Counterfactual Identifiability via Dynamic Optimal Transport

We appreciate the reviewer's positive assessment and for recognising that our main theoretical result is "original and significant" and "has not been proven elsewhere". We devoted substantial effort to presenting all our results clearly, and we are pleased that this was recognised. We address all the reviewer's comments below.

Q1: The authors should expound upon the baselines and other estimators without consistency proofs: are these clearly incompatible [with] the approach proposed by the authors?

We appreciate the opportunity to elaborate on this. Baselines and other estimators without consistency proofs cannot safely be relied upon to support causal claims. In other words, a lack of identifiability, full or partial, implies the impossibility of making a precise causal claim. This means that the answer can be completely unrelated to the ground truth, and providing such guarantees is what causal methods are designed to do.

For example, the $\mathrm{Naive}$ OT coupling flow baseline can model the observational distribution well, but the inferred counterfactuals are inconsistent, as the model violates Markovianity (c.f. Figure 1). Nasr-Esfahany et al. (2023) reported that their spline flow-based baseline also failed in the Markovian case, only succeeding when the backdoor criterion was applicable. In contrast, we demonstrate that our method produces consistent counterfactual inferences in both the Markovian and Frontdoor criterion settings (c.f. Table 1). We’ll clarify this further in the final version.

Q2: While it is clear strict monotonically makes their approach easier to follow, where is it shown that bijectivity is insufficient alone?

Thank you for pointing this out. We will make this clearer in the final version.

In short, a vector-valued function can be bijective but not monotone. As a simple example, consider the negative identity function:

$T: \mathbb{R}^d \times \mathbb{R}^k \to \mathbb{R}^d, \quad T(u;\mathbf{pa}) = -u$,

which is bijective in $u$, but not vector-monotone:

$\langle T(u_1;\mathbf{pa}) - T(u_2;\mathbf{pa}), u_1 - u_2 \rangle = (-u_1 + u_2)^\top(u_1 - u_2) = -\lVert u_1 - u_2 \rVert ^2_2 < 0$,

as it violates the condition

$\langle T(u_1;\mathbf{pa}) - T(u_2;\mathbf{pa}), u_1 - u_2 \rangle \geq 0$.

Another example is the $T^{(3)}$ mechanism of Eq. (29) in Appendix A.1, which is bijective but not monotone; thus, its counterfactual estimates are shown not to be consistent. In Appendix A.1, we provide a gentle example showing that monotonicity is necessary for consistent counterfactual inferences in the Markovian setting.

Q3: Bounding, hypothesis testing, and structure discovery are all valuable approaches in causal inference, even when a causal model is only partially identified. The authors make clear, however, that consistent counterfactual estimation requires point identification.

We certainly agree that partial identification methods, such as Manski-style bounds (Manski 2003), constitute valuable causal inference approaches. Our intention was not to discount such efforts, but rather to stress that identification, partial or otherwise, is essential for valid counterfactual inference and is a prerequisite for making precise causal claims from observational data. Unfortunately, this fact remains underappreciated in ongoing empirical work on high-dimensional counterfactual inference, posing an operational risk which we highlight and seek to address.

To avoid misunderstanding, we will add the following to the text:

“In settings where full identification is impossible, researchers can still draw valuable causal conclusions through bounding, hypothesis testing, or structure discovery. In this work, however, we concentrate on scenarios that require point-identification to obtain consistent counterfactual estimates from observational data alone.”

Q4: The authors' results apply to counterfactuals regarding atomic interventions and atomic conditions on evidences. Do the authors expect similar results to apply under soft interventions or distributional constraints on evidences?

Thank you for the insightful question. While further theoretical work is needed to accommodate arbitrary soft interventions, as they can alter the underlying causal mechanism and potentially violate (vector) monotonicity, the current method supports a suitably well-defined subset of soft interventions so long as they preserve those properties.

Similarly, distributional constraints on evidence (e.g. $X \sim Q(X)$) are more general than atomic observations $X = x$, and may not yield point-identifiable counterfactuals without additional assumptions. We see this as fertile ground for future research and expect optimal transport to play a valuable role in the analysis. We will highlight this further in our Discussion section.

References

Manski, C.F., 2003. Partial identification of probability distributions. New York, NY: Springer New York.

We thank the reviewer for their positive assessment, noting our integration of ideas across fields, rigorous theoretical claims, clear structure, and thorough coverage of related work. We also appreciate their view that the experiments are a core strength: "very cleverly executed and convincing in validating theoretical claims". Please find our responses to all the reviewer's comments below.

Q1: I am familiar with the work from Nasr-Esfahany et al. (2023) which I understand does similar work for models with one-dimensional variables. What are the challenges going from the results from this work to the multi-dimensional problem? Can the authors highlight which contributions are novel and to what extent?

We thank the reviewer for their feedback and for the opportunity to clarify our contributions.

