Exogenous Isomorphism for Counterfactual Identifiability
Authors: Yikang Chen
Dehui du
Venue: ICML 2025 spotlightposter (2025)
Keywords: causality
causal inference
counterfactual identifiability
Primary area: general_machine_learning->causality

Abstract: This paper investigates $\sim_{\mathcal{L}\_3}$-identifiability, a form of complete counterfactual identifiability within the Pearl Causal Hierarchy (PCH) framework, ensuring that all Structural Causal Models (SCMs) satisfying the given assumptions provide consistent answers to all causal questions. To simplify this problem, we introduce exogenous isomorphism and propose $\sim_{\mathrm{EI}}$-identifiability, reflecting the strength of model identifiability required for $\sim_{\mathcal{L}\_3}$-identifiability. We explore sufficient assumptions for achieving $\sim_{\mathrm{EI}}$-identifiability in two special classes of SCMs: Bijective SCMs (BSCMs), based on counterfactual transport, and Triangular Monotonic SCMs (TM-SCMs), which extend $\sim_{\mathcal{L}\_2}$-identifiability. Our results unify and generalize existing theories, providing theoretical guarantees for practical applications. Finally, we leverage neural TM-SCMs to address the consistency problem in counterfactual reasoning, with experiments validating both the effectiveness of our method and the correctness of the theory.

Exogenous Isomorphism for Counterfactual Identifiability
Yikang Chen' Dehui Du
Abstract
thus confirm the reliability of the constructed model 1odlel-11 if counterfactual identifiability is guaranteed.
This paper investigates ~v -ddnttiiiifiabillttyy. a form of complete counterfactual identifiability within the Pearl Causal Hierarchy (PCH) framework, en- suring that all Structural Causal Models (SCMs) satisfying the given assumptions provide consis- tent answers all causal questions. To simplify this problem, we introduce exogenous isomor- phism and propose "egriidetiffiaility.. reflecting the strength of model identifiability required for - -ic-iieeniiiabiiiityy We explore sufficient assump- tions for achieving -iittiiiiiii in two spe- cial classes of SCMs: Bijective SCMs (BSCMs), based on counterfactual transport, and Triangu- lar Monotonic SCMs (TM-SCMs), which extend ~e, -cc--identiiin -iientiiiabiiity. Our results unify and general- ize existing theories, providing theoretical guaran- tees for practical applications. Finally, we lever- age neural TM-SCMs to address the consistency problem in counterfactual reasoning, with exper- iments validating both the effectiveness of our method and the correctness of the theory.
(Pearl & Mackenzie, 2018) categorized causal questions into three levels of human cognition, termed the Pearl Causal Hierarchy (PCH) (Bareinboim et al., 2022). Counterfactual reasoning corresponds to human imagination, situated at the highest level L3 of the PCH, encoding the most intri- cate and nuanced information. Structural Causal Models (SCMs) provide specific semantics for addressing counter- factual questions. Counterfactual identifiability on SCMs guarantees that all SCMs adhering to the assumptions yield consistent results for La quantities. Prior research has con- strained causal structures to identify specific counterfactual effects (Shpitser & Pearl, 2008; Correa et al., 2021; Xia et al., 2023), or limited causal mechanisms to determine counterfactual outcomes (Lu et al., 2020; Nasr- fasr-Esfahany et al., 2023; Scetbon et al., 2024). Further studies investigate model identifiability for counterfactuals, offering empirical evidence that counterfactual outcomes are identifiable (Khe- makhem et al., 2021; Javaloy et al., 2023).
This work introduces a novel identification target: identifica- tion across the entire counterfactual layer of the PCH. This requires that all SCMs satisfying the assumptions yield con- sistent results for any counterfactual statement. Within the PCH, since the counterfactual layer L3 encodes all causal information, the identifiability of the SCMs in L3 implies that these SCMs provide consistent answers to all causal questions, rendering them indistinguishable for any causal statements. Thus, identifiability over the counterfactual layer represents the most stringent and comprehensive goal for causal quantity identifiability within the PCH.
1. Introduction
The purpose of counterfactual reasoning is to answer ques- tions about hypothetical, unobserved worlds. has been applied in tasks such as fairness evaluation (Kusner al., 2017), explanation generation (Karimi et al., 2020), harm quantification (Richens et al., 2022), and policy optimiza- tion (Tsirtsis & Rodriguez, 2023). Counterfactual identifi- cation a subtask of counterfactual reasoning, aiming to determine whether all models satisfying certain reasonable assumptions yield the same answer to a specific counter- factual question. It is critical ensure the dependability of counterfactual reasoning results, as the reasoning out- comes can only be consistent with the assumptions - and
To achieve this goal, we first examine the identifiability prob- lem causal inference from the perspective of model identi- fiability in Section 2. In this context, identifiability over the counterfactual layer is denoted as ~ ~c, -idettifiabiiltyy. To simplify the ~ca ~ca- -identifiabbiytty problem, ternative form of identifiability in Section 3, denoted ~Er identifiability, which is induced by an equivalence relation termed exogenous isomorphism. This form of identifiability demonstrates that fully recovering the exogenous variables of SCMs is unnecessary to ensure consistent results for any counterfactual quantities, clarifying the strength of assump- tions required for counterfactual layer identification.
establish al-
"Shanghai Key Laboratory of Trustworthy Computing, East China Normal University. Correspondence to: Dehui Du <dhdu@ @ ecw.c.du.uu..
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Exogenous Isomorphism for Counterfactual Identifiability
We then explore sets of assumptions that induce ~er- identifiability, focusing on two special classes of SCMs. The first class, termed Bijective SCMs (BSCMs), is stud- ied in Section 4, where we induce exogenous isomorphism between BSCMs from the perspective of counterfactual transport (Theorem 4.6), offering a novel interpretation for counterfactual identifiability. The second class, termed Tri- angular Monotonic SCMs (TM-SCMs), is examined in Sec- tion 5. We identify a simple method to induce exogenous isomorphism between TM-SCMs (Corollary 5.4), which can be viewed as a strengthened version of ~ L. -identifiability for Markovian Causal Bayesian Networks. This corollary unifies and generalizes prior theories for proving counter- factual outcome identifiability, indirectly demonstrating that several models used in previous works for counterfactual outcome identifiability are theoretically -c -idettifiable This provides theoretical guarantees for their safe applica- tion in practice. Finally, leveraging this theory, we utilize TM-SCMs parameterized by neural networks in Section 6 to address counterfactual identification and estimation prob- lems. Experimental results in Section 8 on these models empirically support the validity of our theory.
Px. defined as the pushforward of P under X, such that Px = X,P A set X E Fx represents random event, with the associated probability term P(XX E x) = Px (2).
2.1. Structural Causal Model
Definition 2.1 (Structural Causal Model (SCM)). An SCM is tuple MM- = (I, Sv Lu, f. Pv), where I is a finite index set. The product space Siv = II,ez Sly, and Qu Ihe Sty denote the domain of endogenous and exogenous variables, respectively. The measurable function f : Sly x Du + Gy = (fe)izz specifies the causal mechanisms, where each fo Sly x Sty + Sly, is associated with a set of indices pa(i) C I. The probability measure Pu is called the exogenous distribution. If Po = I Pv, is product measure, the SCM is said to be Markovian.
In this work, we primarily focus on recursive SCMs, which are defined such that a partial order = exists on the index set I, and for any pair of indices i,j E I, if j 3 i, then & pa(j). If the partial order = admits a linear extension S. we call it the causal order of the SCM, which can be constructed using a topological sorting algorithm and is not necessarily unique. Recursiveness ensure that the solution of the SCM exists and is unique. According to (Bongers et al., 2016), if an SCM has unique solution, then there exists a function I such that for almost all u E Pu and v E Ωy v = [uu if and only if v = f(v, u). We refer to I as the solution mapping of the recursive SCM.
2. Preliminaries
In this section, we provide the background for understand- ing the problem and describe the problem setting. We begin by reviewing the relevant concepts of SCMs. In Section 2.1, we present the definition of SCM and introduce key notions such as do-intervention and causal order. Subsequently, in Section 2.2, we leverage these concepts to logically charac- terize Li, L2. L3 in PCH and define La-consistency. Next, in Section 2.3, we introduce counterfactual identification problem. Finally, in Section 2.4, we elaborate on model identifiability, establish the connection between causal iden- tifiability and model identifiability, and motivate the re- search objective of this work: ~c, -identifiability.
SCMs also allow for operations called do-interventions. For subset of indices Ix S I. performing an intervention to set X = Vix to a specific value x corresponds to deriv- ing a new SCM MMx] = (I, Su. fl). Pu), called a submodel. In M fix] = (fipx)kez is defined such that
x i Ix,
Si i i
Notation We study the problem from a more general measure- perspective. Let the probability space be F.. where the set Ω is the sample space, the a- algebra F on Q is the event space, and P is the probability measure. We assume all measurable spaces are standard Borel. We denote by uppercase X random variable tak- ing values from the measurable space Fx), which is a measurable function from Ω to Ωxx where 2x is also called the domain of X. For a given finite index set I, we use uppercase bold X to denote a collection of random vari- ables (Xx)xzz with the associated space (Πx, Fx), where Ωx = -Ther Ωx and Fx = 1ze Fx, are the product space and the product o-algebra, respectively. For subset of indices I C I. (X;)ier r is abbreviated as X, or renamed as Y with Y C X. where the corresponding index set is denoted by Iy. The distribution of X is represented by
Endogenous variables in Mix) will be denoted as V(x)-
Another property of recursive SCMs is that any submodel is also recursive SCM. This implies that, in the submodel, given a value u, the endogenous variables Vix are deter- mined as I (u). Under this premise, the value of the endogenous variables Y(s] under a given u is represented as Y May (u) = (T [x] (u)) called the potential response.
2.2. Pearl Causal Hierarchy
(Pearl & Mackenzie, 2018) categorized causal questions into three levels of human cognition: seeing, doing, and imagin- ing. To formalize these questions and delineate the logical differences between the three levels, symbolic languages L1, C2. L3 are introduced. Each language consists of L;- statements yp, where every yp E L; a Boolean combination
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Exogenous Isomorphism for Counterfactual Identifiability
of inequalities between polynomials over probability terms P(a:). For L1, (P((a) takes the form P(Y E }), represent- ing the probability of y occurring, making Ci standard probabilistic logic. Lg introduces interventions, with P(az) in the form IP(Y, [x] E y), denoting the probability of y occurring when X is intervened to x. L3 further encodes conjunctions under different interventions, making P((a) take the form P(Y. E ).), where Y. 1x,1)) jeim rep- resents the collection of endogenous variables Y [(x]] from
constructing a proxy M' that satisfies this partial knowl- edge and performing equivalent reasoning on M'. Causal identification is the task of proving or reasoning whether any proxy constructed from partial knowledge provides con- sistent answers to specific causal questions.
For example, when the true SCM .M" is assumed to be Markovian and both observational distribution and the causal graph are available, constructing a Markovian Causal Bayesian Network (CBN) suffices to answer any causal question in L2 via truncated factorization (Pearl, 2009). Thus, Markovian CBNs can be regarded as identifiable at the intervention level. When the Markovian assumption fails, semi-Markovee CBNs can still answer a subset of causal questions in L2 using do-caleulus.
submodels Mpxi] MMaa) (x.l....... ;. M,.. Thus, Y. E Vv.in is
logically equivalent to the conjunction A Y [b,]] E j-
An SCM M provides semantics for the symbolic languages L1. L2. L3. For L3. any term of the form P(Y, E V).) can be evaluated by .M as PVM(N.) where PsM is called the counterfactual distribution. Let YM.. = (Y,j), Detm.
