Structural Causal Models (SCMs)

• Structural causal models (SCMs) are rigorous frameworks that encode causal relationships as deterministic or stochastic structural equations linking system variables.
• They distinguish causation from mere association, supporting counterfactual reasoning and interventional analysis across diverse fields like epidemiology, economics, and machine learning.
• SCMs extend traditional models by handling cycles, latent confounders, and feedback, thereby enhancing causal discovery and dynamic equilibrium analysis.

Structural causal models (SCMs) are rigorous mathematical frameworks that describe how variables in a system are generated by deterministic or stochastic mechanisms, and how interventions on these variables propagate causal effects throughout the system. By encoding the relationships among variables as structural equations, SCMs formalize both the statistical dependencies and the causal semantics necessary for reasoning about interventions, counterfactuals, and the identification of causal effects. SCMs generalize and subsume traditional statistical models, allowing precise distinctions between mere association and causation, and they provide the logical foundation for a wide variety of causal inference tasks in fields as diverse as epidemiology, economics, machine learning, and the natural sciences.

1. Formal Structure and Interventional Semantics

An SCM is specified as a tuple $\mathcal{M} = (V, U, F, P_U)$, where $V$ is a set of endogenous variables, $U$ is a set of exogenous variables, $F = (f_i)_{i \in V}$ is a collection of structural functions (or mechanisms) such that each $f_i$ generates $X_i \in V$ from its direct causes (parents) and corresponding exogenous noise $U_i$, and $P_U$ is a joint distribution over $U$. Each structural equation is of the form

$X_i = f_i(\text{pa}_i, U_i),$

where $\text{pa}_i \subset V \setminus \{X_i\}$ denotes the endogenous parents of $X_i$. The collection $(f_i)$, together with the graph induced by the parent assignments, defines a causal diagram—typically a directed acyclic graph (DAG), though cyclic and latent-variable generalizations exist (Bongers et al., 2016).

Interventions are modeled using Pearl's do-operator. For an intervention $\text{do}(X_I = \xi_I)$ (where $I \subset V$), the equations for $X_i, i \in I$ are replaced with constants: $X_i := \xi_i$. The interventional distribution is computed by solving the modified system and propagating the effect through the remaining equations.

2. Relationship with Dynamical Systems and Equilibrium Analysis

SCMs can be systematically derived from equilibrium analysis of ordinary differential equation (ODE) systems under suitable stability conditions. Given a dynamical system

$$\dot{X}_i(t) = f_i(X_{pa_\mathcal{D}(i)}), \qquad X_i(0) = (X_0)_i,$$

one obtains a set of equilibrium equations by setting $\dot{X}_i = 0$: $0 = f_i(X_{pa_\mathcal{D}(i)}), \qquad \forall i.$ Provided the ODE system is globally stable (i.e., converges to a unique equilibrium from any initial condition), and the equilibrium system is structurally solvable (meaning the equilibrium can be uniquely solved for each variable as a function of its parents), the equilibrium conditions can be recast as structural equations

$X_i = h_i(X_{pa_\mathcal{M}(i)}),$

where $$pa_\mathcal{M}(i) = pa_\mathcal{E}(i) \setminus \{i\}$$. Thus, a deterministic SCM is formally induced at equilibrium by the underlying ODE system (1304.7920, 1408.2063). Interventions at the ODE level (such as fixing $X_i$ to $\xi_i$ via an infinite-strength control) correspond precisely to replacing the corresponding structural equation in the SCM ($X_i := \xi_i$). This construction also applies when the original ODE—and thus the induced SCM—contains cycles, so even models with feedback (cyclic dependencies) can have meaningful SCM representations at equilibrium.

3. Solvability, Cycles, and the Simple SCM Class

In general, not every SCM—especially those with cycles or latent variables—admits a unique solution for the endogenous variables. The existence and uniqueness of solutions (solvability, unique solvability) are foundational requirements for causal semantics. Formally, an SCM is solvable with respect to a set $O$ if there exists a measurable function $g_O$ such that, for almost every realization of exogenous variables, the equations for $O$ can be uniquely solved in terms of $pa(O) \setminus O$ and the corresponding exogenous noise. In the linear case, unique solvability reduces to the invertibility of certain submatrices

$A_{LL} = I_L - B_{LL},$

where $L \subseteq V$ (Bongers et al., 2016).

The class of “simple SCMs” comprises models that are uniquely solvable with respect to every subset of variables. Simple SCMs always have well-defined joint, interventional, and counterfactual distributions, are closed under intervention and marginalization, and preserve a generalization of graphical Markov properties (via $\sigma$-separation rather than d-separation in the presence of cycles). These guarantees formally extend the “convenient” properties of Bayesian networks (which require acyclicity) into more general, cyclic, or confounded systems.