As pointed out by Nasr-Esfahany et al. (2023) in their Section 5.1:

“It is not clear how to generalize the monotonicity condition to BGMs with multidimensional $V$, which is a known issue in Markovian causal structures (Nasr-Esfahany & Kiciman, 2023).”

This generalisation to $d > 1$ in Markovian causal structures was indeed an open problem, with close connections to both the (non-linear)-ICA (Hyvärinen and Pajunen 1999) and disentanglement (Locatello et al. 2019) literature. In multi-dimensional settings, one’s model is subject to indeterminacies, partly caused by symmetries of the prior $P_U$, which are unresolvable without further constraints.

To the best of our knowledge, we are the first to identify the constraints needed to provably solve this problem. We develop a new theoretical foundation for counterfactual identification of multi-dimensional variables by connecting dynamic Optimal Transport (OT) and causality. Building on this foundation, we introduced a Markovian OT coupling flow which enables the estimation of high-dimensional, unique and monotone causal mechanisms. We demonstrate that our method enables consistent counterfactual inferences of high-dimensional variables ($d > 1$), given only observational data.

Establishing counterfactual identification of high-dimensional Markovian SCMs is both timely and important for the responsible use of causal AI, as many recent works rely on such assumptions to support causal claims (Pawlowski et al. 2020; Sanchez and Tsaftaris 2021; Sánchez-Martin et al. 2022; Monteiro et al. 2023; Ribeiro et al. 2023; Y. Chen, L. Tian, et al. 2024).

Although not the main focus of our analysis, in Appendix A.4 we also extend counterfactual identifiability results from observational data to non-Markovian multi-dimensional ($d > 1$) settings when the following common criteria apply: (i) Instrumental Variable (IV), (ii) Backdoor Criterion (BC), and (iii) Frontdoor Criterion (FC). We prove counterfactual identifiability when the FC applies under similar conditions to the BC setup in Nasr-Esfahany et al. (2023), and expand the instrumental variable results to $d > 1$. These results show our method is widely applicable and useful for all these common causal structures.

To conclude, in the context of the most closely related work by Nasr-Esfahany et al. (2023), we can argue that our contribution represents a breakthrough in causal modelling, with significant implications for practical applications involving counterfactual image generation.

Q2: To my understanding, the proposed approach does the following assumptions: exogeneity, dim(endogenous) = dim(exogenous), and exogenous distribution is uniform. It seems that this setting is too restrictive. Could the authors comment on this?

We thank the reviewer for raising this point, and invite them to refer to the dedicated section on Assumptions & Plausibility in Appendix A.5. In addition, we provide a more succinct response below.

We would like to point out that, aside from the uniform prior, these assumptions are commonplace in the literature (Pawlowski et al. 2020; Monteiro et al. 2023; Ribeiro et al. 2023; Nasr-Esfahany et al. 2023). What sets our work apart is that we provide counterfactual identifiability guarantees for the multi-dimensional case, a set of results that was previously missing for the Markovian, Instrumental Variable and Frontdoor criterion cases. These results are of vital importance to ensure that any causal claims derived from our models are properly supported.

We only require that the prior $P_U$ be a uniform measure for our most strict result. Under different priors, a counterfactual equivalence relation is obtained, in the spirit of many existing identifiability results (Khemakhem et al. 2021; Nasr-Esfahany et al. 2023; Hyvärinen et al. 2024). Thus, relaxing the uniformity constraint on the prior is permissible in exchange for less strict guarantees.

We close with Pearl and Bareinboim (2022):

"Assumptions are self‑destructive in their honesty. The more explicit the assumption, the more criticism it invites. [...] Researchers therefore prefer to declare 'threats' in public and make assumptions in private."

Our work counters this practice: we intend to make our assumptions and constraints explicit so they can be better understood, challenged, and ultimately relaxed in light of new evidence or insight.

Q3: Can the results from this work be easily extended to more complex counterfactuals, e.g., nested counterfactuals?

In Appendix A.4.3, we prove counterfactual identifiability for ($d>1$)-dimensional variables when the Fontdoor Criterion (FC) applies. In the FC setting, we already compute nested or cross-world counterfactuals of the sort found in mediation analysis, as described below.

For example, consider the graph: $X \to M \to Y$, with unobserved confounder $Z$, such that: $X \leftarrow Z \to Y$.
To compute a cross-world counterfactual, we nest two operations.

Given observed evidence $\\{M = m, X = x\\}$, we compute the counterfactual mediator $M_{x'}$ had $X$ been $x'$:

$M_{x'} \mid \\{M = m, X = x \\}$.

In structural equation form, using abduction-action-prediction, we compute:

$m' = f_M(x', f_M^{-1}(x, m))$.

