Consolidated SCMs

• Consolidated SCMs are structural causal models that integrate marginal models across overlapping datasets while preserving counterfactual semantics.
• They utilize response-function parametrization and linear constraints to maintain consistency in discrete settings and optimize polytope-based formulations.
• This approach enables multi-context causal inference, robust falsification via variable addition, and the derivation of tighter bounds on counterfactual influences.

Consolidated Structural Causal Models (SCMs) generalize and extend the classical SCM framework to enable integrated reasoning across multiple marginal models, datasets, or system boundaries. In the discrete (categorical) setting, consolidation addresses the challenge of merging partially overlapping, possibly independently developed, SCMs into a joint model with coherent counterfactual semantics. This construction underpins advanced methods in multi-context causal inference, federated learning of causal mechanisms, and variable-based falsification, and relies on explicit response-function parametrizations and convex polytope characterizations.

1. Foundational Problem: Structural Causal Marginal Problem

The formal consolidation problem is the Structural Causal Marginal Problem (SCMP), introduced and rigorously treated for categorical SCMs by Bongers, Forré, and Mooij (Gresele et al., 2022). Given $m$ marginal SCMs

$$M_i = (V_i, U_i, F_i, P_{U_i}), \quad i=1,\dots, m,$$

each defined over a distinct or overlapping variable subset $V_i \subset V$, the SCMP asks whether there exists a joint SCM $M = (V, U, F, P_U)$ whose (counterfactual) marginals on each $V_i$ agree with those of $M_i$ for all $i$. The focus is on categorical (finite-state) SCMs, where the state and exogenous-value spaces are discrete and where model equivalence is cast in terms of interventional and counterfactual distributions.

Counterfactual consistency of the consolidated model requires that for all interventions or counterfactual queries posed within any $V_i$, the distributions induced by $M$ match those of the corresponding $M_i$. This is a stricter requirement than simple marginalization, encompassing post-intervention and counterfactual worlds.

2. Response-Function Parametrization and Linear Constraints

Consolidation leverages the response-function representation for categorical SCMs. For each variable $V_j$ with domain $\mathcal{V}_j$ and parents $PA_j$, all possible deterministic functions $f_j: PA_j \to \mathcal{V}_j$ are enumerated, and an exogenous response function $R_j$ is sampled to select the instantiation. The structural equations become:

$V_j := \tilde{f}_{j, R_j}(PA_j),$

with $R_j$ drawn from $P_{R_j}$.

Given all $R = (R_1, \ldots, R_n)$, the observational distribution is

$$P(V = v) = \sum_{r} P_R(r) \prod_j \mathbf{1}\{v_j = \tilde{f}_{j, r_j}(pa_j)\}.$$

Marginal SCMs correspond to restrictions or projections onto subsets of $(V, R)$.

Imposing counterfactual consistency between the consolidated model and each marginal SCM yields a collection of linear constraints on the joint response-function probability vector $c = (c_k)$, where each $c_k$ corresponds to a particular instantiation of the response functions (i.e., a vertex of the full behavioral response space). Each marginal parameter must match its induced value in the consolidated model—e.g., for Boolean $A,B,C$ with marginals $(A,C)$ and $(B,C)$:

$$a_j = \sum_{b} \sum_{k:\;h_k(\cdot, b) = f_j(\cdot)} c_k, \qquad b_j = \sum_{a} \sum_{k:\;h_k(a, \cdot) = f_j(\cdot)} c_k,$$

where $h_k$ denotes a response function from $(A, B)$ to $C$.

The feasible set of consolidated SCMs is the polytope:

$$\mathcal{C} = \{c \in \Delta^{K-1} \mid A c = a(\lambda^A),\; B c = b(\lambda^B),\; \lambda^i_{min} \leq \lambda^i \leq \lambda^i_{max} \}$$

where $A$, $B$ are constraint matrices mapping joint response probabilities to marginals, and the simplex constraint $c_k \geq 0$, $\sum c_k = 1$ enforces a valid probability mass.

