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Causal Consistency of Structural Equation Models
Paul K. Rubenstein, Sebastian Weichwald, Stephan Bongers, Joris M. Mooij, Dominik Janzing, Moritz Grosse-Wentrup, Bernhard Schölkopf
TL;DR
Causal models at different levels of description should agree on the effects of interventions, but such agreement is not automatic. The paper formalises this requirement through exact transformations between SEMs and uses them to relate model levels, showing that intervention ordering is central to causal consistency. The framework covers marginalisation, macro-level aggregation, stationary behaviour, and cyclic SEMs, while ambiguous manipulations can result from inexact transformations.
Problem
Causal models describing the same complex system at different levels should agree on intervention effects, but differing descriptions may lack a consistent causal correspondence.
Method
The paper introduces exact transformations between SEMs, using a surjective order-preserving mapping between interventions to formalise causal consistency.
Results
Exact transformations provide a framework for relating marginalised, macro-level, stationary, and cyclic SEM descriptions while preserving consistent causal reasoning.
Takeaways & Limitations
Well-specified interventions and their ordering are essential for interpreting whether different SEM descriptions represent the same causal system.
Takeaways & Limitations
Not every induced collection of interventional distributions corresponds to an SEM, and surjectivity without order preservation can map observations to interventions.
Abstract
from arXiv · showhide
Complex systems can be modelled at various levels of detail. Ideally, causal models of the same system should be consistent with one another in the sense that they agree in their predictions of the effects of interventions. We formalise this notion of consistency in the case of Structural Equation Models (SEMs) by introducing exact transformations between SEMs. This provides a general language to consider, for instance, the different levels of description in the following three scenarios: (a) models with large numbers of variables versus models in which the `irrelevant' or unobservable variables have been marginalised out; (b) micro-level models versus macro-level models in which the macro-variables are aggregate features of the micro-variables; (c) dynamical time series models versus models of their stationary behaviour. Our analysis stresses the importance of well specified interventions in the causal modelling process and sheds light on the interpretation of cyclic SEMs.
1 INTRODUCTION
Causal models of complex systems may describe different levels of detail, but consistency requires agreement on intervention effects. The paper introduces exact SEM transformations to assess such consistency across marginalised, macro-level, and stationary descriptions.
- Motivation: Complex physical systems are often modelled macroscopically because measuring and modelling every microscopic component is impractical.An ideal gas may contain approximately 10^22 particles per litre.
- Three settings: Different model levels arise through marginalising variables, aggregating micro-variables into macro-variables, or describing stationary behaviour instead of dynamics.Examples include blood chemistry, neuronal activity, and final reactant-product ratios.
- Causal consistency: Causal models at different levels should agree in their predictions of intervention effects.This consistency requirement motivates the paper’s formal analysis.
- Illustrative problem: Aggregating LDL and HDL into total cholesterol can produce contradictory conclusions because equal changes in total cholesterol may correspond to opposite effects on heart disease.Raising LDL and raising HDL both raise TC but have different effects on HD.
- Contribution: The paper introduces exact transformations between SEMs as a framework for evaluating whether two models are causal descriptions of the same system.The framework explicitly uses a natural ordering on interventions to align causal reasoning across model levels.
- Applications: The framework is applied to subsystem modelling, micro-to-macro descriptions, and the emergence of cyclic SEMs.The paper proves exactness for transformations in all three settings.
2 STRUCTURAL EQUATION MODELS
The paper defines SEMs as intervention-aware mathematical models that associate structural equations and exogenous-variable distributions with well-defined outcomes. Its formulation allows restricted interventions, dependent exogenous variables, and cyclic structures.
- General formulation: The formulation does not require all perfect interventions, independent exogenous variables, or acyclicity.The unique-solution condition is retained so cyclic SEMs can still be considered.
- SEM definition: An SEM consists of structural equations, an intervention set with a natural partial ordering, an exogenous-variable distribution, and unique solutions under interventions.These components together ensure that each permitted intervention induces a well-defined distribution.
- Interventions: Perfect interventions replace the structural equations of the intervened variables with equations fixing those variables to specified values.Multiple-variable interventions replace each corresponding structural equation.
- Intervention ordering: The intervention ordering places an intervention below another when the latter extends it by intervening on additional variables without changing the original assignments.For example, do(X_i = x_i) ≤ do(X_i = x_i, X_j = x_j).