Regarding counterfactuals, as they represent the highest level in the PCH, achieving counterfactual identifiability requires stricter conditions and more complex methodolo- gies. For instance, even Markovian CBNs cannot ensure consistency when answering certain L3 questions (Nasr- Esfahany & Kiciman, 2023). Prior research on counterfac- tual identification often focuses on identifying a subset of counterfactual statements op L3. These efforts can be categorized into constraining causal structures to identify counterfactual effects, such as (Shpitser & Pearl, 2008), or constraining causal mechanisms to identify counterfactual outcomes, such as "Nasr-Esfahan et al., 2023).
Since each Y ||د, MaJx, o U almost surely, the PV. is the pushforward measure (Y..), ).. yielding:
yie.diiing
PY. - Pu ((VM.))
(1)
Equation is called L3 La-valuation. After the La-valuation, a statement yp E C3 evaluated by M, denoted as (PM). can be checked for validity. If (MM) holds, we write M = y.
The Pearl Causal Hierarchy (PCH) is defined as the com- bination of the above syntax and semantics. Specifically, when an SCM M is given, the collection of observational, interventional, and counterfactual distributions defined syn- tactically by L1; L2, La and semantically by L1, L2, La- valuations constitutes the PCH. In this sense, the PCH en- capsulates all causal information entailed by an SCM.
2.4. Model Identifiability
SCMs are special class of generative models, where the exogenous distribution and causal mechanisms can be inter- preted as the latent distribution and generative function of the model, respectively. Identifiability has been continu- ously discussed topic in the literature, appearing in represen- tation learning (Bengio al., 2013), causal discovery (Gly- mour et al., 2019), causal representation learning (Schölkopf et al., 2021). and causal quantity inference (Pearl, 2009). Providing a broad definition of identifiability may lead to ambiguity. Borrowing from the definition of identifiability in (Khemakhem et al., 2020), these notions of identifiability can be unified through equivalence classes:
Furthermore, the PCH exhibits strict hierarchy. In terms of syntax, the representations of the terms P((;)) imply that L1 Ç L2 & La. In terms of logical expressiveness, similar theorems have been established, known as the Causal Hierarchy Theorem (CHT) (Bareinboim et al., 2022).
To illustrate the expressiveness of L.s consistency is in- troduced. Let (,M {x(M) o E L.) denote the C.-theory of M, then C.-consistency is detined as follows: Definition 2.2 CC,-Ssssstten)).. SCMs M(l) and (2 are Li-consistent, denoted (2 ~c. M if their Li- theories are identical, i.e., ((( (.))))).
Definition 2.3 (>=-Ideniiffaiiiitt). Let A denote a set of assumptions, [A] = {M M } A} includes all models satisfying the assumptions, and ~ denote equivalence relation between models. Then model is said to be identifiable from A. or identifiable up to eequivalenec class, if for any (1)) E [A).M() ~ M
The CHT demonstrates that in the PCH, ~c: generally does not imply ~c; for < j. formally indicating that higher levels possess greater expressiveness than lower levels.
2.3. Counterfactual Identification
We now interpret the problem of causal identifiability from the perspective of -iidentifaailltty Suppose the causal question involves a set of L,-statements two SCMs Ma) and (2 are said to be s--consistent if ((MM(1) = (2(M(2)). Since e-consistency is an equiva-
In practice, the underlying true SCM M is almost never fully specified, making direct reasoning on M impracti- cal. Consequently, researchers rely on partial knowledge about M to address causal questions, which necessitates
S C,, and
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Exogenous Isomorphism for Counterfactual Identifiability
Ience relation between SCMs, denoted as M(1) ~ M the problem of causal identification with respect to yp can be reframed as a problem of --piideniiiiabiitty.
the strength of model identifiability required to achieve com- plete counterfactual identifiability.
Let the recursive SCMs M = (I, Sv, 2 f(k), p())) share the index set I and the domain of endogenous vari- ables Gly. For a given endogenous value E flv, abbreviate f(() (Vpattr(s)))) as f!a) (v, Exogenous iso-
The CHT states that when A contains only low-level knowl- edge, -i-iieetiiiabiiity is often unattainable. Achieving this requires assuming higher-level information. For in- stance, under the assumptions of known L observational distribution, causal graph, and Markovianity, the causal graph encodes structural constraints on L2, enabling the constructed CBN to answer all questions in L2. Since the union of all statements 4 S L2 equals L2, this property is also referred to as ~L,- -d--iideniifiabiiityy.
morphism is then defined as:
Definition 3.1 (Exogenous Isomorphism). Recursive SCMs M and M(2 are said to be exogenously isomorphic, denoted M() ~EI ( if there exists a shared causal ordering I and function h S2 + 227 satisfying:
Component-wise Bijection: For each i E I. h = (bi)kez, where h; 0/702 - 2/? is a bijection;
This paper focuses on ~CC-identiffabiiity,, which is an enhanced version of ~c ~ca-identifiabiiity. requiring ~ip" identifiability for any & S L3. If A satisfies ~aa-
Exogenous Distribution Isomorphism: P(P = h, P())
identifiability, then by the definition of L Lx-consistency, any
Causal Mechanism Isomorphism: For each i E I, for almost every ( E s0l) and all v E Sty.
M() E [A] satisfy M(1 ~Ls M Since L3 represents the highest level in the PCH and encodes all causal information of the SCM, if (M()) Lss M the two models are indistinguishable under any causal state- ment. Therefore, ~ -ي -ddntiiifiiiiiittty is the ultimate goal for causal identifiability within the PCH.
(v, = (v,
The implication of "=g--ddetifiabilin for ~c iddetifiabiiitty is formally stated in the following theorem:
Theorem 3.2 (~~gI Implies ~ca). For recursive SCMs ( and (2)), if (() ~EI M then M() ~es M
3. Exogenous Isomorphism
~c "c,-identifiabilitty, by definition, is not straightforward to handle, as it is indirectly defined through the PCH rather than directly based on the model itself. Therefore, we aim to find a simpler madel-based identifiability notion that implies --g-identiffabiiity thereby simplifying the problem.
Sketch of proof The mechanism isomorphism of each fi can be progressively deduced into the potential responses, ensuring VMl, = v oh. Given P hy PyP we have (v M. o h)2 = ( ((") i... (v (V 2 (V Pp Thus, all L3-evaluations for M( and M are identical, and any statement to E L3 yields the same result. Therefore, M(1) ~cg M(2), For a detailed proof, see Appendix A.2.
One possible choice is -identifiabbiiiyy which requires uniquely identifying the generative model. Some prior works have achieved =-identifiability (Xi & Blodm-Reddy, 2023), which indirectly induces ~c.--idenifiaailiii serves as one approach to addressing <, -identifiability, but the required assumptions are often too strong. An alternative property is defined via counterfactual equivalence (Peters et al., 2017, Proposition 6.49), which- while not insisting on uniqueness of the causal mechanisms does demand full recovery of the exogenous distribution.
This
4. Bijective SCM
Once identifiability is defined, the next step is to explore the assumption sets that induce identifiability. In this work, for simplicity, we target some special settings of recursive SCMs. Throughout, the set of positive integers {i | r, i E N+} } is abbreviated as I for 1,r E N+ Moreover, we assume that for each i E Z, the domain of exogenous variables Sy, and the domain of endogenous variables Ry; are both indexed Rdd with 1 d as the index set. That is, both exogenous and endogenous variables are random vectors, and the nested index set I = {(i.j) | i E I.j e 1: d.} indexes the individual dimensions of Pu and Sy.
Between iidentifiabbiii counterfactual equivalence and -eg-identifiabiiityy there should exist other forms of ~u- identifiability, as counterfactual reasoning does not require completely fixed latent representation. This is akin to hu- mans being able to answer "what-iff questions without fully understanding the underlying factors of the physical world. Identifying such a form of --iientifiaabiity is a goal worth exploring. Motivated by this, we have identified an equiva- lence relation between SCMs, denoted as ~ei and referred as exogenous isomorphism, for which -ng-iidettiiaabiity strictly implies ~2,- iiddetiiiiiabbittyy.. yet it is weaker than the previous two. This characterization more precisely reflects
Since the cardinalities of the exogenous and endogenous domains are equal, a bijection can be established between these domains. A bijection implies no information loss. If the causal mechanism is also bijective, then for any obser- vation in the endogenous domain, a distinguishable latent
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Exogenous Isomorphism for Counterfactual Identifiability
encoding can be identified in the exogenous domain. Below, we formally define this specific setting of SCMs:
Proposition 4.5. If the BSCM M is Markovian, then for almost all v, v' E fly, the conditional distributions satisfy P (: v") = (KMil v, v' PViV (c. v)?
Definition 4.1 (Bijective SCM (BSCM)). A recursive SCM M is called bijective SCM (BSCM) if its solution mapping T is bijection.
Combining this with Theorem 3.2, we find that when all counterfactual transports are fixed, ~EI -Er-identifiability can be achieved. Let the following assumptions be defined: (i) ABSCM: M is a BSCM; (ii) Ag; : the total order S. is a causal order for M: (iii) Ap,: the observational distribution of M is Pv with strictly positive density; (iv) Ak: there exists a function K Gy Gy x Gy - fly (K;)er such that counterfactual transport K.M K almost surely.
Proposition 4.2. A recursive SCM . is a BSCM if and only if f((v,-) -) is a bijection for every E Z and all v E Ωv.
For BSCMs M and M since their solution mappings (R))
+00) Syy are bijections, it is evident that the composition (-1 r(1; 20 - 2 is also bijec- tion. Given that exogenous isomorphism requires bijec- tion between 2 and 2 natural question arises: can
Let A (nSCM.5.PV.K {ABSCM A AK): Then:
(())) directly serve an exogenous isomorphism? The following theorem provides precise answer.
Theorem 4.6 (EI-ID from Counterfactual Transport), An SCM is -i--ieentffiable from AqBSCM. S.P.K)
Theorem 4.3 (BSCM-EI). two BSCMs M(2 and M share a common causal order S and the same observational distribution Pv, then M(1) ~EI M2 if and only if for every i I, there exists a bijection h4 00. - 22. such that for all v E lvv
Verifying the assumption set Ak in practice challeng- ing, making Theorem 4.6 difficult to apply. By combining Proposition 4.5 and considering Markovian BSCMs, nat- ural direction is to identify specific types of transport that are always well-defined between conditional distributions. One such transport, the KR transport, is constructed using one-dimensional conditional cumulative density functions in 1 : d order, ensuring that the mass at each point of the distribution follows lexicographical order, and:
(v.))) (f(t) Of
h
almost surely.'
4.1. -er-Identification for BSCM
Lemma 4.7 (Santambrogio 2015, Proposition 2.18). Given any two distributions P and P' on Rd with strictly positive densities, the KR transport T Rdd - Rdd such that P = T TP always exists and is almost surely unique.
To derive assumption sets that imply -gg-identifiabiiiiy, we need to restrict our perspective to a single SCM M. Consider the conditions in Theorem 4.3, where there exists function h;
22/ + 22( such that for all (f( -)=-) (f(")(v (v,-)) = h, holds almost surely. Now, consider different v, v' E fty. Since both f( (v, and 1! (v.-) are bijections, by the associativity of composition,
Ωv,
The KR transport induces a special case of A(BSCM, P. in Theorem 4.6. Let the following assumptions be defined: (i) Aui M is Markovian; (ii) AKRi for each i E I, the counterfactual transport component KM, M, is almost surely equivalent to the KR transport.
