction of unknown mechanisms.

A prior or added restriction may select among compatible answers; retain that dependence. It is not new evidence.

MMP.15:4.6 - Return the sufficient result and propagate consequential changes

Return the target and the assumptions under which its identifying expression, bound or ambiguity result holds. Make the data factors recoverable for estimation and uncertainty under MMP.13. Numerical evaluation belongs to an appropriate computational method; a completed optimization or simulation is not an identification proof.

Check the derivation as needed: verify a finite case, inspect a disputed graph condition, or calculate a countermodel’s observational and intervention laws. These checks establish the mathematical consequence of assumptions. Assurance that the assumptions describe the subject requires relevant subject evidence; successful fitting or self-consistent simulation does not provide it by itself.

When an actual change admits a direct pathway, changes an intervention’s version, alters selection, or changes the target population, propagate it through the affected derivation and calculation. Compare an alternative assumption when that comparison can change the intended use. Do not invent a sensitivity exercise for an already sufficient result under unchanged conditions.

C.11.DUA selects whether remaining uncertainty warrants more evidence, an explicitly conditional answer, a narrower agreed question or no further work. MMP.16 can design an observation if that further work is chosen. A mathematical identification result does not authorize or physically carry out the intervention.