Counterfactual Causal Spaces

• Counterfactual Causal Spaces are rigorous mathematical frameworks that combine factual and hypothetical worlds through probability measures and causal kernels.
• They extend traditional causal models with structural equations, graphical representations, and non-deterministic formulations to enable cross-world reasoning.
• These frameworks find practical use in domains such as fairness, clinical decision-making, and generative modeling while presenting open challenges like learning in cyclic systems.

Counterfactual causal spaces formalize the joint consideration of factual and counterfactual worlds within rigorous mathematical structures that support the modeling, identification, and computation of causal effects under hypothetical scenarios. These frameworks range from measure-theoretic probability spaces, product causal spaces equipped with explicit kernels, and system representations accommodating spillover/interference, to graphical constructions for cross-world reasoning and novel semantic generalizations. This entry summarizes foundational definitions, axioms, modeling alternatives, identification theory, computation, and practical implications of counterfactual causal spaces, drawing upon measure-theoretic, algebraic, statistical, and algorithmic perspectives.

1. Measure-Theoretic Foundations of Counterfactual Causal Spaces

Counterfactual spaces are constructed as products of world-specific measurable spaces, providing a joint domain for "parallel worlds." Formally, the underlying space is

$$\Omega = F\Omega \times CF\Omega,\qquad \mathcal{H} = F\mathcal{E} \otimes CF\mathcal{E},$$

where $(F\Omega, F\mathcal{E})$ and $(CF\Omega, CF\mathcal{E})$ encode the factual and counterfactual worlds (Park et al., 1 Jan 2026).

A Counterfactual Probability Space (CPS) is $(\Omega, \mathcal{H}, P)$ with $P$ a probability measure. A Counterfactual Causal Space (CCS) is a CPS augmented by a family of causal kernels $\{K_S\}_{S \subseteq T}$, where each $K_S$ encodes transition probabilities respecting world- and interventional structure.

Key axioms for CCSs:

• Trivial intervention: $K_\emptyset(\omega, A) = P(A)$.
• No cross-world effect: Kernels on factual events depend only on the factual coordinates and vice versa.
• Interventional determinism: Interventions pin down coordinates, ensuring probabilistic consistency.

Such spaces generalize classical causal models by separately specifying the stochastic relationships (probability measure $P$) and causal dependencies (kernels $K$), enabling the representation of varying degrees of shared information (ranging from independence to synchronisation of worlds) (Park et al., 1 Jan 2026). These constructions extend beyond Pearl’s ladder of causation by treating interventions and counterfactuals as orthogonal mathematical operations.

2. Structural Equation and Graphical Model Extensions

Structural Causal Models (SCMs) and associated graphical structures are central to operationalizing counterfactual causal spaces.

Consider a deterministic SCM $(U, V, F)$, with exogenous $U$, endogenous $V$, and functions $F$ assigning values based on parents and exogenous variables. Counterfactual queries are typically answered via an "abduction–action–prediction" sequence: first inferring exogenous context, then intervening on $F$ (or $U$ in backtracking semantics), and finally propagating to derive outcomes (Kügelgen et al., 2022).

Cross-world SCMs explicitly model real and counterfactual worlds, possibly via "teleporters"—variables unaffected by intervention and functionally invariant—linking the two worlds in a graphical representation. Teleporter Theory introduces rigorous criteria (d-separation and functional invariance) for variable identification and yields plug-and-play modules enabling cross-world symbolic reasoning and adjustment formulae derivation (Li et al., 2024). Twin- and k-world network constructions underpin algorithmic treatment of multiple counterfactual scenarios while controlling computational complexity via bounded treewidth (Han et al., 2022).

Partial equilibrium, local interaction, and network equilibrium regimes articulate different counterfactual adjustment rules in systems with interdependent units (e.g., firm networks), each mapping to distinct potential outcome objects and requiring different exogeneity for identification (Mate, 1 Jan 2026).

3. Canonical, Nondeterministic, and Probabilistic Generalizations

Canonical representations of SCMs formalize the equivalence class of all models compatible with observed and interventional data by specifying the "one-step-ahead counterfactual process" $S^{(i)}$ for each node. Normalization procedures further separate the learnable kernel $\psi_i^\mathcal{C}$ from latent dependence $N^{(i)}$, enabling analysts to simulate arbitrary admissible counterfactual worlds by freely choosing normalization while preserving observational/interventional constraints (Lara, 22 Jul 2025).