(P (V).)))) (v. ( (v. (f)
Let (BS.M.,.M.PV.KR) = {ABSCM, AM Akct Then:
almost surely, where both sides come from the SCM M, and we explicitly name this concept as follows:
Theorem 4.8 (EI-ID from KR Transport). An SCM is WEI- identifiable from M.
Definition 4.4 (Counterfactual Transport). For a BSCM M, the function KM : Ωy x Gy x By - Sy is called the counterfactual transport if KMM = (KM and for every i E and all v, v' E Ωl, the component K v.M((,,v') (fi(v'))) (fi(v.-))-
Nevertheless, in most practical scenarios, conditional cumu- lative density functions are unknown, and one often only has access to samples from the distributions. Therefore, KR transport cannot be constructed as defined, motivating the need for a tractable approximation of KR transport. This leads more practical subclass of BSCMs.
Under Markovianity, the practical meaning of counterfactual transport is the transport between conditional distributions:
'The BGM proposed in EsfahaEs et al., 2023) corre- sponds to the fi of a BSCM. Furthermore, component-wisg wise bi- jection and causal mechanism isomorphism in exogenous isomar- phism are described as BGM equivalence. Therefore, Theorem 3.2 extends (Nasr- Esaa-ass et al., 2023, Proposition 6.2).
ZA more general notion of counterfactual transport was defined in (Lara et al., 2024) from the perspective of coupling between conditional distributions. In this paper, the counterfactual transport in Markovian BSCMs (Definition 4.4 and Proposition 4.5) is special instance of their more general framework.
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Exogenous Isomorphism for Counterfactual Identifiability
5. Triangular Monotonic SCM
for any (i,5", E I and ((i,j) < (") for any (1,j), 3') I with i < i', we call it vectorization under S. In other words, vectorization merges all random vectors into unified order. Based on whether the solution mapping I is a TM map under vectorization, we define a more specialized class of SCMs:
We call a function T(x) = (x) where T: R - R. a triangular mapping if the j-th component T3 depends only on X1.-- The name originates from the fact that the Jacobian matrix of such mapping is triangular. For a triangular mapping T, we define its monotonicity signature Ş(T)= = (E(T))) as:
Definition 5.2 (Triangular Monotonic SCM (TM-SCM)). A recursive SCM M called triangular monotonic SCM (TM-SCM) if there exists a vectorization i under a causal order such that the re- re-indexed P, o I is TM mapping.
Vx1---1 E Ri-1, (x1:5-11; s.m.i.
E(T))=
-1
Vx1.j-1 E Ri-1, (x15-115) is s.m.d.
0
otherwise
Proposition 5.3. A recursive SCM M is a TM-SCM if and only if fv.. is TM mapping and there exists & such that {(fi(v,)) = for every i E I and all v E Stv.
where s.m.i. and s.m.d. abbreviate strictly monotonically increasing and decreasing, respectively. If d - = (TT)), i.., every component is consistently s.m.i., we call T a trian- gular monotonic increasing (TMI) mapping. TMI mappings are closely related to KR transport:
5.1. ~EI -deettiiifiiiiioo for TM-SCM
As subclass of BSCMs, TM-SCMs allow the assumption set in Theorem 4.8 to be further simplified. The new iden- tifiability strategy is specifically tied to TM-SCMs. Let
Lemma 5.1 (Jaini et al. 2019, Theorem 1). Given any two distributions and P' on R2 with strictly positive densities, if a TMI mapping T _2 + Rd satisfies P TyP', then T is almost surely equivalent to the KR transport.
A S(MM.SCM.g...F = Ag AM: AM. }, where
ATM-SCM states that M is a TM-SCM. Then:
TMI mappings can be further generalized. For a triangular mapping T, if d Σز= [E(T)),, i.e., every component is either consistently s.m.i. s.m.d., we call T triangular monotonic (TM) mapping. TM mappings, in addition be- ing bijective due to monotonicity, exhibit special properties: they remain TM mappings under inversion, composition, and selection of contiguous components. Moreover, their monotonicity signatures adhere to specific rules, which are elaborated in Appendix A.4. Crucially, for two TM map- pings (1) and (2) with identical monotonicity signatures, (22) (T(1)-1 is always TMI mapping.
Corollary 5.4 (EI-ID from Triangular Monotonicity). An SCM is ~g -EL-identifiable from A(MM.SC..S.M.PY)
Sketch of Proof By combining the definition of TM- SCMs with the properties of TM mappings, any counterfac- tual transport in a TM-SCM is a TMI mapping. Furthermore, under Markovianity, given the causal order S and observa- tional distribution Pv, the KR transport is almost uniquely determined by Lemmas 4.7 and 5.1, which then implies Theorem 4.8. A detailed proof is provided in Appendix A.4.
Corollary 5.4 (together with Theorem 3.2) can be viewed as a strengthened version for Markovian CBNs, which ex- hibits ~c, -idetifiabbiity, i.e., SCMs are ~e, -cg-ieentifiable from A(S.M.Pv) Corollary 5.4 augments A(<.MPPv) with the additional assumption ATM.SCM. strengthening ~>2-" identifiability to ~2,- "'n-iientifiibiiity.
Next, we aim to relate the solution mapping I of an SCM to TM mappings. However, TMM mappings are defined over the index set 1 d, whereas SCMs are defined over the nested index set = {(v,5) i E j E 1 1: d.). To bridge this gap. we introduce the concept of flattening, a generalization of permutation. For finite index set I, we call bijection 6 I + 1: |Z| flattening mapping of Z, and flattening refers to re-indexing according to 1. Define the re- re-indexing P. Ilier Ax + +11 Zx such that for every i I,
Moreover, this extends and unifies the theoretical results in (Lu et al., 2020, Theorem 1), (Nast-Esfahany et al., 2023, Theorem 5.1), and (Scetbon et al., 2024, Theorem 2. 14). Specifically, these theorems focus on counterfactual identi- fiability from the perspectives of bijectivity, monotonicity, and Markovianity, demonstrating identifiability based on abduction action-prediction Compared to these results, our theoretical contributions offer the following advance- ments: (i) Generalizing the measurable space of each en- dogenous variable from scalar IR to a vector Rd, enabling support for broader class of SCMs; (ii) Demonstrating ~eg-iieetiifability based on exogenous isomorphism theory (i.c. Theorem 3.2), which implics ~cs -ddentiiiability rather than merely the identifiability of counterfactual outcomes.
P.((x.)aez) (E.-16))sE4())
where [[] is the image set. Since is bijective, P. is also bijective, and for any J C | we have
(P. (())))()))) (x.0))ie 1))
where - [Z] is the pre-image set.
For the nested index set I and a total order < on I, if
a flattening mapping/ satisfies ((i.j) - i(i,j') = - j
6

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Exogenous Isomorphism for Counterfactual Identifiability
6. Neural TM-SCM
the solution map Ty is constructed by composing multi- ple mappings, it is named Composed Mapped Solution Mapping, such as CausalNF (Javaloy et al., 2023).
Problem formulation Consider the true underlying SCM M = (I, Gy. y ,P)), where Di f*, and Pt are unknown. Assuming M Markovian TM-SCM with causal order S, how can one construct parameterized proxy SCM Ma (I, Sv SU, So, PUa) based on the observational dataset Dv = [v. | v(0 ~ Pv } such that MM is as La-consistent with M as possible?
TVSM: Constraints the re-indexed solution mapping P. o T to be a TM mapping by defining it as the flow constructed from the solution to the ODE
(dx(() = vo (z))))), dl, (x(0) = Vo.
(5)
According to Corollary 5.4 and Theorem 3.2, this prob- lem is equivalent to constructing MM such that satis- fies AIIMESCM.S.MM.F) as closely as possible. This entails four construction objectives: (i) G1: Mg += ArM-SCM- meaning that Ma belongs to the TM-SCM class; (ii) G2: MMa- += Ac, meaning that S is one of the causal orders of Ma. (iii) G3: Ma }= AM. meaning that Mg satisfies Markovianity. (iv) G4: Ma += Apv. meaning that Ma induces the observational distribution Pv.
where the vector field by is Lipschitz continuous triangu- lar mapping. According to the Picard-Lindelis theorem, such flows are TMI, see Lemma A.21 in Appendix A.4, Since the solution map Ta is implied by a triangular velocity field, it is named Triangular Velocity Solution Mapping, such as CFM (Khoa Le et al., 2025).
The relationship between these related works and TM-SCM is detailed in Appendix B.4, and the implementation details of the four prototype models can be found in Appendix C.
To satisfy G1, Ma must be TM-SCM one of its spe- cial subclasses. Furthermore, in this paper, we primarily focus on neural networks that provide the parameters 0 for MM. thereby referring to it as a neural TM-SCM. Previ- ous works have constructed neural SCMs that satisfy this requirement in the scalar case; we categorize these works into four categories and propose corresponding prototype models as extensions to the vector case:
To satisfy G2, the causal mechanisms fo in the constructed SCM need to maintain a specific causal structure to de- rive the causal order S. When paraneterizing the causal mechanisms f&, as in (Xia et al., 2023), we address the issue of inappropriate additional independence assump- mions-ariston from only knowing the causal order- by setting pa(i) ( j S i,j I}. Similarly, when indi- rectly modeling the solution map T6 using an autoregressive generative model as in (Javaloy et al., 2023), we align the autoregressive order with the causal order.
DNME: Constraint the causal mechanism fi,e to be a TM mapping, such that the causal mechanism
fr.e (Vmtt) u.) = bv,o(Vp((s) + ai (Vp((t) @ u, (2)
To satisfy G3, the modeled exogenous distribution is re- quired to satisfy Pt; = II,z Pu..- To enhance the expres- siveness of the exogenous distribution and to indirectly indi- cate that identifiability is independent of the specific imple- mentation of the exogenous distribution, we employ uncon- strained normalizing flows. Specifically, the log-likelihood of each PUs,o is given by the change of variables formula:
where
bi,e pa(s)) E Rd with vector a,,
always strictly positive, and @ denotes component-wise multiplication. Since the Jacobian matrix of fú, w.r.t. u, is diagonal, is named Diagonal Noise MEchanism, such as LSNM (Immer et al., 2023).
TNME: Constraint the causal mechanism fie to be a TM mapping, such that the causal mechanism
log pU. (u.) log. pz. (T, (u:)) + log det J (u:)
(voo(Va(()) u:) = bs,o(Vpa(t) + (Ai., uf, (3)
(6)
where Tí, : Zi + U1 is MAF (Papamakarios et al., 2017), and det J T (us) is the Jacobian determinant at uq-
where A (Vpr(())
IRd, b (V,o(V) Rdd with matrix A, always a strictly positive lower triangular matrix. Since the Jacobian matrix of Si,o w.r.t. u, is lower triangular, it is named Triangular Noise MEchanism, such as FiP (Scetbon et al., 2024).
