Title: Counterfactual Identifiability via Dynamic Optimal Transport
Venue: NeurIPS 2025
Accepted: yes
Keywords: generative models

causality

counterfactual inference

optimal transport

continuous normalizing flows

flow matching
Abstract: We address the open question of counterfactual identification for high-dimensional multivariate outcomes from observational data. Pearl (2000) argues that counterfactuals must be identifiable (i.e., recoverable from the observed data distribution) to justify causal claims. A recent line of work on counterfactual inference shows promising results but lacks identification, undermining the causal validity of its estimates. To address this, we establish a foundation for multivariate counterfactual identification using continuous-time flows, including non-Markovian settings under standard criteria. We characterise the conditions under which flow matching yields a unique, monotone and rank-preserving counterfactual transport map with tools from dynamic optimal transport, ensuring consistent inference. Building on this, we validate the theory in controlled scenarios with counterfactual ground-truth and demonstrate improvements in axiomatic counterfactual soundness on real images.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider counterfactuals of an outcome with respect to its parent set, assumed causally sufficient. Under strict monotonicity and regularity conditions referenced in the manuscript, the authors prove nonparametric identifiability from observational data using dynamic optimal transport. They construct an estimator and show that it provides consistent results in experiments (simulated and real world) where nonparametric baselines without rigorous identifiability proofs appear to be inconsistent.\n\nSTRENGTHS AND WEAKNESSES:\nTo the best of my knowledge, the authors are correct in saying that nonparametric counterfactual identification under monotonicity with multidimensional treatments has not been proven elsewhere. This result is therefore original and significant. The authors communicate their approach very well. \n\nThere are a few points which were not entirely clear to me: \n- The authors should expound upon the baselines and other estimators without consistency proofs: are these clearly incompatible the approach proposed by the authors?\n- While it is clear strict monotonically makes their approach easier to follow, where is it shown that bijectivity is insufficient alone? \n\nI disagree with the authors in the second paragraph of their introduction that point-identifiability is necessary for valid causal inference. 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.\n\nQUESTIONS:\nThe 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?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work studies counterfactual identifiability from observational data in models with no unbserved confounders and high-dimensional variables. The proposed approach is is a dynamic optimal transport flow mechanism strictly monotone in the target variable. On the practical side, the authors describe a specialized Batch OT coupling approach suitable for Markovian model, i.e., exogeneity is satisfied. Extensive experiments are performed on real data to validate the the paper's claims.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- The paper combines ideas from several fields and provides a rigorous theoretical analysis. I did not manage to follow all ideas and results presented, but regardless I recognize that this paper is well-written and nicely structured. Necessary background is well-covered given the space constraints. The authors have also made a very meticulous job in covering related work.\n- Experimental results are the main strength of this paper. Very cleverly executed and convincing in validating theoretical claims.\n\nWeaknesses\n\n- (Contributions) It feels that most results are derived or similar to prior work. 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?\n- (Assumptions) 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?\n\nQUESTIONS:\nPlease see weaknesses from above.\n\nAdditionally,\n\n(Q1) Can the results from this work be easily extended to more complex counterfactuals, e.g., nested counterfactuals?\n(Q2) 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.).\n\nLu, Chaochao, et al. \"Sample-efficient reinforcement learning via counterfactual-based data augmentation.\" arXiv preprint arXiv:2012.09092 (2020).\nTriantafyllou, Stelios, et al. \"Agent-specific effects: A causal effect propagation analysis in multi-agent MDPs.\" ICML, 2024\n\nLIMITATIONS:\nMentioning more clearly assumptions in conclusion and how they limit the pracictability of the work would be beneficial for readers.