We then compute the counterfactual outcome (nested) given the counterfactual mediator:

$Y_{x',M_{x'}} \mid \\{ Y = y, M = m, X = x \\}$,

which in functional form is given by the nested operation:

$y' = f_Y(m', f_Y^{-1}(m, y)) = f_Y(f_M(x', f_M^{-1}(x, m)), f_Y^{-1}(m, y))$

So, yes, our approach already works for complex counterfactuals.

Q4: Can you comment on the potential connection/overlap of the result from Proposition 4.3 and counterfactual identifiability results under noise monotonicity (Lu et.al. Triantafyllou et.al.).

We thank the reviewer for the pointers. We will add a reference to Triantafyllou et al. (2024).

Definition 4.2 for noise monotonicity in Triantafyllou et al. (2024), as first specified by Pearl (2009) for binary SCMs, specifies a total order $\leq$ on the domain of $V_i$, which naturally applies to scalars ($d=1$), but not vectors ($d>1$). In the absence of a meaningful natural order, readers have thus far only considered $V_i$ as a scalar quantity (Nasr-Esfahany et al. 2023), to the best of our knowledge.

For the $d=1$ case in Proposition 4.3 (c.f. Appendix A.2), our result is practically equivalent to noise monotonicity, in line with Theorem 5.1 in Nasr-Esfahany et al. (2023).

For the $d>1$ case in Proposition 4.3 (c.f. Appendix A.2), we provide a new result by identifying a generalised notion of monotonicity for vectors and its relation to Optimal Transport (OT). We find that since the OT map is the gradient of a convex function, it is a monotone operator, and thus the counterfactual transport map must be ‘vector-monotone’ in $x$; a desired property of counterfactual functions. This is not our main result, but it represents an important motivating step for our study of OT maps as causal mechanisms.

References

Triantafyllou, Stelios, et al. "Agent-specific effects: A causal effect propagation analysis in multi-agent MDPs." ICML, 2024

Lu, Chaochao, et al. "Sample-efficient reinforcement learning via counterfactual-based data augmentation." arXiv preprint arXiv:2012.09092 (2020).

Pearl, J. and Bareinboim, E., 2022. External validity: From do-calculus to transportability across populations. In Probabilistic and causal inference: The works of Judea Pearl (pp. 451-482).

We sincerely appreciate the reviewer’s time in preparing their feedback. We are pleased that the empirical evaluations were found to be "clear and practically relevant". Although we respectfully disagree that clarity is a significant issue, as corroborated by the other reviewers, we commit to incorporating all the suggested changes to further improve clarity. We consider these to be minor and straightforward to address. Please find below our detailed responses to each concern.

Q1: It is unclear to me what precise assumptions are made in Proposition 4.5 and Proposition 4.11. Do we assume monotonicity of f_i? It would also be good to see a discussion about the assumptions and what they mean intuitively.

For an intuitive discussion about the assumptions, we invite the reviewer to refer to the dedicated Assumptions & Plausibility section in Appendix A.5, which appears to have been missed.

To clarify, we treat 'vector-monotonicity' (c.f. Eq. (36)) as a maintained hypothesis about the true data-generating function when applying the model to data. However, it was not obvious that the counterfactual transport map $T^*$ would be strictly vector-monotone in $x$, and we do not assume it a priori in the theory; we prove it is true given only the standard assumptions outlined below.

The assumptions made to prove Proposition 4.5 are stated at the beginning of the proposition, precisely:

1. Equal dimension $\operatorname{dim}(X) = \operatorname{dim}(U)$;
2. Markovian setting $U \perp\\!\\!\\!\perp PA$;
3. $P_U$ and $P_{X|\mathbf{PA}}$ are absolutely continuous w.r.t. the Lebesgue measure, with strictly positive and bounded densities on a bounded, open, convex domain.

Item 3 above corresponds to the assumptions needed for Caffarelli (1992)’s regularity theorem, so the result applies directly, ensuring that the OT map is bijective. For a helpful summary of Caffarelli (1992)’s results, the reviewer may refer to Theorem 12.50 in Villani (2008). In combination, our assumptions ensure that $P_U$ and $P_{X|\mathbf{PA}}$ admit well-behaved densities and guarantee the existence and uniqueness of a dynamic OT map (Brenier 1991; Caffarelli 1992).

The assumptions made to prove Proposition 4.11 are stated at the beginning of the proposition, namely:

1. Assumptions 1, 2 and 3 as listed above for Proposition 4.5;
2. $P_U$ is the continuous uniform measure on $[0,1]^d$;
3. $T$ is the time-1 dynamic OT map that pushes $P_U$ forward to $P_{X|\mathbf{PA}}$ described in Proposition 4.5.

We then prove that the counterfactual transport map $T^*$ must be strictly vector-monotone in $x$. To the best of our knowledge, this is a new result, and it extends the classical monotone-quantile notion to multivariate settings ($d > 1$), thereby guaranteeing counterfactual identifiability from observational data.

