Input-Output Structural Causal Models (ioSCMs)

Updated 2 July 2025

Input-Output Structural Causal Models (ioSCMs) are generalizations of classical SCMs that encompass cycles, feedback, latent variables, and dynamic input-output relations.
They leverage operator-theoretic state representations and σ-separation in directed mixed graphs to identify causal effects in complex systems.
ioSCMs enable robust causal inference and abstraction, supporting applications across engineering, biology, and artificial intelligence.

Input-Output Structural Causal Models (ioSCMs) are a comprehensive generalization of traditional structural causal models, developed to rigorously represent, analyze, and identify causal relationships in systems where cycles, feedback, latent confounders, selection bias, and dynamic or high-dimensional input-output relationships are essential. The ioSCM framework extends beyond acyclic graphical models and incorporates advanced operator-theoretic, algebraic, and information-theoretic perspectives to support both theoretical understanding and practical modeling requirements for complex dynamical, engineered, biological, and social systems.

1. Mathematical and Graphical Formulation

An Input-Output Structural Causal Model (ioSCM) generalizes the classical SCM by incorporating cycles, inputs, outputs, latent variables, and, when needed, time-dynamics or operator-valued state representations. Formally, an ioSCM consists of:

A set of observed (endogenous) variables,
A set of input (intervention/context) variables,
A set of latent (unobserved) variables,
A (possibly cyclic) set of structural equations or operator assignments,
A joint distribution for latent variables,
A directed mixed graph (DMG) that may include both directed and bidirected edges.

The central device in the input-output operator-theoretic approach is the natural state $\xi_t$ at time $t$ : an operator mapping all possible future inputs into (future) outputs, given the past inputs up to $t$ : $\xi_t^u(v_{t,\infty}) := [L_t F(u^t + v_{t,\infty})]_{[0,\infty)}$ Here, $u^t$ is the past input, $v_{t,\infty}$ the future input, $F$ is the system operator, and $L_t$ is the left-shift.

The induced DMG captures directed (possibly cyclic) dependencies via arrows, and latent confounding is encoded with bidirected edges. This structure provides a graphical basis for reasoning about conditional independencies and the flow of causal effects, with cycles and complex mediation or confounding handled intrinsically.

2. Causal Calculus with Cycles, Confounders, and Selection Bias

The ioSCM framework generalizes all classical tools of causal inference—specifically, do-calculus, adjustment, identifiability, and algorithmic procedures—beyond the acyclic and fully observed setting (1901.00433). The main technical advance is the replacement of d-separation with σ-separation for DMGs: $A \Indep^\sigma_G B \mid C \implies X_A \Indep_P X_B \mid X_C$ where $G$ is the DMG induced by the ioSCM.

Do-calculus maintains its three fundamental rules when σ-separation replaces d-separation:

Insertion/deletion of observations,
Action/observation exchange,
Insertion/deletion of actions.

Conditional independencies required for these rules' application are checked using σ-separation, even in the presence of cycles and feedback loops.

The classic backdoor and selection-backdoor criteria and adjustment formulas are likewise generalized. The generalized adjustment formula for estimation of conditional causal effects in arbitrary ioSCMs is: $P(Y | C, \doit(X), \doit(W)) = \int P(Y | X, Z, C, S=s, \doit(W))\, dP(Z | C, \doit(W))$ Here, all independence requirements are in terms of σ-separation in the corresponding (possibly cyclic, confounded) DMG.

The ID algorithm—which determines whether a causal effect is identifiable from observed data and the graph—is generalized to work for ioSCMs via consolidated districts (generalizations of c-components) and apt-orders (generalization of topological orders for graphs with cycles).

3. Operator-Theoretic State Representation and System Identification

A rigorous feature of the ioSCM approach is the natural state operator, encoding the system's entire "conditional memory" of past inputs and enabling minimal prediction of future outputs from future inputs (1009.5277). Crucially:

The natural state is a minimal operator mapping possible future inputs to outputs, given the input history.
For time-invariant, continuous systems with bounded and tapered input spaces, or polynomial integral operator systems with integrable kernels, the natural state set uniquely determines the system—that is, system identification is possible from the collection of natural states.
For pathological cases with "infinite memory," two distinct systems may share natural states, so identifiability is lost.

The differentiability properties of the natural state enable representing state trajectories via (possibly infinite-dimensional) differential equations: $\frac{d}{dt} \xi_u(t) = [L_t G(u + R_t)]_{0,\infty} + [L_t (L(u + R_t, d + 0))]_{0,\infty}$ This provides a bridge between input-output operator models and classical state-space ODEs, making ioSCMs amenable to variational and sensitivity analysis, and robust simulation.

4. Dynamic, Stochastic, and Frequency-Domain Extensions

ioSCMs encapsulate a broad range of settings beyond deterministic, static, or acyclic models:

ODE and Dynamic Structural Causal Models (DSCMs): ODE systems—possibly with cycles—can be systematically mapped to SCMs describing equilibrium or asymptotic states. DSCMs generalize this to entire trajectories, modeling asymptotic (e.g., periodic, oscillatory) dynamic responses to time-varying interventions (1304.7920, 1608.08028).
Stochastic and Measurement/Driving Noise: The theory extends to both measurement and driving noise, via SDEs or process noise, modeling interventions at the level of initial conditions, drift/diffusion terms, or observed measurements (2001.06208).
Markov Processes and Equilibrium SCMs: CTMPs or other Markov models, when at equilibrium, are mapped to equilibrium SCMs whose structural assignments are chosen to match the conditional distribution laws implied by the process, thereby enabling mechanistically consistent counterfactual inference (1911.02175).
Frequency-domain Causal Effects: ioSCMs can be represented as process SCMs for time series, supporting algebraic identification of spectral transfer functions or causal effects in the frequency domain using process graphs and path/trek rules (2305.11561, 1804.03911).