To satisfy G4, in generative model learning, the observa- tional dataset Dy is commonly used to optimize learning objective, ensuring that the proxy SCM MM induces an observational distribution Pv.o that closely approximates the true observational distribution Pv. To achieve this, we adopt the traditional maximum likelihood (MLE) approach, corresponding to the negative log- likellikoooo (NLL) loss:
CMSM: Constraint the re-indexed solution mapping P. o To to be a TM mapping by composing multiple TM mappings
Po T T1,00 Tn,
(4)
N
where each T, can be an autoregressive affine transfor- mation in normalizing flows (Dinh et al., 2017). Since
arg min
log Pv. (v.(0))),,
7

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Exogenous Isomorphism for Counterfactual Identifiability
where Pv, is the log-likelihood of the modeled observa- tional distribution. Since TM-SCM ensures bijective To, the log-likelihood lo, (v()) at v!( can be computed using the change of variables formula.
approaches obtain identifiability for a subset of counter- factual queries; for example, CVAE-SCM (Karimi al., 2020) is identifiable for specific counterfactual queries from causal sufficiency and observational distribution, and NCM (Xia et al., 2023) shows duality between the identifiability of crructure-oonsrrinee proxy SCMs and non- -on-paraetrric identification results. Once parametric assumptions are in- troduced, complete counterfactual identifiability becomes attainable, exemplified by any subclass of neural TM-SCMs. Additional examples are provided in Appendix B.4.
7. Related Works
Isomorphism and counterfactual identifiability Several prior works have introduced notions similar to exogenous isomorphism and shown that SCMs satisfying these equiva- lence relations enjoy counterfactual consistency, mirroring the result of Theorem 3.2. This body of work includes coun- terfactual equivalence as defined in (Peters et al., 2017), which requires identical exogenous distributions; BGM equivalence, introduced for BGM within BSCM in (Nasr- Esfahany et al., 2023); LCM isomorphism, proposed by (Brehmer et al., 2022) for causal representation learning and proven, under additional conditions, to coincide with LCM counterfactual consistency; and domain counterfac- tual equivalence between ILDs, together with necessary and sufficient conditions, as defined in (Zhou et al., 2024). The latter three lines of work additionally assume that the solu- tion mapping Ií is bijective, whereas Theorem 3.2 applies to arbitrary recursive SCMs.
8. Experiments
To demonstrate that neural TM-SCM can effectively ad- dress the counterfactual consistency problem in practice, we conducted experiments on synthetic adatasets. These experiments were designed to showcase the model's ability to generate counterfactual results that are consistent with the test set, using only the endogenous samples drawn from the observational distribution as the training set." 3
Datasets
The experiments involve the following synthetic datasets, with details described in Appendix D.1.
TM-SCM-SYM: A collection of four small datasets (BARBELL, STAIR, FORK, BACKDOOR) with up to 4 causal variables, using exogenous distributions that are standard or Markovian multivariate normals and manually defined TM causal mechanisms.
TM-SCMs and their identifiability Previously, we clas- sified four construction prototypes of TM-SCMs from a construction perspective; a complementary perspective con- siders the tasks these models address and the corresponding identifiability guarantees. For cause-effectt identification, special TM-SCMs -i-SCMs-incluuingg LiNGAM (Shimizu et al., 2006), ANM (Hoyer et al., 2008), PNL (Zhang& & Hyvärinen, 2009), LSNM (Immer et al., 2023), and CAREFL (Khe- makhem et al., 2021)-have been proven identifiable with respect to causal direction. For causal effect estimation, NF (Javaloy al., 2023) establishes representation identifiability of TMI, and similar TM-SCM constructions are adopted by StrAF (Chen et al., 2023). CCNF (Zhou et al., 2025), and CFM (Khoa Le et al., 2025), which therefore inherit the same theoretical guarantees. In counterfactual identification, (Lu et al., 2020, Theorem 1), et al., 2023, Theorem 5.1), and (Scetbon al., 2024, The- orem 2. 14) each demonstrate that TM-SCM entails coun- terfactual identifiability under different settings, and these theorems are all special cases of Corollary 5.4.
ER-DIAG-50 and ER-TRIL-50: Each contains 50 datasets with 3-8 causal variables, Markovian multivariate normal exogenous distributions, and Erdōss-Rnyy causal graphs (edge probability 0.5). ER-DIAG-50 ensures diag- onal Jacobians, while ER-TRIL-50 ensures lower trian- gular Jacobians for TM mappings.
Metries To evaluate trained Me. we compute OBSwD (Wasserstein distance) for the fit to the observational dis- tribution and CTFRMSE (root mean square error) for L3- consistency with ground truth counterfactual outcomes.
Ablation on TM-SCM-SYM To provide empirical ev- idence for Corollary 5.4 and Theorem 3.2, and show that neural TM-SCM addresses counterfactual consistency, we performed small-scale ablation study TM-SCM-SYM. including: (i) w/o O: Reverses causal order S, ablating Ag; (ii) w/o M: Uses non mon-Markovian exogenous distributions, ablating Aw; (iii) w/o T: Constructs non-triangular SCM, ablating ATM-SCM (supported only by CMSM and TVSM).
Counterfactual inference with neural SCMs Proxy SCMs built with neural network components are widely employed for counterfactual reasoning by directly learning from observational or interventional data. Several methods focus on the tractability of inference rather than identifiabil- ity by adopting different neural modules, including DSCM (Pawlowski et al., 2020), Diff-SCM (Sanchez & Tsaftaris, 2022), and VACA (Sánchez- Martin al., 2022). Other
Figure 1 shows scatter plots of OBSwD and CTFRMSE on the validation set. Configurations w/o O and w/o T fail to converge, highlighting the necessity of Ag and ATM-SCM-
*oode is available at: hutps//githa.ccomeyiss/mmmm
8

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Exogenous Isomorphism for Counterfactual Identifiability
Table 1. Ablation results neural TM-SCM on ER-DIAG-50
ER-TRIL- TRIL-50. The values shown are the means of 50 experiments. subseript representing the 95% % CI. The best performnf results are highlighted in bold.
METHOD
ER- DIAG-50
ER-TRIL-50
0.53
0.
DNME
W/O O
0.78
0.89
W/o M
0.62
0.58
0.47
0.55
TNME
W/O O
11.24
6.41 +9.84
(b)
w/o M
0.62
0.73
0.37
0.42 12
O
2.64
2.12
CMSM
W/O M
1.69
0.75
W/O T
O. 0.64
1.25
0.46#0.05
0.50
C171655
TVSM
W/O O
O. 0.79
0.88
w/O M
0.53
0.53
w/O T
0.67
0.78 a..82
OB5WD
(C)
theories, providing reliability guarantees for counterfactual reasoning. Our empirical evaluations further validate the practicality of the proposed approach, paving the way for reliable applications in counterfactual modeling.
Figure 1. Ablation results of neural TM-SCMs on TM-SCM-SYM. Colored curves depict sliding-window regressions, with shaded areas showing 95% CL. (a) DNME for BARBELL; (b) TNME for STAIR; (c) CMSM for FORK; (d) TVSM for BACKDOOR.
Acknowledgement (Camera ready)
Most w/o M settings result in higher CTFRMSE. emphasizing the role of Am. CMSM shows slightly lower stability than other methods. Additional results are in Appendix D.4.
Acknowledgements
This work was supported in part by the National Key R&D Program of China (No. 2022ZD0120302),
Ablation ER-DIAG-50 and ER-TRIL-50 We further conducted more comprehensive evaluation on ER-DIAG- 50 and ER-TRIL-50, reinforcing the generality of the theory through experiments on a wide variety of synthetic SCMs. The ablations in these experiments also targeted Ag, As. and SCM-
Impact Statement
This study aims advance the field of causal inference by providing both theoretical and practical support for the con- sistency and trustworthiness of the results of counterfactual reasoning. In future applications, the proposed methods can enhance the reliability of reasoning in relevant prac- tical scenarios. However, we emphasize the necessity of maintaining close collaboration with domain experts and stakeholders when applying these models in practice. is particularly important to assess whether the theoretical assumptions proposed in this paper satisfied to mitigate the risks of potential negative consequences.
Table presents the final CTFRMSE on the ER-DIAG-50 and ER-TRIL-50 test sets for each method and its ablations. The results demonstrate that violating Aç, AM, ArM SCM significantly degrades L3-consistency, emphasizing the im- portance of these assumptions and the validity of the theory to ensure consistent counterfactual inference. Additional results on these datasets can be found in Appendix D.4.
9. Conclusion
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By simplifying the problem through the introduction of exogenous isomorphism and --g-identifiability we devel- oped concrete methods to achieve identifiability in BSCMs and TM-SCMs. These findings unify and extend existing

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Exogenous Isomorphism for Counterfactual Identifiability
A. Proofs
Definition A.4 (UCCM)
Definition A.3 (CCM)
Definition A.1 (UCM)
Definition 2.1 (SCM)
Definition 3.1 (Exogenous Isomorphism)
Definition A.6
Lemma A.5 (UCCM-COL CCM)
Lemma A.2 (T UCMC))
Definition A.7 (PCCM)
Lemma A.9 (Isomorphism for CCM)
(AUCCM)
Lemma A.8
Definition 2.2 (C--Consistency)
Lemma A.10 (Isomorphism for AUCCM)
(Potential Response PCCM)
Definition 4.1 (BSCM)
Theorem A.I2 (Decomposition Theorem)
Lemma A.11 (Isomorphism for PCCM)
Proposition 4.2 (BSCM Decomposition)
Lemma A.13 (Pushforward as Conditional)
Theorem 3.2 (EI-ID = L3-ID)
Definition 2.3
(~-ID)
Theorem 4.3 (BSCM-a --II)
Definition 4.4
Lemma A.14 (Partial Pushforward as Conditional)
(Counterfactual Transport (CT))
Theorem 4.6 (EI-ID from CT)
Proposition 4.5 (CT by Markovianity)
Lemma 4.7 (KR- -KR-Transport Uniqueness)
Theorem 4.8 (EI-ID from KR- (R-Transport)
Definition 5.2 (TM-SCM)
Lemma A.15 (TM Inverse)
Lemma A.16 (TM Compose)
Lemma A.18 (TM Component)
Lemma A.20 (Vectorize an)
Lemma A.19 (Vectorize- 1,)
Lemma 5.1 (TMI KR Transport)
Remark A.17 (TM TMI)
Remark A.17 (TMI-SCM Decomposition)
Lemma A.21 TMI)
Corollary 5.4
(EI-ID from Triangular Monotonicity)
Figure 2. Overview all theorems discussed in the main text and appendix, along with their dependency graph. Nodes represent theorems, and edges indicate dependencie, directed from top to bottom. Different colors denote different topics: theorems related to recursive SCMs are marked blue; theorems related exogenous isomorphism are marked in green; theorems related to BSCMs are marked yellow; and theorems related to TM-SCMs are marked in red.
A.1. Background and Preliminaries
Definition 2.1 draws characteristics from both (Bongers et al., 2016) and (Bareinboim et al., 2022), but differs in the following aspects: (i) we use a shared index set Z for exogenous and endogenous variables, rather than using an additional index set J for endogenous variables, because this paper does not focus on the relationships between exogenous variables; (ii) partially compensate for the limitation introduced in (i), allow Pu ILezz Pu, to not necessarily be product measure. This relaxation permits endogenous variables to be dependent unless the SCM is Markovian.