Q2: $T$ seems to be a function $T : \mathbb{R}^d \times \mathbb{R}^k \to \mathbb{R}^d$ (line 187) and $T : \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}^k \to \mathbb{R}^d$ (for example line 181). If you abuse notation, please indicate this clearly.

Thank you for raising this. Firstly, we presume the reviewer meant to state that the map on line 181 is $T : \mathbb{R}^k \times \mathbb{R}^k \times \mathbb{R}^d \to \mathbb{R}^d$ rather than $T : \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}^k \to \mathbb{R}^d$. To clarify, we intended to redefine $T$ on line 187 but recognise that this can be made clearer by avoiding an overload in notation with the map on line 181, for example.

To resolve this ambiguity, we will define the counterfactual transport map using $T^* : \mathbb{R}^k \times \mathbb{R}^k \times \mathbb{R}^d \to \mathbb{R}^d$ before its first use in Eq. (8), also ensuring consistency with its use in Proposition 4.11.

Q3: Sometimes you use $X$ and $U$ and sometimes $\boldsymbol{X}$ and $\boldsymbol{U}$. It seems like $X$ and $U$ were never defined. It is unclear to me whether both notations mean the same thing.

As per Definition 3.1, we use $\mathbf{X}$ and $\mathbf{U}$ to denote the set of all the endogenous and exogenous variables in the SCM, respectively. Without loss of generality, we then define $X$ and $U$ on line 142, as relating only to the $i^{\text{th}}$ causal mechanism in the SCM.

Q4: In line 612 the authors require “smooth respective densities” which is an assumption that does not appear in the propositions of the main text.

Thank you for spotting this. We inadvertently stated this in the proof, but it is not necessary to obtain this result, so we will omit it in the final version. Please refer to our response to query Q1 above for an intuitive discussion of the assumptions required to obtain the result.

Q5: Does the fact that the “velocity field is Lipschitz continuous” follow from a previously made assumption? Because it sounds like it is something that is assumed on the spot. If so please include all such assumptions in the proposition.

Thank you for the suggestion. We had treated Lipschitz continuity as a standard standing assumption in the Neural ODE literature. As in R. T. Chen et al. (2018) and Kidger et al. (2020), this condition holds when the velocity field is parameterised by a neural network with finite weights and Lipschitz activations (e.g. ReLU, Leaky-ReLU, tanh, softplus etc). Other smooth activations with bounded derivatives, such as GELU and SiLU, have finite global Lipschitz constants and can be scaled to 1-Lipschitz without changing qualitative behaviour.

To avoid ambiguity, we will state the above explicitly and explain that it follows immediately from our network parameterisation.

Q6: The proof of Proposition A.8 is not quite clear to me. In the proposition, it says that there is a function g for every combination of u and pa. In general these functions can be different. It is therefore unclear to me how the authors arrive at equation (60) as $g_{u_2}$ and $g_{u_1}^{-1}$ seem to not necessarily be equal.

Thank you, we are happy to clarify. Our Proposition A.8 result mirrors Nasr-Esfahany et al. (2023)’s Proposition 6.2, but here it is extended to an optimal transport context.

First, we prove that two equivalent transport maps $T^{(1)} \sim_{\mathcal{L}_3} T^{(2)}$, in the sense of Definition 4.7, produce the same counterfactuals. Transport equivalence implies that there exists an invertible transition map $g$ between the exogenous variables of the two transport maps $T^{(1)}, T^{(2)}$. That is, we have that:

$u^{(1)} = g(u^{(2)}), \quad \text{and} \quad u^{(2)} = g^{-1}(u^{(1)})$.

Then, by the equivalence condition in Definition 4.7, the equation in question (i.e. Eq. (60)) follows directly by substitution:

$T^{(1)}(g(u^{(2)}); \mathbf{pa}^\*) = T^{(2)}(g^{-1}(u^{(1)}); \mathbf{pa}^\*)$.

Q7: If I understand Theorem 5.1 of Nasr-Esfahany et al. (2023) correctly, their results extends to the multivariate case when there is one noise variable for each x_i. In this context, it would be interesting to know what the significance is of modelling multiple noise variables for a single observed variable?

As pointed out by Nasr-Esfahany et al. (2023) in their Section 5.1:

“It is not clear how to generalize the monotonicity condition to BGMs with multidimensional $V$, which is a known issue in Markovain causal structures (Nasr-Esfahany & Kiciman, 2023).”

Our contributions include precisely this missing generalisation to $d > 1$ for the Markovian, Instrumental Variable and Frontdoor Criterion settings. This unlocks counterfactual identifiability for multi-dimensional variables, such as images.

