Structural Causal Model Distance (SCMD)

Updated 24 October 2025

Structural Causal Model Distance (SCMD) is a metric that quantifies differences in interventional distributions between structural causal models.
It employs both kernel-based embeddings and adjustment strategies to capture discrepancies in graph structure and functional parameters.
SCMD is used to benchmark causal discovery algorithms, assess model transferability, and evaluate interventional as well as counterfactual performance.

Structural Causal Model Distance (SCMD) is a family of metrics designed to quantify the (dis)similarity between structural causal models (SCMs), with a core focus on differences in interventional or causal effect predictions, rather than solely on graphical structure or marginal distributions. SCMD measures play a central role in evaluating the performance of causal discovery algorithms, benchmarking model transferability across environments, and providing a principled approach to assess model generalization in tasks involving interventions or potential distribution shifts.

1. Foundational Principles and Definitions

SCMD is defined to capture both structural and functional (parametric) discrepancies between SCMs, particularly in terms of their implied interventional distributions. While various distances have been used in the literature to compare models—such as the Structural Hamming Distance (SHD) or Structural Intervention Distance (SID)—SCMD is distinguished by its explicit goal: to measure the difference between the full causal semantics (the collection of causal queries an SCM can answer) that two models embody.

Given two SCMs, $\mathcal{M}^1$ and $\mathcal{M}^2$ , over variables $V_1, \ldots, V_d$ possibly under different causal graphs and mapping functions, SCMD quantifies the aggregate “distance” between all relevant interventional distributions predicted by each model. A canonical form, as introduced in recent kernel-based approaches (Goff et al., 23 Oct 2025), is:

$\mathrm{SCMD}(\mathcal{M}^1, \mathcal{M}^2; v^1, v^2) = \sum_{i,j=1}^{d} \left\| \mu_{P_1(\cdot | do(V_i = v^1_i))}(V_j) - \mu_{P_2(\cdot | do(V_i = v^2_i))}(V_j) \right\|_{\mathcal{H}}$

where $\mu_{P(\cdot | do(V_i = v_i))}(V_j)$ denotes the RKHS mean embedding of the interventional distribution of $V_j$ under $do(V_i = v_i)$ , and the summation aggregates over all variable pairs.

This kernel-based approach enables SCMD to be sensitive to both structure (e.g., which variables are parents, edge orientation) and functional form (e.g., parameter or noise distribution changes), in contrast to classical graph-edit distances.

2. Methodologies for Quantifying SCMD

Two principal strands exist for the quantitative formulation and computation of SCMD:

(a) Graph- and Adjustment-Based Distances

Earlier proposals such as the Structural Intervention Distance (SID) (Peters et al., 2013, Henckel et al., 13 Feb 2024) define SCMD-like metrics as the count of (ordered) node pairs $(i, j)$ for which the predicted intervention distribution (typically using parent adjustment) differs between a learned and ground-truth SCM across all Markov-compatible distributions:

$\mathrm{SID}(G, H) = \#\{ (i, j) : \text{H's prediction of } P(X_j | do(X_i)) \text{ is incorrect relative to G} \}$

Extensions consider more sophisticated adjustment sets (ancestor adjustment, optimal adjustment) (Henckel et al., 13 Feb 2024), or even all valid formulas returned by a complete identification strategy, and extend this counting approach to CPDAGs and partial orders.

(b) Functional/Distributional and Kernel-Based Distances

Kernel-based SCMDs (Dhanakshirur et al., 2023, Goff et al., 23 Oct 2025) embed interventional distributions as elements of a reproducing kernel Hilbert space (RKHS), then employ Maximum Mean Discrepancy (MMD) or its conditional/interventional variants to continuously quantify differences:

For sets of interventions and targets, SCMD computes the normed difference between RKHS embeddings of the interventional distributions.
SCMD summarizes per-intervention/pairwise effects (for all $i,j$ ) and may also average over multiple intervention values (E-SCMD) or focus selectively on key targets (P-SCMD).

This approach captures not only wrong or missing interventions but also functional changes in causal mechanisms and noise, even when these do not induce observable differences in the marginal or conditional distributions.

(c) Counterfactual and Abstraction-Oriented Notions

The causal distances “ladder” (Peyrard et al., 2020) formalizes the hierarchy:

Observational Distance (OD): difference in observed data distributions.
Interventional Distance (ID): difference in do-intervention distributions.
Counterfactual Distance (CD): difference in counterfactual queries after conditioning on evidence.

SCMD subsumes ID in this hierarchy, while higher-rung CD becomes relevant in settings involving personalized interventions.

3. Theoretical Properties and Guarantees

Recent SCMD instantiations have established key theoretical properties:

Metricity: SCMD is a proper metric—non-negative, symmetric, and obeying the triangle inequality—under standard kernel choices (Goff et al., 23 Oct 2025).
Consistency: Kernel-based SCMD admits plug-in estimators that are universally consistent; estimation risk converges at $O(N^{-1/4})$ for conditional embeddings with regularization, compared to $O(N^{-1/2})$ for marginal embeddings (Goff et al., 23 Oct 2025).
Discriminative Power: SCMD subsumes SID’s discriminative power—if SCMD is zero, SID must also be zero, but not vice versa—which means SCMD detects subtler differences such as functional/parametric changes undetectable by graph structure alone.
Handling of Markov Equivalence: Separation-based distances (Wahl et al., 7 Feb 2024) offer a metric on the set of Markov equivalence classes by comparing the (in)dependence structure induced by d-separation/m-separation, bridging structural SCMD and distributional effect-based SCMD.