Following Definition 2.1, an SCM describes the functional relationships between endogenous and exogenous variables through a set of deterministic equations the form v; = f((pm(()) u.), known as structural equations. (Bongers et al., 2016) discuss the solvability of this system equations. Specifically, if there exists a pair (V, U) such that V = f(V, U) holds almost surely and the distribution of U exactly Pt. then the SCM is said have solution. If there exists a function
13

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Exogenous Isomorphism for Counterfactual Identifiability
I' such that for almost all ay and all E fy, v r(u) implies =f(v, u), then the SCM is said be solvable. According to (Bongers et al., 2016, Theorem 3.2), the SCM has solution if and only if it is solvable.
A recursive SCM is special type of SCM with certain desirable properties. In this subsection, we will prove a fundamental lemma on recursive SCMs to facilitate subsequent proofs and introduce some notations. For an index i E I in recursive SCM, the lower set an(i) = (j j < i} of the partial order N is called the causal ancestors of i, while the lower set pr(i) = ( j < it its linear extension I is called the causal prefix of Additionally, an" an) = an(i) U {i} and pr (1) pr( U {i}. Consider recursive SCM M (I.Ωy. Qu. f. Pu). we can recursively expand the endogenous
part of f, obtain a new function:
Definition A.1 (Unrolled Causal Mechanism (UCM)). For each causal mechanism fi Syymo
pro x ly - Ωv, in M, let f QU (2) Sly; be the result of recursively unrolling the endogenous part By puc) of the causal mechanism fi, such that for any Su.
(1)a Ji ((Fs(Mar) k
(4)
This is referred to as the Unrolled Causal Mechanism (UCM).
The following lemma demonstrates that the tuple of UCMs constitutes the solution mapping I of a recursive SCM, thereby indirectly proving the existence and uniqueness of solutions for recursive SCMs.
Lemma A.2. For all Lu. f(v, u) if and only if v I(u), where r(u) (f)
Proof. Let u E Dy be arbitrary.
: Suppose the structural equations v = f(v, u), meaning each component satisfies nn: = Ji(V (() uz). We proceed by induction based on the partial order 3. When pa(i) = 0. we have o; = fi(u:), and by the definition of UCM, (u,) = f((u,) Uz. When pa(i) + 0, assume that for each &s E pa(i), Us (um) holds. Then,
(() (2)) (( (x)))()))) ui)
Definition A.I Induction
pa(ppa uz)-
Thus, for each i E I, we have es = (t)) implying that (ui)sez = (f(werr))) i... v= I(u).
Suppose v I(u), meaning each component satisfies vs = (()) Then, for each i I,
f fi(aan-()) (i)
= fi((fi
u.)
Definition A.1
fV(p()))) u.).
Ug fx (u (1))
where the last equality follows from Uh = (K) for each & E pa(i). Therefore, the structural equation e; fe (Vputt), ui) holds for every i e I, and thus the system of structural equations = f(v, u) holds.
We decompose the problem and, based on the causal mechanisms, define the following compound structure as an extension of the causal mechanisms at the counterfactual level, establishing a connection with the potential responses in the PCH. Suppose there are n submodels of recursive SCMs. Mpxx)
and let the UCM corresponding to fi(x,) in
each submodel M [xy] be denoted by frix)]
M(xx]
Definition A.3 (Compound Causal Mechanism (CCM)). Let fi(x.) IT Dyme Sv. I[ Ry, be the tuple of causal mechanisms indexed by E I across the submodels, denoted as (J where x, = (x))je1=n- This satisfies, for each j E 1 n, all Vj E Ωy, and all E Lu..
((x.))(Vpuu))) JEi:: u,) (Va u:)
This is referred to as the Compound Causal Mechanism (CCM).
14

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Exogenous Isomorphism for Counterfactual Identifiability
IT'=1 Sy be the result of Definition A.4 (Unrolled Compound Causal Mechanism (UCCM)). Let Sape.1 Pumnco recursively unrolling the [T Sypcs part of the CCM filx.]- such that for all u E Qu.
(ixx.)(unn = Salxe.] (fNAxx)(MMa ( kEpu(s)) ("e)
This is referred to the Unrolled Compound Causal Mechanism (UCCM).
Lemma A.5. For each E I and all u E Cu, the UCCM ((x.)(uur (0) is equal to the tuple of UCMs (Fipe) (Uip,)
Proof. Given any u E Qy, we proceed by induction based on the partial order 3.
When pa(t) = 0, according to the definition of UCCM, have fapc.] (u)n (u.)))(u)) = fi(x.) (u.). Additionally, according to the definition of UCM, we also have [ui (0) ()== fitx,] (uz). Furthermore, based on the definition of CCM, (4,) When (a( + 0, assume that for each kt E pa(t). the following holds: fa(x. (ua = (fr(pe)) (Mmn (k)) (k))) Then,
Therefore, (unx. (4)) = (fitx,) (uan (1) (6))))) DeEmm
(um..)(Unnt())) (1)) (alx.) (um (k))) uz)
Definition A.4
Tapxa (((e(((())))))) (Man (1) (4)))se (epptt))) ub)
Induction
(Jitx,) ((fbx,) (inn) us) jel:n
Definition A.3
(fi(e,) (m))
Definition A.1
Therefore, for each E I and all E Qu. we have fafx.. (um=)))))))( = (fite,) (k)))
Definition A.6 (Augmented Unrolled Compound Causal Mechanism (AUCCM)). Augmenting the domain of fiix.] from Dunico to (ay resulting in Di[x.) Q0 1[ j- Ωv. This ensures that fipx.1 (Mmn (8)) (Upr (6) for all E flu, which is referred to as the Augmented Unrolled Compound Causal Mechanism (AUCCM).
Definition A.7 (Prefix Compound Causal Mechanism (PCCM)). Define E (0)xx. : Qu Iī== Dy tuple of multiple AUCCMs,
(i)- This is referred to as the Prefix Compound Causal Mechanism (PCCM).
Lemma A.8. For all fy, the potential response V M. (u) equals the PCCM (z)
Proof. Given any E Ru, consider the component indexed by in the j-th submodel, namely (VM, (u))i, and (8z[x.) (u)):.. According to the definition of potential response, (VM. M. (u))a, = (V M (u)); = (T (Txx (u)); where the i-th component of the solution mapping I [xx]] (x, is Fixx,](uun (1)) according to Lemma A.2. On the other hand, by the
definition of PCCM, (x))), (0.(x.) (anx,((u)), (u));- and by the definition of AUCCM, ((fxx.(()),
(fipx.)(umn (u)n where, according to Lemma A.5, (fiix.) (uam (1))s is the UCM filx,) (um))))))))) (6)) in the submodel M|x,]- Therefore,
(VM. (u)).. = [x, Lemma A.2
(Man
Lemma A.5
(fitx.) (Man (i))) = (877x.] ( (u))i.-
Since this equality holds for every component i,j, the potential response VM.. (u) equals the PCCM 8r|x.] (u).
A.2. Exogenous Isomorphism
The Definitions A.3, A.4, A.6 and A.7 enable us to progressively derive the isomorphism of causal mechanisms in exogenous isomorphism to the potential responses. Specifically, consider two recursive SCMs ((1) and M such that M() ~ M22), The following lemmas then hold in succession. To prevent ambiguity during the proof process, we use the full notation p(k) (Vpatt))))) (V
instead of the abbreviated f( (v.
Lemma A.9. For each for almost all (1)) 50 and all Ωv, the CCM satisfies
(Voor))))) (V uf
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Exogenous Isomorphism for Counterfactual Identifiability
Proof. For each E I and any E Sv, consider u St such that (V py (r = (V ). Then.
(File) (p)) M
Definition A.3
(xy)s (v (ymerr h.(w,)
i Lx,
Definition of submodel
iE I Ix, jeIm
(xx))
i Ix; i iET 1x, jElm
M ~EE M
flll ( (yme))))))))) ())
Uix) (2 (Vpact)66) "4 (1) Jeen
Definition of submodel
Jipx.] (VPrt)))))) "4 ())
Definition A.3
(1) Since 1 (v pa halu helu = 1.0 (u)p u holds for almost all u, e 0 the proposition is thus established.
Lemma A.10. For each i E I and for almost all ( E Di the AUCCM satisfies
[ix. h (1) (u (1) ((1)) (u (r)) (a)) 9ilx.1
Proof. Consider (1) E 20 for which Lemma A.9 holds. We proceed by induction based on the common causal order S (hence the necessity of the causal order).
When pr(t)] = 0, we have pr (t) = -tt). By Lemma A.9:
(2) (gx...] hpr (5) (u)
9isx. 9 (h.(u, (1)
(2)
pr pr' (i) {i}
(12) (hr )))
Definition A.6
[18)) (n((())) ()
Definition A.4
[(x.)((4) (u(
Lemma A.9
(() (ux (
Definition A.4
(4.
Definition A.6
Dixx.) (u (1)
(1)
pr (i) ={i}
When pr(1)) O, assume that for each ls E pr(i), [0
(a)
(1)
Oklp.] (u (1
(1)
holds. Then,
(2)
(hr
(())))
(1)
((2)
((xx) (h (2) an (2)
Definition A.6
f(xx.) k(x.) (h (0)) (u ( keppp ((((,))))))
(2) (2)
Definition A.4
(C
(hpr (1)) (u ())))))))))))))))))) ti(u
Definition A.6
pr Deepa (z(4x())
Induction
(u ( (())
Lemma A.9
[L [[x.).)
((1)
( ) (epp (1)
Definition A.6
F(L) (M ())
Definition A.4
0 (1))
(1)
Definition A.6
Since Lemma A 9 holds for almost all (1) E ( the proposition thus established.
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Exogenous Isomorphism for Counterfactual Identifiability
Lemma A.11. For each E I and for almost all u E 00 the PCCM satisfies
(2) Bpr (0[x) (1) (u) (2) e (1x. pr( (1))
(r))
Proof. Consider ( E 1 for which Lemma A.10 holds. Then, (2) (1(x)) (b (1 (1) (o)) (91(x.] (h (() (1))()))
(2)
Definition A.7
-0 (u (1)
pet pr)(i) (k))))))))))
Lemma A.10
(1](x. (u (2) (())
(r)
Definition A.7
Since Lemma A.10 holds for almost all a()) 0 the proposition is thus established.
Theorem 3.2 (~EI Implics ~Cg)) For recursive SCMs (( and M M ~EI M then M Nes M(2)
Proof.
Consider (1) E 2 for which Lemma A.8 and Lemma A.11 hold simultaneously. Then, when pr" pr"(1) = I, we
have
M. (u())
Lemma A.
(1) 87xx (u()) Lemma A.11
(BI[x. o ()((())
(2)
Lemmna A.8
(V o h)((())). M.
Since Lemmas Lemma A.8 and Lemma A. hold for almost all ( E 2 it follows that v M. almost surely equals (2 M. o h. Moreover, since p(2) = hy ((1) it follows that
PM (v.. M. ) P = (v M. o P (V M. (V M.
p( P
PM
Therefore, for any counterfactual random variables Y. V. and any event v.. we have P.M()) (V.) (MM (V.), as they are obtained by marginalizing PV. = PV. P. According to the La-valuation, any term of the form IP(Y, E V.) has equal assignments in (1)) and M Since for any sp E L3. ip consists of terms of the form PPYY. V.). it follows that (p((()) ((M Therefore, according to the definition of Ly-theor, we have
()( = (P(M()) I Cs} = (p(M()) I ee Ca} = L2(M
ie., M()) ~C, (2), according to Definition 2.2.