The Markovian setting in particular was an important open problem with close connections to both the (non-linear)-ICA (Hyvärinen and Pajunen 1999) and disentanglement (Locatello et al. 2019) literature, rendering our theoretical results broadly relevant and widely applicable. In the context of the most closely related work by Nasr-Esfahany et al. (2023), we can argue that our contribution represents a breakthrough in causal modelling, with significant implications for practical applications involving counterfactual image generation.

There seems to be a misunderstanding, as we never use multiple noise variables per observed variable. In all cases, we make the equal dimension assumption that $\operatorname{dim}(X) = \operatorname{dim}(U)$, for any multi-dimensional variable $X$ within an SCM $\mathfrak{C}$ (see line 142). For an intuitive discussion about this assumption, please refer to the Assumptions & Plausibility section in Appendix A.5, and the response below.

Q8: It would be nice to have a discussion about what it means that $X_i$ and $U_i$ are assumed to have the same dimension $d$, and about the implications of the bounded support assumption for the noise $U_i$.

The equal-dimension assumption $\operatorname{dim}(X) = \operatorname{dim}(U)$ is commonplace in flow-based generative mode, as it ensures that a bijective transport map is mathematically possible. When this assumption does not hold for the true data-generating process, a bijective model can still be used by, for instance, augmenting the lower-dimensional variable with dummy coordinates.

Lastly, $P_U$ must be absolutely continuous w.r.t. the Lebesgue measure, with a strictly positive and bounded density on a bounded, open, convex domain for Caffarelli (1992)’s regularity theorem to apply. The implication of this is that the OT map is bijective provided that the target measure $P_{X|\mathbf{PA}}$ meets the same criteria. We will make this clearer in the final version.

References

Villani, C., 2008. Optimal transport: old and new (Vol. 338, pp. 129-131). Berlin: springer.

Kidger, P., Morrill, J., Foster, J. and Lyons, T., 2020. Neural controlled differential equations for irregular time series. Advances in neural information processing systems, 33, pp.6696-6707.

We appreciate the reviewer's time and feedback. We've done our best to address each point in detail below, given the limited available space.

Q1: We thank the reviewer for the helpful suggestion. We have added the following definition of $d>1$ monotonicity in the main text (based on Eq. (36)), before Proposition 4.3 where it is first invoked.

Definition (Monotone Operator in $u$). Fix $\mathbf{pa} \in \mathbb{R}^k$. A mapping $f : \mathbb{R}^k \times \mathbb{R}^d \to \mathbb{R}^d$ is monotone in $u$ if

$\langle f(\mathbf{pa},u_1)-f(\mathbf{pa},u_2), u_1-u_2\rangle \geq 0 \quad \text{for all} \\, u_1, u_2 \in \mathbb{R}^d$.

We also now include the following implications paragraph after Proposition 4.3 based on the material previously in Remarks A.2 & A.4.

"Monotonicity of the counterfactual transport map $T^*$ w.r.t. $x$ ensures rank preservation: individuals with higher factual outcomes retain higher counterfactual outcomes (no rank inversions), which supports fairness and consistent counterfactual inferences (c.f. App A.1). As we will show, the Optimal Transport (OT) specialisation of this map generalises quantile and rank functions for $d>1$, ensuring multivariate rank preservation. Intuitively, mass is pushed along gradients of a convex potential; the gradient structure prevents "folding'', and its monotonicity enforces directional consistency so that points that start close remain close under the map, thereby preserving order and preventing overlaps."

Q3: In Definition 3.1, we deliberately do not specify the variables as scalars explicitly, to allow for multivariate ($d>1$) generalisations of $X_{i}, U_{i}$. We then clarify our intentions on line 107, stating that we focus on counterfactual identification for multi-dimensional variables $X_i \in \mathbf{X}$. Without loss of generality, on line 142 we reiterate our intent and drop the $i$ subscript on $X, U$ to avoid notational clutter. We hope this clarifies our intention, and we are happy to consider suggestions for improvements.

Q6: We appreciate the reviewer's observation and identify that the placement of the existence quantifier in Eq. (57) for the function $g$ as the cause for the confusion. To address this, we have amended it to correctly state that the existence quantifier precedes the universal ones, i.e. it now reads: $\exists g, \forall \\, u, \mathbf{pa}$, etc.

This aligns with Definition 6.1 in Nasr-Esfahany et al. (2023), where a single, invertible function $g$ is required to ensure equivalence across all values of $u$ and $\mathbf{pa}$. There is no dependence of $g$ on $\mathbf{pa}$, nor is there a need for a different $g$ for each value of $u$.

Importantly, this amendment is purely notational and does not affect any of the theoretical results. We have revised the text accordingly to clarify this point and avoid further ambiguity.

Q7: Thank you for the question; we are happy to clarify. The reviewer is correct, and our Markovian SCM may have $n$ such structural assignments. In our analysis, we focus on the $i^{\text{th}}$ mechanism in the Markovian SCM without loss of generality.