4. Graphical Properties and Abstraction

Standard SCMs induce a causal diagram (typically a DAG), but with cycles or latent variables, the graphical semantics become more nuanced. For acyclic SCMs, the global directed Markov property aligns d-separation in the DAG with conditional independence in the induced distribution. When cycles or latent confounding is present, d-separation may fail, and generalized separation criteria like σ-separation apply (Bongers et al., 2016).

Abstraction between SCMs—mapping one SCM to another, possibly at a different level of description—requires careful treatment to preserve interventional consistency. Proposed formalizations require mapping node sets and functional mechanisms while ensuring that the result of a sequence of abstraction and intervention commutes with the operations' order (Zennaro, 2022). Properties such as surjectivity, injectivity, and functoriality at the structural and distributional level delineate families of abstractions, supporting applications in causal representation learning and multi-scale modeling.

5. Extensions and Generalizations

For dynamical systems whose equilibrium behavior depends on initial conditions or functional invariants (e.g., systems with conservation laws or semi-stability), SCMs can be insufficient. Causal Constraints Models (CCMs) generalize SCMs by encoding causal relations as constraints that are only active under specific interventions, thereby capturing dependencies on constants of motion or functional laws such as the ideal gas law or reaction network invariants (Blom et al., 2018). CCMs allow explicit modeling of when each constraint (not just equation) should be enforced, broadening the scope of causal modeling for systems where classic SCMs do not provide a complete description.

Other notable extensions include:

• Modeling latent selection mechanisms via a conditioning operation that preserves simplicity, acyclicity, and linearity of SCMs, allowing adjustment for selection bias while retaining causal semantics in the selected subpopulation (Chen et al., 12 Jan 2024).
• Internally-standardized SCMs (iSCMs), which perform z-scoring at each variable during generative sampling to remove artifacts such as increasing variances or pairwise correlations, thus preventing causal order leakage in synthetic benchmark data (Ormaniec et al., 17 Jun 2024).
• Methods for consolidating mechanisms in large SCMs, producing simpler composite structural equations that preserve the full set of interventional behaviors, going beyond traditional marginalization approaches that may strip away useful causal information (Willig et al., 2023).
• The formulation of SCMs for extremes (i.e., modeling causal dependencies among rare, large events) using extremal graphs and Pareto limits, which can lead to the disappearance of some causal links in the tails of the distribution (Engelke et al., 9 Mar 2025, Jiang et al., 12 May 2025).

6. Practical Implications and Methodological Advances

SCMs provide the foundation for a broad range of methodological advances:

• Constraint-based structure discovery in both acyclic and cyclic settings requires careful attention to solvability and can leverage separation criteria like $\sigma$-separation or extremal partial correlations for extremes.
• Identification algorithms for linear SCMs, including randomized polynomial-time procedures for tree-shaped models, use algebraic techniques to determine which parameters are (uniquely or k-) identifiable from observational data (Gupta et al., 2023). Necessary and sufficient conditions for identification with deterministic relations and latent confounders are established using combinatorial arguments involving unique components and the marriage theorem (Yang et al., 2021).
• The systematic integration of data-driven, expert, and literature-based causal knowledge is possible via frameworks like the Causal Knowledge Hierarchy (CKH), which weights and aggregates multiple sources of evidence to derive a high-confidence causal graph (Adib et al., 2022).
• SCMs are foundational to recent machine learning research, including the use of graph neural networks (GNNs) for neural-causal modeling (Zečević et al., 2021), the learning of latent SCMs from high-dimensional data using variational methods (Subramanian et al., 2022), and the formulation and estimation of individual causal effects by individualizing the population via a formal (“indiv-operator”) abduction process (Chang, 17 Jun 2025).

7. Impact and Applications

The impact of SCMs spans scientific disciplines and methodological innovation:

• SCMs formalize and underpin the logic of intervention, counterfactual reasoning, and identification in epidemiology, economics, social sciences, and engineering.
• The framework has been directly applied to precision medicine (e.g., counterfactual MR imaging for disease modeling (Reinhold et al., 2021)), market dynamics, environmental systems (such as river network hydrology (Engelke et al., 9 Mar 2025, Jiang et al., 12 May 2025)), and causal discovery in neuroscience and biology.
• Recent work extends SCM reasoning to individual-level inference (individual causal effect estimation) (Chang, 17 Jun 2025), as well as to new data domains (e.g., benchmark datasets generated via iSCMs to avoid shortcut bias in algorithm evaluation (Ormaniec et al., 17 Jun 2024)).
• The theoretical and computational developments in SCMs continue to drive progress in both classical causal statistics and modern data-driven causal representation learning.

SCMs remain a central construct in contemporary causal inference, offering both technical rigor and the flexibility to address a growing array of settings—including those with cycles, feedback, selection bias, extremal behavior, and multi-scale abstraction—supported by a robust body of mathematical theory and a growing ecosystem of methodological extensions.