How Causal Models Compose

The Composition Problem

Consider two models, each correct in its own frame:

Model A — Labor supply. Predicts hours worked from wage, education, and household composition. Variable: income, defined as monthly post-tax household income.

Model B — Health outcomes. Predicts mortality from income, age, and lifestyle. Variable: income, defined as annual pre-tax individual income.

A user asks: "How would expanded employment affect population mortality?" The LLM finds both models in the library, sees they share an income node, joins them at that node, and produces an answer. The answer reads fluently. It is also wrong — not by a small fraction, but in ways that would change a policy recommendation.

Three things differ between the two incomes: time grain (monthly versus annual), tax basis (post-tax versus pre-tax), and unit of aggregation (household versus individual). Each is a separate translation problem. None of them is recoverable from the surface label "income."

Three operations the word "compose" hides

This is one face of a more general problem. The verb compose conflates three distinct technical operations, each of which fails differently:

Most composition failures conflate these. Engineers think they are doing graph union; what they are actually doing is referent alignment by hope and assumption reconciliation by inheritance.

Four Primitives

The primitives are not a complete solution to composition. They are the minimum vocabulary that makes composition checkable rather than fluent. Each one converts a category of silent assumption into a machine-readable artifact that the user — or another model in the library — can challenge.

1. Referent declaration

Every variable in every model carries a structured referent, not just a name. The referent specifies — at minimum — unit, basis, population, and time grain, along with a pointer to the empirical distribution from which the variable's prior was drawn.

When two referents differ, the library detects the mismatch automatically. Composition does not proceed until the mismatch is acknowledged. The point of the referent is not pedantic precision; it is to make naming unable to launder ambiguity.

2. Bridge nodes

When two referents differ but the underlying quantity is in principle convertible, a bridge node is inserted between them. The bridge is itself a small SCM: it takes the source referent as input, applies an explicit transformation, and outputs the target referent.

The bridge has its own assumptions and its own scope conditions. A bridge from monthly post-tax household income to annual pre-tax individual income requires assumptions about household composition (a four-person household is not four times a one-person household) and tax incidence (the marginal tax rate is not a constant). Each assumption is declared. Each is checkable. Each can fail.

The closest precedent is the input/output node of object-oriented Bayesian networks (Koller & Pfeffer, 1997) and the related fragment-composition framework of Laskey & Mahoney (1997). Bridge nodes generalize that idea: they carry their own assumption sets and scope conditions rather than serving as pure passthrough connectors.

The library's job is not to prevent you from joining models. It is to make the joining honest.

3. Assumption pushforward

When two SCMs are joined, the new graph's identifiability has to be recomputed from scratch. Inheriting the parents' identification claims is the most common bug in composition.

Suppose Model A has a clean identification path for P(Y | do(X)), and Model B has one for P(Z | Y). The joined model lets the user ask P(Z | do(X)) — but the path through Y might pass through a node that is a confounder in one parent's frame and a collider in the other's, or might open a backdoor that neither parent had reason to address.

Identifiability does not compose. It has to be re-derived in the joined graph. Pushforward is the primitive that forces this re-derivation rather than letting it be assumed. The data-fusion literature (Bareinboim & Pearl, 2016) develops the rigorous theory; the primitive here is the operational form a library has to expose to its users.

4. Scope intersection

The joined model is valid only where both parents' scope conditions hold simultaneously. The library computes this intersection and reports it before producing an answer. The technical term for the underlying problem in causal inference is transportability (Pearl & Bareinboim, 2014); scope intersection is the practical operation a library performs to enforce it.

The intersection can be empty. If Model A is valid for working-age adults in formal employment 2010–2020 and Model B is valid for retirees across multiple countries 1990–present, the intersection contains no one. The library should say so, rather than produce a number.

The intersection is often much narrower than either parent. A reader can then decide whether the narrowed scope still answers their question. They cannot make that decision if the library does not tell them what was narrowed and where.

A Worked Example

Apply the primitives to the failure case from §1. The user wants to ask: how would expanded employment affect population mortality? The naive join produced a fluent wrong answer in milliseconds. The principled join takes longer, narrows the question, and produces something the user can act on.

The naive join took milliseconds and produced a number. The principled join took longer and produced a smaller number, scoped, qualified, and traceable to four declared assumptions. Only one of the two answers is useful.

What This Asks of the Mediator

Composition is exactly the place where the LLM is most tempted to paper over mismatches with fluent language. When a user describes a situation, and the model retrieves an SCM with an income variable, the model can produce a confident explanation of why the model fits. The explanation will sound principled. It will not be principled. It will be fluent.

The composition primitives' real job is not data plumbing. It is converting silent assumptions in the LLM's surface fluency into machine-readable artifacts that the user can interrogate. Each primitive forces the mediator to declare something it would otherwise leave implicit:

• Referents force the mediator to commit to a specific quantity, not a word.
• Bridges force the mediator to declare the conversion model and its assumptions.
• Pushforward forces the mediator to recompute identification in the joined graph, not assume it.
• Scope intersection forces the mediator to report the narrowed valid range alongside any answer.

The primitives sit in the lineage of model cards (Mitchell et al., 2019) and dataset documentation more generally — checkable artifacts that travel with the model — but with one important difference. Model cards are passive: a reader consults them. Composition primitives are active: the library checks them, the mediator is forced to populate them, and downstream queries depend on them resolving.

What Gets Harder

The primitives address a category of failure. They do not exhaust the problem of composition. Three classes of difficulty resist the primitives — and pretending otherwise is its own dishonesty.

Cross-boundary structural assumptions

When two models are joined, the union DAG sometimes reveals causal paths that neither parent specified. The primitives can detect that the path exists and flag the missing adjustment set. They cannot supply the missing structural assumption itself. This is where domain experts are irreplaceable, and where the library has to know its place: it can frame the question for a human, but it cannot answer it.

Cyclic dependencies between models

Two models can each be acyclic on their own and produce a cycle when joined: Model A's output feeds Model B's input, which in turn feeds Model A's. The primitives detect the cycle. Resolving it requires either dynamic modeling — turning the cycle into a temporal sequence with explicit lags — or rejecting the join entirely. Neither is automatic, and neither is always available.

Identification under disagreement

Two models can specify conflicting graphs over shared variables. Model A says income → health. Model B says health → income, perhaps mediated by labor-force participation, perhaps by some shared driver that neither names. Both models may have empirical support; both may be valid in their original scope. The primitives detect the disagreement. They cannot adjudicate it. This is where causal inference becomes scientific argument, and the library's role is to surface the argument cleanly, not to resolve it pretending to be neutral.

Key Terms

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