3 SEMS FOR CAUSAL MODELLING
SEMs serve as causal models by describing both variable distributions and how those distributions change under interventions. Their intervention semantics can represent physically implementable, restricted manipulations and their compositions.
- Causal interpretation: SEMs describe distributions of variables together with how those distributions change under interventions.Do-interventions are interpreted as actual or potentially hypothetical physical manipulations.
- Well-defined outcomes: A unique-solution condition ensures that each permitted intervention produces a well-defined distribution, including in cyclic SEMs.For acyclic SEMs this condition is automatic, but it is imposed because cyclic models are also studied.
- Compositional interventions: The intervention ordering captures whether one physical manipulation can be followed by another without undoing the first.The bulb-removal example maps sequential removals to a combined intervention.
- Restricted interventions: Restricted intervention sets can model situations in which some physical actions, such as using a light switch, are unavailable.Removing light bulbs is used as the available intervention in the example.
4 TRANSFORMATIONS BETWEEN SEMS
The paper defines exact transformations between SEMs to formalize when different model levels preserve causal reasoning. The framework maps variables and interventions so that corresponding intervention-induced distributions and their composition remain consistent.
- Distributions implied by an SEM: An SEM implies a family of distributions, one for each intervention, rather than a single joint distribution.The intervention index set and its partial ordering record how interventions relate and can be composed.
- Transforming distributions: A variable transformation produces distributions on the target space indexed by source-level interventions, which are not automatically interventions on the target variables.Such transformed distributions need not be realizable by any target-level SEM.
- Exact transformations: Exact transformations view one SEM as a transformed description of another through a variable map and a surjective, order-preserving intervention map.The transformation is defined only when the transformed distributions can also be represented by an SEM.
- Properties: Exact transformations compose transitively, while identity mappings and variable relabellings are exact transformations.These results provide basic closure and sanity checks for the framework.
- Causal consistency: Order preservation ensures that intervention effects and compositionality transfer from the source SEM to the transformed SEM.If one source intervention can be added to another, their mapped interventions retain that ordering; Theorem 6 expresses this as causal consistency.
- Causal interpretation: If a macro-level SEM is an exact transformation of a more complex model, reasoning about interventions on macro-variables can use the macro-model while retaining causal consistency.The paper therefore treats variables such as temperature or pressure as causal entities only when the exactness condition has been established.
- Failure of exactness: Dropping order preservation can map an observational source distribution to an interventional target distribution and create incompatible intervention histories.The examples include a source intervention that can be extended at the micro-level while its mapped target intervention conflicts with an intervention already performed.
5 EXAMPLES OF EXACT TRANSFORMATIONS
The paper applies exact transformations to three settings: simplifying SEMs by marginalisation, aggregating micro-variables into macro-variables, and replacing dynamical processes with equilibrium models. These transformations preserve causal consistency under the specified interventions, while their validity depends on assumptions such as intervention scope, acyclicity, averaging conditions, or contraction.
- 5.1 MARGINALISATION OF VARIABLES: Marginalising childless or never-intervened variables yields exact transformations that simplify SEMs without losing causal content for the remaining variables.Childless-variable marginalisation applies generally; marginalisation of never-intervened variables is stated for acyclic SEMs.
- 5.1 MARGINALISATION OF VARIABLES: These marginalisation operations formally justify causal models that focus on a subsystem of a more complex system.Successive application of the operations produces a causally consistent simpler model.
- 5.1 MARGINALISATION OF VARIABLES: Theorem 10 relies on acyclicity, although marginalising non-intervened variables in cyclic SEMs can be allowed with additional technical conditions.The paper also notes that its treatment relies on allowing dependent exogenous variables; imposing independence would invalidate the stated result in general.
- 5.2 MICRO- TO MACRO-LEVEL: Averaging specified micro-variables can produce macro-variables whose macro-level SEM is an exact transformation of the micro-level SEM.The construction is presented for a linear SEM under a matrix condition in which each column of A sums to the same value.
- 5.3 STATIONARY BEHAVIOUR OF DYNAMICAL PROCESSES: A discrete-time linear dynamical process with identical noise can be exactly transformed into a potentially cyclic SEM describing the distribution of its equilibria.The transformation is well-defined under a contraction assumption on the linear dynamics.