5. Macro-level, Clustered, and Abstraction-based Causal Identification

ioSCMs support analysis at macro or cluster levels:

C-DMGs (Cluster-Directed Mixed Graphs): Clusters of variables, possibly arranged in cyclic or partially observed structures, enable causal identification at a coarse abstraction. The do-calculus (with σ-separation) is sound and complete for identifying macro causal effects (i.e., effects among clusters) in C-DMGs over DMGs, unconditionally (2506.19650). Non-identifiability is characterized via SC-hedge structures in the SC-projection of the cluster graph.
Abstraction and Consistency Across Levels: ioSCMs formally support exact transformations between models at different resolutions—e.g., micro to macro, fine to coarse, dynamic to stationary—ensuring that causal statements and intervention effects are preserved across levels, provided intervention sets are suitably specified (1707.00819, 2306.14351).
Comparison to RCMs: ioSCMs, rooted in SCM algebraic principles, provide a representational backbone to population-level potential outcomes models, and enable a precise algebraic and structural control of representability, composition, and reversibility (2306.14351).

6. Latent, Neural, and High-dimensional ioSCMs

Recovering Latent Causal Structure: Recent methods recover latent variables and their causal wiring from high-dimensional data by joint Bayesian inference of variables, structure, and parameters, explicitly supporting input-output settings with known interventions (2210.13583).
Neural and GNN-based ioSCMs: Novel architectures such as redundant input neural networks (RINNs) or interventional GNNs enable scalable causal discovery, estimation, and interpretability of latent ioSCM structure from data, matching graph-based intervention logic and supporting both observational and interventional queries (2003.13135, 2109.04173).
Measure-theoretic Generalizations: Information Dependency Models (IDMs) extend ioSCMs beyond graphical functional frameworks, using information fields and topological separation to generalize causal reasoning for settings with cycles, context-specific edges, or non-graphical dependence (2108.03099).

7. Practical Applications and Limitations

Applications

Engineering and Control: ioSCMs generalize state-space and system identification models, supporting simulation, control design, and perturbation analysis in feedback-rich or high-dimensional engineered systems.
Biology, Neuroscience, Economics: Modeling feedback, cyclic regulation, counterfactual and macro/micro-level causal analysis in systems biology, climate dynamics, market modeling, or neural activity.
AI and Explainability: ioSCMs and their neural variants enable explainable, robust prediction and counterfactual reasoning in AI/ML systems subject to interventions or distributional shifts.

Limitations and Challenges

Model Assumptions: Identification and inference require that model properties (e.g., memory structure, differentiability, stability) match sufficient conditions for uniqueness or well-posedness.
Computational Tractability: Algorithms for identifiability, adjustment, or structure learning can face computational challenges in large, cyclic, or high-dimensional regimes; the validity of certain simplifications (e.g., input-output elimination, transfer function identifiability) depends on model class (single output, strong connectivity, etc.).
Abstraction and Causal Consistency: The precise preservation of causal semantics across abstraction levels or between micro/macro models requires careful specification of intervention sets and mapping transformations.

Summary Table: Key Features of ioSCMs

Aspect	ioSCM Characterization	Reference
Cycles	Allowed (DMG with directed cycles)	(1901.00433, 2506.19650)
Latent Confounding	Bidirected edges in DMG, explicit latent modeling	(1901.00433)
Adjustment/Backdoor Criterion	σ-separation-based, generalizes d-separation	(1901.00433)
Macro (Cluster) Causal Effects	C-DMGs permit macro-level identification, do-calculus is unconditional	(2506.19650)
Operator-Theoretic State Representation	Natural state operator, minimal and uniquely identifying under conditions	(1009.5277)
Dynamics (ODEs, Markov, etc.)	ODE/DSCM, SDE, Markov models systematically mapped to ioSCM framework	(1304.7920, 1608.08028, 1911.02175)
Frequency-Domain Causal Effects	Path/trek rules and transfer functions in process SCMs	(2305.11561, 1804.03911)
Abstraction and Consistency	Exact transformations between levels; algebraic conditions for representability	(1707.00819, 2306.14351)
Neural/High-dimensional Inference	RINN, GNN, Bayesian deep learning for latent and high-d causal structure	(2003.13135, 2109.04173, 2210.13583)

Conclusion

Input-Output Structural Causal Models (ioSCMs) constitute a mathematically robust, operator- and graph-theoretic extension of SCMs, enabling comprehensive causal reasoning—including identification, adjustment, and counterfactual analysis—in systems with cycles, latent confounding, dynamic feedback, macro-level aggregation, and high-dimensional structure. The ioSCM framework integrates and generalizes methods from functional analysis, information theory, time series analysis, and modern machine learning, providing both a solid theoretical foundation and a practical modeling toolkit for real-world complex systems in science, engineering, and artificial intelligence.