4. Applications in Causal Inference and Generalization

SCMD is deployed in several analytical and empirical settings:

Benchmarking Causal Discovery: By measuring the causal “distance” between a discovered and a ground-truth SCM, SCMD enables evaluation that is sensitive to both structural correctness and predictive fidelity for interventions (Peyrard et al., 2020, Goff et al., 23 Oct 2025).
Transferability and Domain Alignment: High SCMD between source and target environments signals likely generalization difficulties—whether due to structural non-invariance or changes in intervention response—guiding model selection and adaptation in domain generalization or transfer learning (Goff et al., 23 Oct 2025).
Interventional and Counterfactual Evaluation: SCMD provides a foundation for model validation that moves beyond observed-data likelihood, relevant for settings with intervention data or causal reasoning tasks.
Bridge to Model Abstraction: Mapping between SCMs at different levels of granularity (micromodels to macromodels) can be evaluated by decomposing SCMD into structural and distributional components, with interventional consistency providing an additional ‘commutativity’ constraint (Zennaro, 2022).

5. Advances, Implementations, and Computational Aspects

Advances in computing SCMD focus on scalability and expressiveness:

Algorithmic Efficiency: Efficient reachability algorithms for identification-based SCMD (adjustment distances) allow for polynomial-time or even linear-time computation relative to graph size, enabling applicability to large-scale graphs (Henckel et al., 13 Feb 2024).
Software and Estimation: Open-source implementations (e.g., the gadjid package) have brought these metrics to practical causal model evaluation. Modern RKHS-based estimators with regularization ensure robust empirical estimation even when only observational data are available.
Empirical Utility: Synthetic and real-world experiments (e.g., on the Sachs protein network) demonstrate that SCMD can distinguish between models with identical graph structure but differing functional form, as well as detect subtle but practically significant changes that would be missed by classic structural distances (Goff et al., 23 Oct 2025, Dhanakshirur et al., 2023).
Continuous Metrics: Innovations such as continuous Structural Intervention Distance (contSID) provide nuanced, scale-invariant measures suitable for quantitative assessment in settings where effect magnitude matters (Dhanakshirur et al., 2023).

6. Extensions and Open Directions

SCMD continues to serve as a unifying concept for several threads in causal inference:

Dynamics & Equilibrium: Mappings from ODEs and dynamical systems to SCMs at equilibrium naturally induce SCMD as a metric to compare physical and causal models according to their equilibrium interdependence (Mooij et al., 2013, Mooij et al., 2014, Bongers et al., 2018).
Extreme Causal Structures: In the tails (extremes), SCMs may collapse to sparser “extremal” SCMs, with SCMD quantifying the loss or preservation of causal connectivity under rare events (Engelke et al., 9 Mar 2025).
Counterfactual Flexibility: Distribution-consistency SCMs (DiscoSCMs) and individual causal inference via SCM abduction operators generalize the notion of SCMD to measure “distance” between models in their counterfactual predictions, reflecting uncertainty or capacity constraints beyond classic SCMs (Gong et al., 29 Jan 2024, Chang, 17 Jun 2025).
Joint Dependence & Mechanism Shifts: Distance covariance and related metrics have been employed for model selection in SCMs by testing joint independence of residuals; these provide alternative/auxiliary dependence-focused SCMD implementations (Chakraborty et al., 2017, Chen et al., 2023).

7. Comparative Table of Principal SCMD Approaches

Metric/Approach	Basis	Captures
SID (Peters et al., 2013)	Adjustment criterion, graph-only	Qualitative graph structure: correct/incorrect intervention statements
Adjustment Identification (AID)	Adjustment strategies, graph	Identifiability/formula-based differences; scalable computation
Ladder Distances (Peyrard et al., 2020)	Distributions under interventions/counterfactuals	Fine-grained, rung-wise causal difference (OD, ID, CD)
Kernel-based SCMD (Goff et al., 23 Oct 2025)	Interventional RKHS embeddings	Structural + parametric differences; continuous metric
contSID (Dhanakshirur et al., 2023)	RKHS, interventional distributions	Effect magnitude sensitivity, continuous and quantitative
Separation-based (Wahl et al., 7 Feb 2024)	d-separation/m-separation structure	Differences in implied conditional independence/Markov equiv. class

This table synthesizes foundational methods with recent continuous and empirical approaches, reflecting the breadth of SCMD applications and formulations.

Structural Causal Model Distance thus provides a rigorous framework and a practical suite of metrics for comparing and analyzing SCMs, sensitive to both graph structure and interventional behavior, and underpins a wide span of developments from causal discovery benchmarking and evaluation of model transfer, to new directions in counterfactual and individualized causal inference.