A.3. Bijective SCM
From the perspective of measure theory, (regular) conditional distribution is defined as probability kernel Pyix Fx x 2x > 0, 1]. where it is required that the index sets satisfy Ix n Iy = If for any X E Fx and y E Fy there exists joint distribution vx, y) = fx PY) x) x) where Px the marginal distribution, then such a probability kernel Pyxx is referred as regular conditional distribution given X. The Decomposition Theorem establishes its existence and uniqueness:
Theorem A.12 (Decomposition Theorem). If the space Ωx is Polish space, then the regular conditional distribution PY|x Fy x Ωx + [o, 1] exists and is almost surely unique.
Lemma A.13. Y f(X), then for all y Fy and for almost every X E Qx the conditional distribution x (v.x) 6 (2) (f((x)),, where 81(x) (Y) the Dirac measure and y (x()) is the indicator function.
Proof. For any X Fx and y Fy, according the definition of the joint distribution,
(x y) ((X, Y) y) P((W w N
=P([W wollwee (f(x)))
= [[tW|WEE (
-))( nx).
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Exogenous Isomorphism for Counterfactual Identifiability
Then, for the Lebesgue integral of the indicator function (or Dirac measure).
J (x(x(x)) / 1 5- (xx
1-1D) Px(dx)
-xx(f ())
= =)r x xx).
According to Theorem A.1 since Pyix that satisfies Pr. X) = 1x P Px is almost surely unique, it follows that for all y Fy and for almost every X E Pxx Prix ))) = ly((xx)).
Lemma A.14. Y f(x, Z), then for all y E Fx and for almost every z Uzz the conditional distribution Pxiz(D,2) (U(,, (((,)))))x(( z)) ()).
Proof. For any y E My. consider the condition that Lemma A.13 holds for E Ptz. Then,
(x = Pxxx(i(X) x 02x,2)
Factorization
3x
1 1y([(x,2)) (x/zxxx)))
Lemma A.13
Jax
- (1) Px)zxxx
- Zxzz(dx, z)
Pxiz (ff(:z))~ [>),
(ff(, Px(z((,z) (27).
Since Lemma A.13 holds for almost every E Ωz the original proposition follows.
Proposition 4.2. recursive SCM M is a BSCM if and only if J<(v, is a bijection for every E I and all E dv.
Proof. = Suppose that the SCM a BSCM. Consider each E I. According to Definition 4.1, this is equivalent to proving that (V(p(())) -) is both injective and surjective for all fl. under the assumption that r is a bijection.
First, we show that fs ( -) is injective for all v E fl. Given any Gy and any u,, ut, ly such that fe(v (a))))() u:) Ji (Vpa(t)) u(), let vi fi paps))) uz) (note that of may not equal v.). Construct v* (Vpa(t) of VZV where (Pate)ppat) is arbitrary. By the definition of BSCM and since is bijection, there exists unique u" such that v T(u*). Replace the i-th component of u² with u, and u/ to obtain u and u', respectively. Since ʃi /i us) = fs(V pa(t), u4) = v/' it follows that V = r(u)= = r(u'). Given the uniqueness of u" such that v* = T(uu), ), we have u = that u, u'. Therefore, for any 44 de E Qu if fi(yaat): u;) fi(Vpa(), u(). then u, u/. proving that f. (V injective.
u², implying
Next, we show that fe -v() is surjective for all Ωv. Assume, for contradiction, that there exists E Ωy such that fi(va((),)) :) is not surjective. Then, there exists vi such that for all E Qu,, f((mm(): u4) * v.. Construct v" = (Vpa(()) v/. V (patt))))) where (((())pa))) is arbitrary. By the definition of a BSCM and since I is a bijection, there exists u² such that v T(u*). where the i-th component satisfies fs (V(paa)) u²) vy. This contradicts the assumption that for all E Su. ( paats) u:) # v/. Therefore, our assumption is false, and (V(pp())))) :) must be surjective for all E sl..
In summary, for all v E Ωv. fa( (Vp(tt)) :) is a bijection.
= Suppose that fv(paa))))) :) is a bijection for each E I and for all v E Gl. This is equivalent to proving that T is both injective and surjective.
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Exogenous Isomorphism for Counterfactual Identifiability
First, we show that I is injective. Let u, u' ay such that r(u) r(u'). According Lemma Lemma A.2, for each i E I, have v2 (a(()) u.) pa u!). Since fi( (V(paa()):)) -) is bijection, it follows that u4 v//- Therefore, u', proving that I is injective.
Next, we show that I is surjective. For each v E Ωy and each i E I, consider the i-th component ug. Since Sa (Vpaa))) -) is a bijection for any v E fy, there exists i; = (Fi(Vpats). (us) such that & = fa(p)a()) 4,). Let (V/((pa )) (wi))Ee.. According to the definition of the solution function T, substituting u into yields T(u) v. Thus, for any Gl.. there exists u such that r(u) v. proving that I is surjective.
Therefore, T is bijection. By Definition 4.1, M is BSCM.
Theorem 4.3 (BSCM-EI). Iftwo BSCMs M and M share common causal order S. and the same observational distribution Pv, then (1) ~gl M if and only for every? i I, there exists a bijection h4 0.. + St) such that for all Sly,
(v. o (fft) 0 (v.-)) h4
almost surely.
Proof.
+ Suppose that that.M( M( ~EI M According to the defimition of exogenous isomorphism, there exists a bijection (hx)iez such that each h2 is a bijection and the causal mechanisms are preserved. For each i I and given any (1) (v, for some u; E s Since f,²) (v, is
E Ωv, consider the causal mechanism 5 (vvh(((("))
bijection, composing both sides with (J! (2) (v, yields
h.(u: (1)) (V (v, uz (1) )) o (1. (v. ().
Since (v, (v, holds for almost every uj (1)
E 0 it follows that
f( (v. h4
almost surely.
< For each i E I, suppose there exists a bijection h; : Sz such that
(2)
(v. f( (v. h4 almost surely. By the associativity of function composition, for any v E Ωv. we have
( (v. - h, (v. (1F (v, (v) ff (v.-)
almost surely. This equality holds for each i E I and all E Ωy thereby preserving the causal mechanisms. Moreover, since the observational distribution satisfies
Py (T(") = -
and both (1) and (2) are bijections, follows by pullback that
(()))))1 r('); -P(2)
By Lemma A.8 and Lemma A.1I, we have (1) ( h. that is, (T()))-1 r h. Therefore, h,p!! p( which preserves the exogenous distributions. Lastly, the component bijections are satisfied by the assumption of h; 52(7 + 0 Hence, (hx)ez constitutes an exogenous isomorphism. Given the existence of such an exogenous isomorphism and the common causal order, it follows that (11) ~EI (2).
Before proceeding with the proof below, we visually depict the objects of study in Theorem 4.3 and Definition 4.4 in Figure 3, allowing readers to intuitively grasp the strong connection between these two concepts. Intuitively, these objects are related through a "flipping" operation; formally, this relationship is characterized algebraically by the computation of associativity and inverses. Subsequently, we focus on Definition 4.4, which enables the problem to be confined within the same BSCM.
19

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Exogenous Isomorphism for Counterfactual Identifiability
D
U
u
(3))
u
(v.))
(M
(v.)
(V.)
(v.)
1.
Ωy
(
(v', 1) to (
0 (v. (3) 0 (..)))
(b)
Figure 3. (a) The objects of study in Theorem 4.3, (1.²) (v. to.,))) (v. (green) and (,, - (red), constructed across different BSCMs; (b) The objects of study in Definition 4.4, (f, (v',)) (v.-)) (blue) and (f, (v',)) o (f(2) (v,-)) (yellow), constructed within the same BSCM.
Proposition 4.5. If the BSCM M is Markovian, then for almost all v, v E Sby the conditional distributions satisfy PVsIV (s, v') = (KMsl's v, v'))
(.
Proof. According to the recursive SCM, there exists a unique solution such that V; = [v(V pa(t): U2). Based on Proposi- tion 4.2, the causal mechanism fi f((y)()))) in the BSCM is a bijection for each i E I and for all v E Gyy. Therefore, we have U; = (fiVV pu(s)) p)(s):))) (V:).
According to Lemma 14, for any fly, it holds that
PriVveco (s Ypa(s) = (fi(Vpu(s)) ())pp(()))))PPu.. 1)pPud Vonco (s Mpu(s));
PU.V (ss Vpa(o)) = ((Ji(Vpa(s): +))=)) Puilvaas (s Vpa(s))
By the Markovianity assumption, Per Vacc Pu., Therefore, for any v. v E Slv, the counterfactual transport satisfies
(KMsls, v, PV/V/eoi (., Mpa(t) = (JL(V )) (fi(V -)- Puaivenos (ss Vpa(s))
(Ai (((F(VV -)= Pvi (ss Vi((o))
(FiVY Pu.IV Co Mpa(t)
Lemma A.14
(Ji(Y Pu.
Markovianity
(A(V) +)), PuivVeeo (s Vp(co)
Markovianity
Pvil (ss (o))))
Lemma A.14
v. This shows that the conditional distribution satisfies Prilvite (vv)) (KMal.v v,v))), (5 v).
Theorem 4.6 (EI-ID from Counterfactual Transport). An SCM is rgr-identifibble from A P.B.CM.s.Pv.K)
Proof. Consider any pair of models M(), ((2) } A and each i E I. For any pair v E Sv, let the components of the counterfactual transport be K KM v') and K K v, v'), respectively. According to the validity of (MM), it holds that
KMa 2l, v, v,v Ky v. v') Kilsv.v")
almost surely. By the definition of counterfactual transport, this implies
(1. (Y(on) :) (
(Vpa())) -1 (s (v)(()): (f.2) (V(3)((pa(t)))) (Vpat)).
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Exogenous Isomorphism for Counterfactual Identifiability
almost surely. Since the causal mechanisms 1 (Vpat))) .) and f (V (a())) are bijections, we can compose them on the right with f (Vpo(t):)) and on the left with (1/²) (v (a())) -))- respectively. By the associativity of function composition, we obtain
(Ff (V
f ())()))) = ( -)) o (Vp((t)))
almost surely, and both sides of the equation remain bijections. Since this equality holds for any pair v. v' E Svv, they are all equivalent to some bijection h2 00 D That is, for each i E I. there exists bijection h4 2 - 2/? such that for all Slv.
(v,))) 5! (v, = h4
almost surely. Furthermore, a common causal order exists by assumption. Therefore, by Theorem 4.3, it follows that ((1) ~E M(2), Theorem 4.8 (EI-ID from KR Transport), An SCM is reg-identifiable from A A(BSCM.S.M.Pv.K.
Proof. We need prove that if M F (BSCM, M.Pv, then .M M ( and subsequently apply Theo- rem 4.6. Specifically, only need to demonstrate that if M += A KR), then Ax(M) holds.
Assume that M }= ABSSM, (.B.CM....Pv.KR) Then, M is a Markov BSCM with causal order S and observational distribution Pv. For each E Z. consider any pair v' E Pvv. Given the observational distribution Pv. the conditional distribution P (:, pa)( is almost surely uniquely determined by Theorem A. Then, according to Lemma 4.7, the KR transport between Pv.IV pvt) (:s Ypa(t) and Vvi (-. v' (atte) is almost surely unique. Let the partial function K2 v, v') be any version of these KR transports. Therefore, K2(-, v, v") is almost uniquely determined, and hence K = (Kz) is also almost uniquely determined.