3. Decomposition, Space Reduction, and Falsifiability

A primary effect of imposing multiple marginal SCMs is to reduce or carve out the set of joint SCMs admissible under all constraints. This yields more stringent bounds on counterfactual influence parameters and narrows the set of possible response behaviors. The resulting solution space is a convex polytope, amenable to polytope vertex enumeration and projection.

A central result (Gresele et al., 2022) is that in the Boolean two-cause-to-one-effect case, as long as the marginal SCMs are statistically compatible (i.e., there exists some joint $P_{A,B,C}$), the consolidated SCM can always realize maximal marginal counterfactual influences for each cause. Hence models with maximal weights on constant response functions cannot be ruled out by simply adding another variable unless further constraints or datasets are available.

This construction introduces a novel mode of falsification: variable-based falsifiability. Adding new variables via additional marginal SCMs (even without observing all variables jointly) can preclude entire families of models that would otherwise be consistent with observed data on smaller variable sets.

4. Solution Methods and Algorithmics

The consolidated SCM derivation reduces to a linear (and convex) optimization problem or feasibility check, solvable by standard LP or polytope enumeration tools (e.g., pypoman). For operational use:

1. Enumerate response functions for the full set of variables.
2. Impose consistency constraints for each marginal SCM as linear equality constraints on the response function weights.
3. Solve for feasibility (i.e., is the constrained polytope non-empty).
4. Enumerate vertices or optimize for counterfactual quantities of interest across the feasible set.

If multiple marginal SCMs have overlapping variable sets, the approach generalizes by augmenting the constraint matrix. Projections onto parameter subspaces yield tightened bounds for each marginal factor, naturally reducing the parameter intervals.

5. Exemplification: The Boolean Two-Cause-One-Effect Case

Consider variables $A, B, C \in \{0,1\}$, with marginal distributions $P_{A,C}$ and $P_{B,C}$ given from two datasets. Each marginal SCM is parametrized by a “counterfactual influence” parameter (e.g., $\lambda_A$ for the effect of $A$ on $C$ conditional on unmeasured factors).

Given a parameterization,

$$a(\lambda_A) = \text{baseline} + \lambda_A \cdot \text{effect\ vector},$$

the joint SCM probability vector $c$ is determined by the solution to the set:

$$a_j(\lambda_A) = \sum_{b} \sum_{k:\;h_k(\cdot, b) = f_j(\cdot)} c_k, \quad b_j(\lambda_B^*) = \sum_{a} \sum_{k:\;h_k(a, \cdot) = f_j(\cdot)} c_k,$$

subject to simplex constraints. In a worked example, feasibility can restrict $\lambda_A$ to a singleton, fully determining the marginal model and yielding a manifestly consolidated joint SCM (Gresele et al., 2022).

6. Practical and Theoretical Implications

Consolidated SCMs are foundational for multi-source causal inference in settings where only overlapping marginals are available. Their characterization via polytopes and linear constraints enables the integration of evidence across studies, simulates interventions not directly shown in any dataset, and provides strong formal tools for ruling out mechanistic hypotheses without joint data collection.

The mode of falsification rooted in variable addition signals that model adequacy can be challenged by expanding the variable set rather than increasing data per se, representing a new dimension in causal discovery and validation.

7. Relation to Broader SCM Methodologies

While the work of Bongers et al. (Gresele et al., 2022) formalizes consolidation in the categorical SCM setting, the principles extend to continuous variables under suitable discretization or response-surface expansion, although the problem quickly becomes computationally intractable there. The method complements standard model selection, identification, and estimation approaches by leveraging linear-algebraic structure for rigorous consistency checks and enabling direct polyhedral exploration of model space.

In summary, consolidated SCMs provide a mathematically rigorous, computationally tractable framework for merging and interrogating families of marginal causal models in the discrete setting, yielding tighter parameter bounds, new falsifiability strategies, and operational workflows for cross-study causal integration.