"Could the authors clarify the advantage of this [our] formulation?"

We appreciate the reviewer's subtle point. Beyond being the natural parameterisation for multi-dimensional variables, our formulation lets global factors (e.g. image brightness/contrast) act coherently across pixels. From an Independence of Cause and Mechanism (ICM) perspective, the mechanisms stay fixed while global shifts are absorbed by $P_U$. In contrast, the component-wise model $X_i = f_i(\operatorname{PA}_{X}, U_i)$ can only transmit dependence through parent links, so mimicking global shifts requires intricate, coordinated retuning of many $f_i$, undermining modularity.

As pointed out by Nasr-Esfahany et al. (2023) in their Section 5.1:

"It is not clear how to generalize the monotonicity condition to BGMs with multidimensional $V$, which is a known issue in Markovian causal structures (Nasr-Esfahany & Kiciman, 2023)."

To the best of our knowledge, we are the first to identify the constraints needed to provably solve this problem.

Q8: Thank you for your comment. We point out that assuming $X$ and $U$ are random vectors of equal dimension is commonplace in prior work using SCMs, see e.g. Pawlowski et al. (2020)'s invertible/explicit mechanism, Nasr-Esfahany et al. (2023)'s BC result for $d>1$, and (Sanchez and Tsaftaris 2021, Rasal et al. 2025) using diffusion. Assuming $P_U$ has bounded support is also not uncommon in the literature, see e.g. Definition 3 in Xia et al. (2021), and their subsequent line of work. We hope this addresses the reviewer's concern, as such assumptions are not uniquely ours.

Rasal et al. Diffusion Counterfactual Generation with Semantic Abduction. ICML 2025.
Xia et al. The causal-neural connection: Expressiveness, learnability, and inference. NeurIPS 2021.

Please find below our responses to the remaining (minor) remarks in the initial review; we were not able to address these directly in our initial rebuttal due to space constraints. If there are any other points we have missed, please do let us know, and we will be more than happy to clarify.

Q9-10: The term "counterfactual query" is used in Definition 4.2 but never defined mathematically. Definition 4.2 is a definition and a statement in one. Please use the definition environment for definitions only.

Thank you for your feedback. I appreciate your point about separating the definition and the statement. Our definition is designed to be consistent with Pearl (2009)'s Definition 3.2.3, with theirs applying to any query $Q$. Given this established treatment, we chose to structure ours similarly, as counterfactual queries are subsumed therein. I hope this clarifies the reasoning behind our approach, but we are open to further adjustments if needed. For example, we can first define $Q$ to be any computable quantity of a model as in Pearl (2009)'s Definition 3.2.3, then proceed with our Definition 4.2.

Q11: $T$ is defined outside an environment and inside. Specifically, in line 187, 193 and 227. Are these all the same function?

Yes, these are all the same function. In line with our proposal to Q2, we will omit the definition on line 187 to avoid redundancy. If agreed by the reviewer, we can also avoid redefining $T$ on line 227 by writing: "Using the time-1 dynamic OT map $T$ defined in Proposition 4.5, which pushes $P_U$ forward to $P_{X|\mathbf{PA}}$ ...". We would be happy to consider alternative phrasings and suggestions.

Q12: There is no such thing as “satisfying a proposition” (line 181, 228 for example). Please state clearly what assumption you make.

Thank you for the feedback. We will rephrase line 181 to read: "Monotonicity of the counterfactual transport map $T^\*$ w.r.t. $x$ (c.f. Proposition 4.3) guarantees that...".

As per Q11, we will also adjust line 228 to read: "Using the time-1 dynamic OT map $T$ defined in Proposition 4.5, which pushes $P_U$ forward to $P_{X|\mathbf{PA}}$ ...". We trust these changes address the reviewer's concern. If a restatement of the assumptions made in Proposition 4.5 would be more appropriate, please let us know and we will append them to the sentence above.

Q13: There is a “motivating example” in the Appendix under “Proofs: Counterfactual Identifiability”. This seems misplaced.

Thank you for pointing this out. We agree and will move the motivating example to a new, different section directly above the “Proofs: Counterfactual Identifiability” section in the Appendix.

Q14: In Equation (9) $x$ on the left hand side seems to be fixed while on the right hand side this is not the case.

Thank you for spotting this. We can fix $x$ on the right-hand side, which results in a Dirac point mass $\delta_{T^{\*}(\mathbf{pa}^*, \mathbf{pa}, x)}$, for a deterministic counterfactual map $T^\*$. We will amend the paper to reflect this change.

Q15: Propositions in the Appendix do not have the same numbers as the corresponding Propositions in the main text.

Thank you for raising this. We will revise the Appendix so that any Proposition restated there carries the same number as in the main text (e.g., “Proposition 4.3 (restated)”), and we’ll update all cross-references.