Since AKR(M) holds, the components of the counterfactual transport KM, are almost surely equivalent to the KR transports. Moreover, because KMM a transport between conditional distributions by Proposition 4.5, and by Lemma 4.7, the KR transport between two distributions is almost surely unique given the distributions. Therefore, the component K KM,a (1, v, v') of the counterfactual transport is almost surely equívalent to Ki(, V, v'). Since the counterfactual transport KM (Ki)kez, it follows that the counterfactual transport KM = K holds almost surely. Thus, Ak(M) is satisfied.
Consequently, we have proven that if M }= A(BSCM,.M.PP.KRR) then M F= A(BSCM...PV.K)- rem 4.6, for any pair M(MMM F A(BsCM,S.Pv.K) it holds that M(1) ~EI M Therefore, for any pair M + AIBSCM.g...Pv.RRR we also have M ~MI M(2) This implies that the SCM is ~er-identifiable from A S.BCC..S..PV.KK..
By Theo-
A.4. Triangular Monotonic SCM
There are some properties of TM mappings:
Lemma A.15. For a TM mapping T : IRd + Rd, its inverse T-1 is also a TM mapping, and the monotonicity signature satisfies E(T)- = E)T-1).
Proof. To clarify the proof, we denote the domain or codomain by the symbols 2x and f2z, both of which actually refer Rd For TM mapping T .x - fiz, assume it is of the form T(x) = (T, X1:j-1, 23 D)jeia where T, [x,,,- flx, + Oz, According to the definition of a TM mapping, for any Xi:j-1 E Ωx the function (x -): 9x2 fzz is consistently strictly monotonic. That is, for each j E 1: 1:d, there exists &, E (-1, 1} such that E(T)) = Ej.
We construct Tig Pz.., - Ix,, such that is of the form Tij (Tr)Re and
T,(o) = (T,(TTs)(ns)))
(zz).
for any j E 1: d: and any Zizj E Oz... Since Tj depends only on Zi.j. T a triangular mapping.
Let T Ti.. Next, we prove that T = T- meaning that T is the explicit form of the inverse function of T. According to the definition of an invertible function, we need to show that z = (z) for any z E Uzz Consider each component,
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Exogenous Isomorphism for Counterfactual Identifiability
i.e., prove that 2, (T(T(z))) for any j 1:d. First, by the form of ', can expand its j-th component
= ((T)) (T(x),,
According to the previous construction, (T(z)) T1.-- and (T(z)), = T3 (Z1:), 2ÿ). Substituting these into the above equation, we obtain
(T((()))); =T (Tig-1(2)--1,, (T
(8))
Now, we need to show that the above expression equals 2j for each j E 1:d. Let Tı:- = X 1- Since T is TM mapping, the function Tj (x1x1-11))) Ωx + Slz is always strictly monotonic and hence invertible. Therefore, the expression simplifies to
Ty(x(u-11 )) T 11 (z)) = Zj-
Thus, (T)())))) = 2; holds for each j E 1:d, implying that T = T Since T is a triangular mapping, T is also a triangular mapping.
Next, we consider the monotonicity signature. For each 1 1:d, recalling that
(z(1),)))) = (T((T1)(4s)):)) (zz).
for any Xi:j-1 E 1xx the function (x : Qx + Pzy is s.m.i., then its inverse (ireesseTr(xxxii (x1:g-1:)) Ozy + Qx also s.m.i. For any Z1:j-1 - let T 11:3-((11.j-)
X1.5-1- Since x(y--1 E fx the function (T) (x))-111)))) is always s.m.i. Therefore, T3 is s.m.i. with respect to 2j given any 21:--1 P Similarly, when (x1o-11);) is s.m.d. a similar conclusion holds. Therefore, according to the definition of the monotonicity signature, for each j 1:d, there exists E, E {-1, 1} such that E(T)) E(T,) Eg- Consequently.
< {(T) (g(T) sEIid (E,)jeı:d (E(T))) =E(T),
i.e. the monotonicity signature satisfies ((T E(T).
Moreover, since Σ 161l = d and T a triangular mapping, according to the definition of a TM mapping, T-1 is indeed TM mapping.
Lemma A.16.
For two TM mappings T(); Rd - Rd and ( R + Rd. their composition T o (2) is also TM T = (((T(2)) where @ denotes the component-wise
mapping, and the monotonicity signature satisfies ( multiplication.
Proof. To clarify the proof, we denote the domains or codomains by the symbols 9xx Gl and Pzz all of which actually refer to Rdd, For the TM mapping T(1) fy - Pzz assume it is of the form TT) = (T,) (y1:j-1, V5)))e where
T(L)) : - x By + Slz According to the definition of TM mapping, for any Y13-1 E a the function T(()) (y 0l + f2z, is consistently strictly monotonic. Therefore, for each j E 1 :d, there exists cf²) E (-1,1)
such that ((T = (1) The same holds for the TM mapping T
We construct Tiy !x Pzy,, such that is of the form Ti. (TR)RELs and
(x13, T (T() (1:)); T (x13,
where T(2) (T! Since iú depends only X1.j. T is triangular mapping.
According to the form of (1),, for any j 1:d and any x E 9xx let y = (x). Then,
(T o (x))) (y)) = T 1))
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Exogenous Isomorphism for Counterfactual Identifiability
According to the form of T we have
yiy = (T(3) (T (x))=1 -T(.
-T(((11))) (x1:))
and yj = -x1:((x1:),))))))) Therefore,
((T o (x)); = 2150) (y1
= =T (1) (T (1::) T (x1:) 25)) = (x1:1,
for any j 1:d. Let T = Tı.. That is, T = T o T Since T a triangular mapping, T o T(2) is also a triangular mapping. Next, we consider the monotonicity signature. For each 1 :d., recalling that
(x1,, = T/ ( 13 (x1:); T (x13,,
If for any X1:--1 E 9x1g- the function -/ (x1:j11,) : 2x, + Ωy is s.m.i., and for any y1:1-1 E PY the function 27fh) (y1:y11-): : S2y + S2z is also s.m.i.. then the composition T. (y15-1, T (x1:)1, -) is s.m.i. According to the definition of monotonic functions, if both functions have the same monotonicity, the composition is s.m.i.; they have opposite monotonicities, the composition is s.m.d. Therefore, according to the definition of the monotonicity signature, for each j E 1:d, we have E(T,) = 5)' E Consequently,
{ T()) {T) (E(T,)) = (E)) { IeeIdd = (E(T)))) E(T,) )) = <(T(1) ((T(2))
i.e. the monotonicity signature satisfies {(T()) T() (T @ ((T
Moreover, since _ E {-1,1), it follows that |5
(1) ( = 1 for any j E J. Therefore, Ej= (55) -,-01 d.
Additionally, since T()) T2) is a triangular mapping, according to the definition of TM mapping, T()) T2) is indeed
TM mapping.
Remark A.17. For two TM mappings T()) _2 + nd and T RR - Rd, i((()()) ((T(2)), then ((T()) T(2) 11) meaning that T (22) is TMI mapping.
Proof. This result trivially follows from the component-win wise multiplication and Lemma A.16.
Lemma A.18. For TM mapping T TRd Rd, its subcomponent Tej restricted to any X1:i-1 1 E Ri-1, ie., Tay (x1:5,1111 is also a TM mapping for any i, E 1:d with i < j. and its monotonicity signature satisfies (TT (x1i-1, ))s = (E(T))
Proof. To clarify the proof, we denote the domain or codomain by the symbols Qx and f2z, both of which actually refer to Rd For a TM mapping T : f2x + Pzz assume it is of the form T(x) = (x; )))))E,, where T; -x xfx, + 522,. According to the definition of a TM mapping, for any X1:j-1 E Ωx1 1. the function (x )) Ixx - S2z, is
consistently strictly monotonic. Therefore, for each j E 1:d, there exists E E {-1, 1, such that E(T,) = Ey-
The subcomponent Te. is of the form of the form (T& ( Xiy 1, x1 Rejjj Since (x depends only on Xi.j Thy (x -) is triangular mapping.
Xcy-1, keey. For any X1:1 1 E R the restriction Tey (x1:-1. is
Now consider the monotonicity signature. Assume that for each A i: j. according to the definition of {(Tk), Th is s.m.i. with respect to Iy given any X1:h- E .x...... Since 1x E .x.._-.. to given any X E fx
(x( -) is also s.m.i. with respect The same conclusion holds when Th is s.m.d.
Thus, according to the definition of the monotonicity signature, for any X E Ri- and each E i : j. we have ((Tx(x)x-1,;)) = Eu. Consequently,
(X1= = (S(T)(XxI- = (Ex)aeR) (E(T))
Moreover, since El= [xl = - -i -it 1, and because T (x is a triangular mapping, it follows from the definition of TM mappings that Try (X1:1-1;)) :) is a TM mapping.
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Exogenous Isomorphism for Counterfactual Identifiability
Lemma A.19. For the vectorization e under S. for each i I, there exists 4 + Eieprco dy such that the image set (((ii))s = 41 l + de - 1, where pr(i) denotes the prefix of index under S.
Proof. Recall that the vectorization is flattening mapping such that for any (t,5), (1,5") E II (i,,)) (') = - and for any (i, (f', ,') E I, i < i' implies ()) < ('', We prove the result using induction over the partial order S. When |p((i] = 0, we have 4 = 1. Since there does not exist any i' E I such that 1' < i, there are no pairs (1', such that (f,)") < 1). Thus, by the definition of vectorization, <(i, 1) = 1. For any (6.,)))esddd we have (i.,)) ((i, 1) = j 1, so (). = j - + 1 = j. Therefore, the image set ((((i.)))
= d2 = 4 1, + d, 1.
Suppose for any A I with pr(k)) < pp(())), there exists 4 = 1 + Eiepree) d, such that the image set 4: 4/ + d+
Let 4 = 1 1 Eiep((s de.
Since i is a bijection, the preimage 1- 4, - -11 -itt) (1x : l2 + du - 1)] = (p( (kk,)))s by the inductive hypothesis. Therefore, <(i, = 1 Lisprii) di.
For any
have j) - ((i, 1) = j- 1, ((, = =j -1 Eiep(t)) di) j Eisp(t) d. Therefore, the image set is o ((((i,))) = + Eiepr()) di : d; + Eiep()) = 14:44 + +d 1.
Lemma A.20. For the vectorization i under S. if S is a causal order, let an(i) kli,k) j E an(i), k E 1:dj3. Then, for each i I, the image set satisfies (in( C 1:1;- 1.
Proof. We prove this by contradiction. Assume there exists j E an()) such that 1 + dj - 1 2 4, Since j + i i an(i) due to acyclicity), and using the properties of vectorization, it follows that 1 2 4 + de.
Thus, 27 > I2 holds if and only if j > i (by the properties of vectorization). However, since j E an(i), we know j < i (by the definition of causal order). This contradicts the assumption that > i. Therefore, the assumption 1 + dj - 1 2 1, does not hold. Hence, for any E I, we have x(an(i) (i) S 1::,- 1.
Proposition 5.3. A recursive SCM M is TM-SCM if and only if f((,,) is a TM mapping and there exists & such that every E I and all Ωl.