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work provides theoretical guarantees for counterfactual identifiability in structural causal models (SCMs). The main result of the paper is an extension of Theorem 5.1 of Nasr-Esfahany et al. (2023) to the multivariate setting. The authors also provide empirical evaluations of their methodology.\n\nMain assumptions: \n* A SCM with independent noise variables, that is, for all $i \\in \\{1, \\dots, n\\}$ \n\t$$X_i = f_i(X_{pa(i)}, U_i) \\qquad U_i \\perp X_{pa(i)}$$\nwhere $U_i$ and $X_i$ are assumed to be $d$-dimensional random vectors.\n* For all $i \\in \\{1, \\dots, n\\}$ the functions $f_i$ are bijective with respect to $U_i$.\n* Knowledge of the observational distribution of the SCM and its induced graph is assumed.\n\nGoal:\nThe goal is counterfactual identifiability, that is, the authors want to provide conditions under which a counterfactual query $Q$ is identifiable. \n\nUnder the aforementioned assumptions and in the univariate case, that is $d=1$, and assuming strict monotonicity of $f_i$ Nasr-Esfahany et al. (2023) establish counterfactual identifiability. The goal of the present paper is to extend this result to the multivariate case, that is, $d>1$.\n\nSTRENGTHS AND WEAKNESSES:\nQuality:\nI am not familiar enough with the optimal transport literature to verify that the proofs are correct. However, it is sometimes not clear to me what assumptions are made to get the stated conclusion. Specifically, for Proposition 4.5 and 4.11. For example, it is not clear to me what “satisfying a proposition” (line 181, 228) means. I am also unsure about the correctness of Proposition A.8. I go into more detail about the proof of Proposition A.8 in the section \"Questions\".\nI like the empirical evaluations; they are clear and seem practically relevant.\n\nClarity:\nIn my opinion clarity is a significant issue with this paper. Especially in the theoretical part, the paper is missing mathematical clarity.\nIt is often unclear what assumptions are made. For example, 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. Below is a list of sections where I believe there are issues with clarity:\n* The term \"counterfactual query\" is used in Definition 4.2 but never defined mathematically. \n* Please only number the equations that you are actually referencing.\n* It seems like the counterfactual dynamic OT map $T^* $ is used in the main text but not defined. Maybe Equation (77) in the Appendix defines $T^*$? If so, please state this explicitly in the main text.\n* Sometimes \"$\\coloneqq$\" is used for definitions and sometimes not. Please be consistent in your use of notation.\n* Definition 4.2 is a definition and a statement in one. Please use the definition environment for definitions only.\n* $T$ 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.\n* It also seems like $T$ is defined outside an environment and inside. Specifically, in line 187, 193 and 227. Are these all the same function?\n* There is no such thing as “satisfying a proposition” (line 181, 228 for example). Please state clearly what assumption you make.\n* Sometimes you use $U$ and $X$ and sometimes $\\boldsymbol{U}$ and $\\boldsymbol{X}$. It seems like $U$ and $X$ were never defined. It is unclear to me whether both notations mean the same thing.\n* There is a “motivating example” in the Appendix under “Proofs: Counterfactual Identifiability”. This seems misplaced.\n* In Equation (9) on the left hand side $x$ seems to be fixed while on the right hand side this is not the case.\n* “… absolutely continuous with densities strictly positive on a bounded, open, convex domain w.r.t. the Lebesgue measure”. It probably should be “… absolutely continuous w.r.t. the Lebesgue measure and with densities strictly positive on a bounded, open, convex domain.”\n* Propositions in the Appendix do not have the same numbers as the corresponding Propositions in the main text.\n* In line 612 the authors require “smooth respective densities” which is an assumption that does not appear in the propositions of the main text.\n* Line 726: 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.\n\nSignificance:\nI would have liked to see more discussion about the relevance of multivariate noise vectors for counterfactual identifiability. It seems to me that Theorem 5.1 of Nasr-Esfahany et al. (2023) trivially extends to the multivariate case if there is one noise variable for each $X_i$, that is, $X_i = f_i(X_{pa(i)}, U_i)$. For which problems is it important to model multiple noise variables per observed variable?\n\nOriginality:\nThis is difficult for me to answer as I do not fully understand the assumption made in this paper. However, it seems that the proof of Theorem 5.1 of Nasr-Esfahany et al. (2023) can be extended to the multivariate setting by assuming strong monotonicity of the $f_i$ (i.e. the generalisation of monotonicity to the multivariate setting) and then use Brenier’s Theorem to get the unique, monotone transport map that pushes $U$ to $X$. $T$ is then the replacement of the inverse of the CDF in the proof of Theorem 5.1 of Nasr-Esfahany et al. (2023). The theoretical results seem to mirror this idea. However, I cannot find the strong monotonicity assumption for the $f_i$. Could the authors clarify this point?