Dear Reviewer,

Thank you again for the time and effort you have dedicated to our manuscript.

We respectfully disagree with your two central criticisms. Below are our final remarks for your and others' consideration.

1. Scope relative to Nasr-Esfahany et al. (2023)

Nasr-Esfahany et al. themselves note in Section 5.1:

“It is not clear how to generalize the monotonicity condition to BGMs with multidimensional $V$, which is a known issue in Markovian causal structures (Nasr-Esfahany & Kiciman, 2023).”

Applying their $d=1$ result component-wise introduces three key limitations:

1. Impose an arbitrary coordinate order. Scalars have a natural order; high-dimensional variables such as images do not;
2. Force coordinate-wise strong monotonicity, restricting models to pixel-wise autoregressive functions;
3. Global effects (e.g., image brightness/contrast) must be expressed through an intricate coordination of pixel-wise mechanisms, undermining mechanism modularity.

Our new approach represents the natural extension to $(d>1)$-dimensional variables, and avoids all three issues.

2. Mathematical Clarity

We consider the original exposition clear, and other reviewers echoed that judgment:

• "clearly written, with minimal typographical errors"
• "well-organized"
• "Section 4 presents a series of theoretical results in a clear and structured manner"
• "well-written and nicely structured"
• "Necessary background is well-covered given the space constraints"
• "authors have also made a very meticulous job in covering related work"
• "authors communicate their approach very well"

In our view, the suggested edits are clarifications rather than substantive changes, as they leave every theoretical result and conclusion intact. All of your suggestions are nonetheless appreciated and have already been incorporated.

Thank you again for your time and for helping strengthen our paper.

Best regards,
The authors

We thank the reviewer for their positive feedback. We are pleased that our counterfactual identifiability result was deemed "a significant and meaningful contribution". We also appreciate the recognition of our paper's clarity, minimal typographical issues, and well-organised structure, particularly the presentation of theoretical results.

Q1: The strong assumptions (e.g., known DAG, bijective and monotonic SCM) might limit the practical usefulness of the method. Could the authors elaborate on potential application scenarios where these assumptions are reasonable?

In Section 6.2, we demonstrate the practical utility of the method on a high-dimensional medical imaging problem, outperforming existing works by a large margin.

Known DAG: Many causal relationships are known in the medical domain, thanks to a long history of results from randomised controlled trials and meta-analyses. This positions healthcare as the ideal application area for causal modelling, as one can tap into existing medical knowledge to help build causal graphs.

Bijectivity: Bijective causal mechanisms (Pawlowski et al. 2020, Nasr-Esfahany et al. 2023) subsume many model classes with known identifiability results, e.g., (non)-linear additive noise models (Shimizu et al. 2006; Hoyer et al. 2008; Peters et al. 2014), post-nonlinear models (K. Zhang and Hyvärinen 2009), and location-scale models (Immer et al. 2023). As such, they represent an attractive model class with practical instantiations using modern flows.

Monotone Map: Learning high-dimensional monotone maps is challenging. However, this is now provably possible in practice thanks to recent advances (Pooladian et al. 2023; Tong et al. 2024). When the model is misspecified and/or the assumptions are violated, guarantees may not hold. This is not unique to our approach, but holds for all identification results.

Many recent works already rely on such assumptions (Pawlowski et al. 2020; Sanchez and Tsaftaris 2021; Monteiro et al. 2023; Ribeiro et al. 2023), but their causal claims remain speculative without identification, representing an operational risk. Thus, our work represents an important advance for the responsible use of causal AI.

We also provide theoretical results for non-Markovian settings (Section 4.3), when 3 common criteria apply: Backdoor Criterion, Instrument Variable and the Frontdoor Criterion, rendering our approach widely applicable.

Q2: My main concern for the synthetic dataset is about the selection of P(U). If a more complicated P(U) is used in the data-generating process, would the generation result be affected?

Intuitively, a more complex data-generating process (DGP) is expected to be harder to model.

To test this hypothesis, we created new datasets with multimodal $P_U$. We use an indicator $s_i \sim \text{Bern}(0.5)$ to induce a non-linear, bimodal shift $\mu_i \in \\{-2, 2\\}$ into $U$'s mechanism. We also use a categorical shift, with $\mu_i \in \\{-2, 0, 2\\}$, to generate another multimodal setting. More details on all new experiments will be added to the paper.

We confirm that more complex $P_U$'s can affect generation, but importantly, the relative rank of our models remains consistent. Naturally, we expect larger models to reduce performance gaps for complicated $P_U$'s.

Q3: How does the proposed method perform when certain assumptions are violated? Specifically, without the monotonicity assumption or without bijectiveness, to what extent is performance affected?

To thoroughly address this, we ran a new set of experiments.

