Counterfactual Spaces (2601.00507v1)

Abstract: We mathematically axiomatise the stochastics of counterfactuals, by introducing two related frameworks, called counterfactual probability spaces and counterfactual causal spaces, which we collectively term counterfactual spaces. They are, respectively, probability and causal spaces whose underlying measurable spaces are products of world-specific measurable spaces. In contrast to more familiar accounts of counterfactuals founded on causal models, we do not view interventions as a necessary component of a theory of counterfactuals. As an alternative to Pearl's celebrated ladder of causation, we view counterfactuals and interventions are orthogonal concepts, respectively mathematised in counterfactual probability spaces and causal spaces. The two concepts are then combined to form counterfactual causal spaces. At the heart of our theory is the notion of shared information between the worlds, encoded completely within the probability measure and causal kernels, and whose extremes are characterised by independence and synchronisation of worlds. Compared to existing frameworks, counterfactual spaces enable the mathematical treatment of a strictly broader spectrum of counterfactuals.

• The paper introduces an axiomatic foundation for counterfactual probability spaces that rigorously models information sharing between factual and counterfactual worlds.
• Its methodology distinctly separates counterfactual reasoning from interventions by formalizing both as orthogonal extensions to traditional probability spaces.
• The framework generalizes SCMs and potential outcomes, enabling flexible modeling of arbitrary cross-world dependencies and complex causal queries.

Counterfactual Spaces: An Axiomatic Foundation for Stochastic Counterfactuals

Introduction and Motivation

The paper "Counterfactual Spaces" (2601.00507) proposes an axiomatic, measure-theoretic foundation for counterfactual reasoning distinct from standard causal frameworks. Motivated by limitations in dominant approaches—such as potential outcomes (POs) and structural causal models (SCMs)—the authors introduce counterfactual probability spaces and counterfactual causal spaces. Both are probability or causal spaces over products of world-specific measurable spaces, thereby supporting explicit modeling of parallel factual and counterfactual (or multiple "possible") worlds. This construction enables rigorous representation of the information sharing between worlds, a conceptual aspect previously treated only indirectly or under restrictive assumptions in SCM- or PO-based formalisms.

The authors critically diverge from Pearl's paradigm, which structurally nests counterfactuals within causal models, treating interventions as essential. Instead, they axiomatize counterfactuals and interventions as separate, orthogonal extensions to probability spaces. Interventions are thus formalized independently from the cross-world relationships crucial to counterfactuals, with the conjunction of these two yielding the "counterfactual causal spaces" framework.

This orthogonal-decomposition is visually contextualized in their adapted version of Pearl’s ladder of causation (Figure 1).

Figure 1: Left, Pearl’s ladder of causation; right, the view of counterfactuals and interventions as orthogonal axes, which combine to yield counterfactual causal spaces.

Mathematical Framework

Counterfactual Probability Spaces

Given probability spaces $(\Omega, H, P)$, the counterfactual probability space is defined on the product measurable space $$(F\Omega \times CF\Omega, F\mathcal{H} \otimes CF\mathcal{H})$$, with $F\Omega$ and $CF\Omega$ representing the spaces of factual and counterfactual outcomes, respectively.

The probability measure $P$ on the joint space not only gives marginals for each world but, crucially, encodes the coupling between worlds. The degree of shared information is precisely captured:

• Independence: $F\mathcal{H} \indep_{P} CF\mathcal{H}$ means events in the two worlds are stochastically independent—there is no cross-world information transfer.
• Synchronization: $F\mathcal{H} \equiv_P CF\mathcal{H}$ (almost sure equality) gives maximal information coupling—conditioning on one world completely determines the other.

Arbitrary intermediate couplings are also possible via the measure $P$, supporting richer and more flexible modeling—e.g., allowing partial common causes, conditional synchronization, or other cross-world dependencies.

Counterfactual Causal Spaces

A counterfactual causal space $(\Omega, H, P, K)$ adds a structured causal mechanism $K$ (a set of transition probability kernels) to the above product space. The formal axioms enforce:

• No cross-world causal effects: Interventions in one world do not causally influence events localized to another world; causal influence is only within-world. All nontrivial cross-world structure arises from the coupling in the measure, not causal mechanisms.
• Orthogonality: This separation formalizes interventions and counterfactuals as independent axes, and allows their arbitrary recombination.

After intervention, the changed shared information structure ("causal kernels") is also expressible in this framework.