\n\nQUESTIONS:\n* Do you require strong monotonicity of the $f_i$'s? If not, how are your assumptions distinguished from strong monotonicity?\n\n* 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 $g^{-1}_{u_1}$ seem to not necessarily be equal.\n\n* 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? Are there specific applications you have in mind? For example, intuitively, why would multiple noise variables be advantageous for the X-Ray experiment?\n\nLIMITATIONS:\nIt 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 $U_i$.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nCounterfactual identifiability ($\\mathcal{L}_3$) from observational data is both important and challenging. According to Pearl's Causal Ladder, it is generally impossible without (often strong) assumptions. Previous methods for counterfactual identifiability often require model assumptions such as linearity, bijectivity, Lipschitz continuity, or the Markovian property. They also rely on data assumptions such as faithfulness, atomic interventions, interventional data, or paired data (e.g., from randomized controlled trials or counterfactual datasets) obtained under different interventional distributions.\n\nThis paper, primarily extending the work of Nasr-Esfahany et al. (2023), builds theoretical guarantees based on bijectivity and monotonicity assumptions to extend identifiability from one-dimensional endogenous variables $X_i$ to $d$-dimensional endogenous variables $X_i$. The goal is to infer counterfactual queries of $X_i$ given its parents $\\textbf{pa}_i$.\n\nThe theortical contributions of the paper follow this roadmap:\n\n1. Proposition 4.3 shows that if the mechanism is monotone in $u$, then the counterfactual transport map $T$ is monotone in $x$, linking the data-generating process assumption to a learnable property of the function.\n\n2. Proposition 4.5 shows that, under several assumptions, monotonicity leads to a unique mapping $T$ determined by $P_U$ and $P_{X|\\textbf{PA}}^\\mathfrak{C}$.\n\n3. Proposition 4.8 attempts to prove that two transport maps are counterfactually equivalent if there exists an invertible map $g$.\n\n4. Proposition 4.10 shows the existence of a monotone, bijective map $g^*$ that minimizes the quadratic transport loss.\n\n5. Proposition 4.11, building on the previous propositions, establishes counterfactual identifiability for Markovian Structural Causal Models (SCMs).\n\nFurthermore, the paper proposes a practical matching method using dynamic optimal transport (OT) flow and demonstrates its effectiveness on both synthetic datasets and a real-world X-ray dataset.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. The extension of counterfactual identifiability in Bijective Causal Models from one-dimensional to $d$-dimensional endogenous variables is a significant and meaningful contribution.\n\n2. The paper is clearly written, with minimal typographical errors.\n\n3. The paper is well-organized. In particular, Section 4 presents a series of theoretical results in a clear and structured manner.\n\n**Weaknesses**\n1. The strong assumptions and setting (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?\n\n2. 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?\n3. Due to the restrictive assumptions made in this paper, additional experiments might be beneficial. For example, 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?\n\n**Minor Comments**\n\n1. In Definition 4.1 (Markovian SCM.), the period inside the parentheses should be removed.\n\n2. On page 3, line 106, what does $P_X^{\\mathfrak{C}_{x'}|X=x}$ mean? Particular, how is an SCM conditioned on $X = x$ interpreted? Clarification would be helpful.\n\nQUESTIONS:\n1. In Proposition 4.8, two transport maps $T^{(1)}$ and $T^{(2)}$ are equivalent if there exists an invertible transition map $g$, and in Proposition 4.10, two prior distributions can be mapped by a unique, bijective, and monotone transport map $g^\\*$ that minimizes the quadratic transport cost. But in Proposition 4.11, we need to identify/learn the $g$ between the two priors: the uniformly distributed prior and the unknown true prior. We cannot directly learn the exogenous prior from data; thus, minimizing the quadratic transport cost might lead to a $g$ that is not the true $g^\\*$ we want.\n\n2. 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?\n\n3. In the synthetic experiment, what is $Z$? (line 287)\n\n4. 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?\n\nLIMITATIONS:\nthis paper discusses the limitation briefly in the conclusion section.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-04"}