In addition to measuring counterfactual error, we inspect the Composition ($\downarrow$) and Reversibility ($\downarrow$) counterfactual soundness axioms, to reveal performance disparities under assumption violations.

We use the Mean Average Percentage Error (MAPE) metric, and 50/250 steps for ODE/SDE solving.

Experiment 1: Monotonicity violation

We use 20 cycles for composition and reversibility measurements; models which are not vector-monotone exhibit significantly inferior counterfactual soundness.

Experiment 2: Bijectivity violation

We use 3 strategies to stress-test bijectivity violations: (i) stochastic abduction by solving an SDE instead of an ODE; (ii) stochastic prediction by solving an SDE; (iii) classifier-free guidance (CFG). One cycle was used for composition and reversibility.

We observe that bijectivity violations affect reversibility and composition significantly, as it becomes harder to 'undo' interventions.

Experiment 3: Markovianity violation

We observe that composition and reversibility are reasonable for the Naive OT-Flow version, as the map remains bijective and monotone. However, the counterfactual error is significantly higher for the Naive version, as the violation leads to inconsistent counterfactuals.

Q4: In Proposition 4.11, we need to identify/learn the g between the uniform prior and the true prior. We cannot directly learn the exogenous prior from data; thus, minimizing the transport cost might lead to a g that is not the true g we want.

Thank you for the thoughtful question. To clarify, learning the prior transition map $g$ explicitly is not strictly necessary.

Let the true generator be $x = f(u;\operatorname{pa})$, $u \sim P_U$. We train a single OT-Flow $T(u;\operatorname{pa})$, $u \sim U(0,1)^d$, to match the output law.

Although there exists a unique OT solution for $T$, we make no claims that it factorises exactly into $f \circ g$, for any choice of $P_U$. There could be a map with a lower transport cost that only implicitly encodes $g$.

To illustrate this simply, consider a rubber-cube representing the uniform prior $U(0,1)^d$.

1. A warp $g$ stretches the cube so that the embedded points follow the true prior $P_U$;

2. A map $f$ then moulds the cube into data space.

A single map $T$ could enable steps 1 & 2 in unison if this reduces transport cost. In analogous terms, a path A-to-C may be closer than a path A-to-B-to-C. We will make this point clearer in the final version.

Q5: What is the assumption on the interventions: soft intervention or hard intervention? It seems that the paper requires the intervention to preserve the monotonicity of the interventional SCM. Is that too restrictive?

We focus on hard interventions as they enable many useful real-world applications. This is exemplified by our experiments in Section 6.2, and ongoing work in: (i) evaluating the impact of interventions (Kusner et al. 36 2017; Tsirtsis and Rodriguez 2023), (ii) understanding cause-effect relationships (Karimi et al. 2020; Budhathoki et al. 2022), (iii) generating targeted synthetic data (Pitis et al. 2022; Roschewitz et al. 2024), and (iv) quantifying disease severity (Mehta et al. 2025).

While further theoretical work is needed to accommodate arbitrary soft interventions, as they can alter the underlying mechanism, the current method supports a suitably defined subset of soft interventions that preserve vector-monotonicity.

Q6: In the synthetic experiment, what is Z? (line 287)

We thank the reviewer, as their question helped us find a typo. The random variable $Z$ is part of the data-generating process, a confounder satisfying the Backdoor Criterion (BC) w.r.t. $\operatorname{PA} \to X$. We include it to reproduce Nasr-Esfahany et al. (2023)’s BC result. In the text, we inadvertently state that we randomised $Z$ to induce a Markovian setting, but what we did was randomise $\operatorname{PA}$, ensuring $U \perp\\!\\!\\!\perp \operatorname{PA}$. We will amend this in the final version.

Q7: Intuitively, in section 6.1, given a point observation $X=x$, how possible could we learn the whole ellipse? Could you elaborate more on the experiment and connect it to the theoretical assumptions and guarantees?

A point observation of $X = x$ fixes the location but underdetermines the shape of the ellipse; thus, infinitely many ellipses can be drawn through that point. By further observing the angle $\operatorname{PA} = \operatorname{pa}$, it is possible to infer the semi-major and -minor parameters of the ellipse via causal abduction. The whole ellipse for an observed $(\operatorname{pa}, x)$ can then be estimated by answering counterfactual queries for all angles in $(0,2\pi)$.

We can provably learn the correct model by using our proposed Markovian OT-Flow coupling. In contrast, the Naive coupling violates Markovianity, entangling $U$ with $\operatorname{PA}$, and yielding inconsistent counterfactuals. This constructed scenario is crucial for verifying our theoretical claims. It shows that when our assumptions and constraints hold (i.e. Markovian/FC setting, monotone OT-Flow map), counterfactual identification in $d>1$ from observational data is possible.

Mehta, R. CF-Seg: Counterfactuals meet Segmentation. arXiv preprint arXiv:2506.16213.