Generalization to N-way counterfactual probability and causal spaces (multi-world scenarios) is straightforward, recursively applying the product measure and kernel construction.

Comparison to Established Frameworks

Beyond SCMs and Potential Outcomes

Classic SCM-based counterfactual inference always ties factual and counterfactual worlds deterministically via shared exogenous variables and structural equations—synchronizing the worlds except for explicitly-intervened-upon variables. This restricts the framework to situations where all randomness is common except for stipulated interventions, precluding scenarios with, e.g., partial dependence, latent common causes, or more general cross-world stochasticity.

Potential outcomes represent another special case, but again fix the worlds' structures in advance and lack mechanisms for arbitrary cross-world dependencies or for formalizing non-interventional counterfactuals.

By contrast, the counterfactual spaces framework permits modeling arbitrary information sharing pre- and post-intervention. The authors explicitly construct counterfactual spaces that subsume both SCM and PO frameworks, proving strict generalization.

Relaxing Strong Assumptions

Nontrivial modeling implications follow:

• No acyclicity required: Cyclic causal structures, including those relevant for dynamic or feedback systems, are admissible.
• Continuous and discrete time: No restrictions to discrete or finite variables—a significant regime, e.g., temporally continuous systems, becomes accessible.
• Flexible exogenous/endogenous modeling: The independence constraints between endogenous and exogenous variables, entrenched in SCMs for the abduction-action-prediction paradigm, are optional.

Expressivity and Query Handling

A key achievement of the framework is supporting a strictly broader class of counterfactual queries, including:

• Counterfactuals with arbitrary cross-world coupling (not solely via explicit interventions)
• Backtracking counterfactuals (conditioning on alternative "history" scenarios)
• Conditional synchronization (shared randomness conditional on some event)
• Noninterventional counterfactuals (“What would have happened had $X$ occurred, given what we observed, if we observed but did not force $X$?”).

Illustrative model examples demonstrate all these, highlighting the formal, query-agnostic nature of the framework.

Theoretical and Practical Implications

The theoretical implications are substantial:

• Unified axiomatic foundation: Probability, causality, and counterfactual inference (including their interacting forms) are unified as extensions of the Kolmogorov measure-theoretic canon.
• Greater generality: Models previously inexpressible (due to acyclicity, structural determinism, shared exogenous variables, or discrete state limitations) are now subsumed.
• Explicit control of world-to-world relationships: The framework promotes explicit modeling of what precisely is "shared" between worlds and to what extent, opening research avenues for statistical, philosophical, and automated reasoning approaches to counterfactuals.

Practically, the framework can shape the design of:

• Quantitative counterfactual reasoning for machine learning, including generative models, XAI (explainability), and fairness (cf. [kusner2017counterfactual, garg2019counterfactual, rosenblatt2023counterfactual]).
• Stochastic control, policy simulation, or legal/ethical analysis in settings where classical assumptions do not hold.
• Applications in RL, healthcare, or economic modeling where interventions and "alternative histories" must be evaluated under complex or uncertain cross-world dependencies.

Implications for Future Research

The separation of counterfactual and interventional reasoning axes suggests new directions:

• Actual causality: More precise forms of token causality can be formalized by relaxing synchrony or sharpening sufficiency/necessity conditions.
• Mechanistic interpretability: Increased flexibility in world coupling supports robust mechanistic abstraction (cf. [geiger2025causal]), crucial for aligning LLMs and other generative systems with human-interpretable semantics.
• Learning algorithms: Machine learning algorithms can be devised to estimate shared information structure—the counterfactual measure—directly from multi-world data or through sophisticated simulation protocols.

Conclusion

"Counterfactual Spaces" (2601.00507) introduces a mathematically rigorous, general, and axiomatic architecture for counterfactual reasoning situated within standard probability and measure theory, extending and clarifying prior causal modeling paradigms. The framework:

• Allows for independent, flexible specification of interventions and counterfactuals.
• Encodes a principled spectrum of information sharing across worlds, from independence to synchronization and intermediate couplings.
• Strictly generalizes prior frameworks (SCMs, POs), removing restrictive assumptions while supporting a richer set of possible queries and modeling goals.

This axiomatization provides a robust foundation for future work in both the theory and application of counterfactual reasoning, especially in domains where classical assumptions cannot be maintained or empirical data is sparse, incomplete, or drawn from heterogeneous sources.