- 5.3 STATIONARY BEHAVIOUR OF DYNAMICAL PROCESSES: The equilibrium construction supports interpreting SEMs as descriptions of processes that equilibrate quickly relative to their external environment.It also gives cyclic SEMs an interpretation as exact transformations of acyclic dynamical processes, avoiding a required temporal ordering of cyclic variables.
6 DISCUSSION AND FUTURE WORK
The discussion presents exact SEM transformations as a framework for causal consistency across model levels, while identifying unresolved questions about transformation choice, approximation, counterfactuals, and model fitting.
- Exact transformations evaluate when two SEMs can be viewed as causally consistent models of the same system.
- The framework relates differing model levels to subsystem modelling, macro-level causal descriptions, and the emergence of cyclic causal models.
- Inexact transformations can produce ambiguous manipulations, as when LDL and HDL are observed only through their sum, total cholesterol.
- A proposed extension would relax exact equality between intervention distributions to study the tradeoff between model accuracy and model complexity.
- The paper does not address counterfactual consistency or provide a criterion for choosing among all possible exact transformations.
- Given an underlying model, measurement transformation, and restricted model class, exactness can test whether a fitted model has a causally consistent interpretation.
A PROOFS FOR SECTION 4.3: ELEMENTARY EXACT TRANSFORMATIONS
The proofs establish basic closure properties of exact transformations: relabelling variables preserves exactness, and composing exact transformations yields another exact transformation.
- Identity mappings and bijective permutations of variable labels are exact transformations.
- Relabelling constructs the new SEM by replacing each X_i in structural equations and interventions with the correspondingly relabelled variable.
- Composing two exact transformations produces an exact transformation from the initial SEM to the final SEM.
B PROOFS FOR SECTION 5.1: MARGINALISATION OF VARIABLES
The marginalisation proofs show how childless or never-intervened variables can be removed while preserving exact causal transformations under stated structural conditions.
- Marginalising a childless variable removes its structural equation and substitutes its function into the equations of its children.
- The intervention mapping for removing a childless variable drops references to that variable, while retaining the remaining noise variables with their marginal distribution.
- For a never-intervened-upon variable in an acyclic SEM, its equation can be substituted into downstream equations without affecting its ancestors.
- The proof uses combined noise variables for the remaining equations and relies on the childless-variable marginalisation law matching the law obtained by simply dropping that variable.
- The resulting structural equations remain acyclic, and the reduced SEM is an exact transformation of the original under the identity intervention mapping.
C PROOF FOR SECTION 5.2: MICRO- TO MACRO-LEVEL
The proof verifies that averaging micro-variables into macro-variables preserves the intervention distributions required for an exact transformation.
- The intervention correspondence is surjective and order-preserving, in fact an order embedding.
- In the observational setting, the macro-level distribution is derived from the defining equations of the macro SEM.
- The push-forward distribution under do(W = w) agrees with the distribution induced by the corresponding macro-level intervention.
- The same agreement holds for do(Z = z) and the joint intervention do(W = w, Z = z).
D PROOF FOR SECTION 5.3: STATIONARY BEHAVIOUR OF DYNAMICAL PROCESSES
The proof maps interventions between dynamical SEMs and shows that the stationary transformation preserves the intervention-specific distributions. Under the contraction assumption, the limiting random variable is identified with the transformed model’s stationary behavior.
- The proof defines a mapping between interventions in the original and transformed SEMs and compares their intervention-specific laws.
- The intervention mapping is surjective and order-preserving, so corresponding interventions can be compared across the two SEMs.
- For every intervention, contraction of the dynamical update ensures that the sequence of random variables converges to a limiting random variable X*.
- Because the transformation is defined as the limit of the dynamical sequence, τ(X) equals X* and satisfies the corresponding stationary equations.
- The transformed SEM is therefore an exact τ-transformation of the original SEM.
D.1 CONTRACTION MAPPING AND CONVERGENCE
This section establishes convergence of the dynamical SEM under interventions by reducing its update rule to a contraction mapping. The contraction mapping theorem then gives a unique fixed point for every intervention.
- Adding a fixed vector to a contraction preserves the contraction property.
- The second auxiliary lemma likewise establishes that the relevant transformed function remains a contraction.
- If the linear map A is a contraction mapping, the sequence generated by the SEM converges everywhere under any intervention.
- The intervened dynamics can be represented as repeated application of a function f formed by composing two component functions.
- The contraction mapping theorem implies convergence everywhere to a unique fixed point for any fixed intervention and exogenous input.