Nondeterministic SEMs (NSEMs) generalize classical SCMs by allowing multi-valued structural functions $F_X:\mathrm{Pa}_X \to \mathcal{P}(\mathcal{R}(X))$. Solutions preserve observed assignments during interventions, permitting non-uniqueness of counterfactual worlds. Sound and complete axiomatizations govern such models, with probabilistic extensions (PNSEMs) defining conditional PMFs and direct identification in Causal Bayesian Networks (Beckers, 2024).

The possible-worlds semantics of Lewis-Stalnaker are formally related to recursive causal models, with embeddings demonstrating equivalence in acyclic cases, but pointing to fundamental incomparability once cycles or unique-solution systems without recursion are admitted (Halpern, 2011).

4. Identification, Estimation, and Computational Complexity

Identification of counterfactual causal effects in interdependent networks depends critically on the choice of regime and the corresponding scope of exogeneity:

• PE regime: direct effect, requires $D_i \perp \epsilon_i \mid X_i$.
• LI regime: first-order spillover, requires $$D_i \perp \{\epsilon_j : j \in N_i \cup \{i\}\} \mid X$$.
• NC regime: total equilibrium effect, requires global ignorability $D \perp \epsilon \mid X$ (Mate, 1 Jan 2026).

SAR estimators in spatial/network models must be interpreted with care, as the same regression output can map to different causal objects depending on the regime and ignorability assumption. Network feedback amplifies both true effects and any endogeneity bias, showing the necessity of counterfactual specification for valid inference.

From a computational perspective, counterfactual reasoning via twin- or k-world networks is tractable whenever associational/interventional reasoning is tractable, with treewidth inflated by only a small constant or linear factor compared to the original graph (Han et al., 2022).

5. Kernel and Representation-Learning Formulations

Counterfactual causal spaces also encompass RKHS-based and deep generative frameworks.

Counterfactual Mean Embeddings (CME):

Counterfactual distributions are embedded in a reproducing kernel Hilbert space (RKHS), enabling nonparametric inference of entire outcome landscapes. The Distributional Treatment Effect is quantified by the RKHS norm between CME points. Under unconfoundedness and overlap, consistent and rate-optimal estimates are achievable; structured outcome domains such as images or graphs are accommodated via kernel choice (Muandet et al., 2018).

Variational and Generative Models:

Recent neural architectures, including variational causal inference frameworks (Wu et al., 2024) and Causal Diffusion Autoencoders (Komanduri et al., 2024), build explicit latent spaces encoding exogenous noise subject to disentanglement constraints. These models perform end-to-end counterfactual supervision (often without explicit counterfactual data) and enable generation of high-dimensional counterfactuals, supporting both identification theory and empirical benchmarks in genomics or computer vision.

6. Modeling Alternatives: Interventionist vs. Backtracking Semantics

While Pearl's interventionist account answers counterfactual queries by holding $U$ fixed and modifying $F$ via "surgical" do-operations, alternative "backtracking" semantics update $U^*$ (exogenous initial conditions), keeping $F$ unchanged (Kügelgen et al., 2022). Backtracking causal spaces introduce a "backtracking conditional" $P_B(U^*|U)$ specifying the similarity or distance between exogenous assignments across worlds. This approach is particularly relevant to explainable AI, recourse analysis, and other domains where maintaining the data manifold and causal coherence are essential.

The two perspectives differ fundamentally:

Feature	Interventionist	Backtracking
Altered object	$F \to F_{x^*}$	$U \to U^*$ (via $P_B$)
Shared object	$U$ fixed	$F$ fixed
Typical uniqueness	Single solution	May be non-unique
Applicability	SCM, potential outcomes	XAI, realistic recourse

7. Implications, Applications, and Open Problems

Counterfactual causal spaces unify and extend classical models, supporting the mathematical treatment of a broader spectrum of counterfactuals—including those without explicit interventions or singular structural equations. Applications span fairness, clinical decision-making, harm analysis, generative modeling, and explanation. The orthogonality of intervention and counterfactual dimensions in these frameworks provides modularity, flexibility, and mathematical clarity absent in traditional approaches.

Open problems include the generic learning of CCS structure and measure from data, identification in cyclic or non-deterministic systems, characterization of actual causality ("token causation"), and empirical grounding of cross-world dependency mechanisms. The expressivity and tractability of these models provide foundational infrastructure for future research in counterfactual analysis across domains (Park et al., 1 Jan 2026, Lara, 22 Jul 2025, Beckers, 2024, Kügelgen et al., 2022, Muandet et al., 2018, Han et al., 2022).