= E <(fi(v,)))=
Proof. Consider a causal order I with its vectorization 6. Define the reindexing mapping P.ç = (Passhed; By Lemma A.19, the vectorization satisfies P (erd)) For simplicity, when we refer u, u. indexed by I. we actually mean the entire vector (za.))jenad
= (2))) and its inverse P ((x)))eaattdd1)
Assume that the SCM M is TM-SCM. By Definition 5.2, there exists causal order S with its vectorization such that the reindexed solution mapping P. o r is a TM mapping. Let P. o r = T = (Tz))EL:D, where each T, is of the form (u uz), and D = Σiez d,. Then, for each i E I, we have (T') x = P ((T))setattdd-)) = P. Tiok-d--1
By Lemma A.2, r(u)= (f( so fr PP -1 The function Tm, -1 - (1.,,-1, is of the form - 44.44d.-1)) Thus, (P.,! o -1) (1:4, .). is of the form
(P -1) I] (u: (ui,)jExaa)
By Lemma A.20 and the fact that Unn(i) = (u.,) an f is independent of -1waao))) Hence,
f (umn()) u.) = (P..) (1od (us
implying that (M -) is a triangular mapping-
By Lemma A.18, the monotonicity signature satisfies (TT. 11... = (E(T)) )a,+-11- Thus, {(fi(Uan()) -)) (E(T)) Since E ((5(T));) = d, it follows that fi(uan(): -) is TM mapping.
Finally, since TM mappings are bijections, for any given E Ωl, there exists n' E Qu such that u' -v). By Definition 1. ff((ff (1'm (())))=p)(()) (u (um(()))))))) -). Therefore, for all v Ωv fi(v,)) is a TM mapping, and there exists & = (E(T)) such that -)) E.-
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Exogenous Isomorphism for Counterfactual Identifiability
= Let < be any causal order and & its vectorization. Assume [1(v;-) is TM mapping and there exists &, such that ((J((v-)) = & for each E I and all Gy. For any given fly, there exists u' Py such that u' r ---l-v] (v). By Definition A.I.
fe((fu(u'nn (() -) f((((((t)))))
By assumption, [a (( -) is TM mapping with ((fi(v,-)) E.. Using Lemma A.20, we construct didc+ Pi. where {(T,) (5,) for each j 4 + d; -.
By Lemma A.2, r (fi)iez. Thus,
P foiez= = (Tustetd- = (Tz)
Each Т depends only on u1:4,11 and since ΣjEnn (5(T,)) Eiez di = D,P. o I is TM mapping. By Definition 5.2, M is a TM-SCM.
Corollary 5.4 (EI-ID from Triangular Monotonicity). An SCM is ~EI -ddeiiiiif from ATM SMM
Proof. We aim prove that if M += ACM. then M } ( S. Pv,,k. allowing us to utilize Theo- rem 4.8. Specifically, it suffices to show that if M F 5 Pv) then AKR (M) holds.
Assume M -= A(IM-SCM...M. P. Then M is a Markovian BSCM, inducing the causal order S. with the observed distribution Pv. Since M is a TM-SCM, by Proposition 5.3, for each i E I and all v E Gl, the causal mechanism f((v,) ))
a TM mapping, and there exists & such that <(fi(v,)) .)) = E.. Furthermore, by the definition counterfactual transport and Remark A.17, K,, (fi(v',)) (f((v,-)) is TMI mapping for any pair v, v' E Ωv.
Now, for each E I, consider arbitrary pair v' fiv- Given the observed distribution Pv, the conditional distribution Pusly mut) (t, Ypu(i) is almost surely uniquely determined by Theorem A. Consequently, the conditional distributions Pvalero (t, Vpr(s) and PvalV (:, (()) are also almost surely determined.
Given these distributions, by Lemma 5.1 and Proposition 4.5, the TMI mapping KM,:- = (fi(v'. -) (f((v,-)) - is almost surely equivalent to KR transport. Thus, AKR AKR(M) holds. Hence, have proven that if M F A(TMSCM.S.MMPV)) then M = Aus.M._...PvKKR)-
By Theorem 4.8, for any pair M() + M we have M ~EI M(2), Therefore, for any pair M(),((2 } M.M.)) we also have M(1) ~El M Thus, SCMs from A(MASOM.S.MPP) are ~E identifiable.
Lemma A.21 (restated from Khoa Le et al. 2025, Theorem 1). If the velocity field v Rd [0,1] + Rd is a Lipschitz continuous triangular mapping for any E [0, 1), then for the ODE
dx((t) = vxx()), t) dt,
(0) Xo:
the flow T(;t) t) is TMI mapping for any i 1).
Proof. We first show that T., is triangular mapping. Writing T(: as consider dimension j = 1. Since vi(,,0 f) is triangular mapping depending only x (t), the corresponding ODE can be written as
and ztt) as (zx, (t))
dæ (1) = vi vx (1), dl,
x1 (0) -x100-
This is one-dimensional ODE. By Lipschitz continuity and the Picard-Lindelöf theorem, the solution exists and is unique, implying that æ, (t) depends only on X1.0. Therefore, by the definition of the flow, (Xo, t) = x1 depends only on X1,0-
Now, assume by induction that T, (x0, t) = (t) depends only on X1:5-1,0- For dimension j. since (- t) is a triangular mapping depending only on 11 (1), the corresponding ODE can written as
dx. (t) = v, (x dt,
æ:: (0) X1:j.0>
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Exogenous Isomorphism for Counterfactual Identifiability
or equivalently,
dæ, (t) = Uj (1), dt,
20)(0) = x
(1:)-1(0) = X1:j-1.0-
Here, 1t) depends only on X1:5-1.0 and can be treated as constant. Thus, the ODE for æz(() becomes one- dimensional ODE. By Lipschitz continuity and the Picard-Lindelöf theorem, the solution exists and is unique, implying that (t) depends only on x,,0- Consequently, z,(t) depends only on X1:j.0 when X1:5-1,0 is unspecified. By the definition of the flow, Ty (xo, 1) = (() depends only on X1:5.0-
By induction, T(; 6) is a triangular mapping.
Next, we show that T (-.tt is a TMI mapping. For each dimension j. consider two different initial values 2y,0. 2/.00 e R such that 25,0 < T'y..- From the previous proof, given X1:j-1,0, the ODE is one-dimensional. Using proof by contradiction, suppose that at time t = t', we have æ, 2 (t), where (t) and z (t) are the solutions starting at 25,0 and 23.0- respectively. By continuity, there exists 1x < L' such that æz (²((") = a² (L"). However, by Lipschitz continuity and the Picard-Lindelöf theorem, the solution to the ODE is unique for any t. Thus, if (t*) = (t") at some t, then (t(t) (t) for all t, including = 0. This contradicts the assumption 25,0 < 2'5.0> so the hypothesis is false.
Therefore, given X1:5-1,0- if 23,0 < o/.00 then (1) < (1). In other words, E(T(() = [(x,)= =1
In summary, tt) is triangular mapping and Σド= E(T}(; ()) d. By the definition of TMI mapping, T(,2) is TMI mapping.
B. Related Works
B.1. Identifiability
Previous work on counterfactual identification has typically focused on identifying specific classes of counterfactual statements & S L3, which can be categorized as follows: (i) Constraining causal structures to identify specific counterfactual effects. This involves representing counterfactual statements using symbols, graphs, and available observational and interventional distributions. Examples include the identification of counterfactual probabilities (Shpitser & Pearl, 2008), nested counterfactual probabilities (Correa et al., 2021), sufficient or necessary probabilities (Tian & Pearl, 2000), path-specific effects (Avin et al., 2005), discrimination effects (Zhang & Bareinboim, 2018). 8). and identification via neural networks (Xia et al., 2023). (ii) Constraining causal mechanisms to identify counterfactual outcomes. This focuses on deterministic counterfactuals inferred through abduction-action -rretiiti-neiiiionnn (Pearl, 2009), a special class of counterfactual statements of the form Y x',,'. Identifiability is achieved primarily by imposing parametric constraints such monotonicity, in (Lu al., 2020), (Nasr-Esfabany al., 2023), and (Scetbon et al., 2024).
Counterfactual Identification
Identifiability The identifiability exhibited by generative models has been a continuously discussed topic in the literature, spanning tasks such as representation learning (Bengio et al., 2013), causal discovery (Glymour et al., 2019), causal representation learning (Schölkopf et al., 2021), and causal quantity reasoning (Pearl, 2009). Identifiability in representation learning refers to uniquely determining latent factors under certain ambiguities, such as permutation and scaling (Hyvärinen et al., 2023), affine transformations (Kivva al., 2022), or component-wise invertible transformations (Gresele et al., 2021). In causal discovery, identifiability pertains to structural identifiability, which requires determining equivalence classes of causal graphs (Spirtes& & Zhang, 2016; Vowels al., 2022) or identifying the direction of cause and effect (Shimizu et al., 2006; Hoyer al., 2008). In causal representation learning, identifiability involves determining latent causal variables and causal structures, where causal variables must satisfy component-wis wise diffeomorphic mappings, and causal structures require graph isomorphism (Brehmer et al., 2022; von Kügelgen et al., 2023). Finally, in causal quantity reasoning, identifiability refers to the consistency of causal statements (Pearl, 2009), which was previously termed causal identification.
B.2. Triangular Monotonic SCMs
To demonstrate the broad applicability of Corollary 5.4, we summarize several representative works from the past and establish their connection to our theory by proving that they are special cases of the TM-SCM. This highlights the significance of our theoretical contributions. Overall, these works can be categorized based on the criteria outlined in Table 2, which further inspires us to derive four prototypical models of neural TM-SCMs.
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Exogenous Isomorphism for Counterfactual Identifiability
Table 2. Representative works that special cases of the TM-SCM.
Model
Research Domain
TM Object
TM Type
TM Principle
LINGAM
Cause-effect Identification
f
ANM
pp + Mi
ANM
Cause-effect Identification
ANM
(e(Vpatt) us
LSNM
Cause-effect Identification
ANM
((Vpats) + 8: (x(p((s))
BGM
Counterfactual Identification
fi
UMM
Prior
PNL
Cause-effect Identification
fi
UMM
(Vg:(Vruc)) u4)
FIP
Counterfactual Identification
UMM
C² smoothness
CAREFL
Cause-effect Identification
PO
DCF
Affine Transform
STRAF
Causal Quantity Estimation
P
DCF
UMNN Transform
CAUSALNE
Causal Quantity Estimation
POT
DCF
Monotonic RQS Transform Partial Causal Transform
CCNF
Causal Quantity Estimation
P.o
DCF
CKM
Generalization
P.O
CCF
CFM
Causal Quantity Estimation
P.o
CCF
Sicard-Lindelöf
Original Research Domain The related works summarized here originate from various research domains, each with distinct objectives. For instance, many studies specialized SCMs aim to address the task of cause-effect identification in causal discovery. Others, as previously mentioned, focus on identifying counterfactual outcomes by imposing constraints on causal mechanisms. Some works explore how to generalize SCMs to other scenarios. Another line of research, more aligned with deep generative models, seeks to address efficient causal effect estimation. Interestingly, despite their differing objectives, the SCM formulations of these works uniformly fall under the TM-SCM framework.
Triangular Monotonicity Object The TM-SCM framework admits two equivalent definitions. One, as described in Definition 5.2, requires the solution mapping P, I, obtained by reindexing under the causal ordering vectorization 1, be TM mapping. The other, given in Proposition 5.3, ensures that the causal mechanism fi(v, for each i I and any fy is TM mapping with consistent monotonicity signatures. Therefore, these works can be categorized based on whether they focus on constraining the global solution mapping P, I or individual causal mechanisms ft.
Triangular Monotonicity Type they achieve:
Existing works can be further categorized based on the types of triangular monotonicity
Additive Noise Mechanism (ANM): Additive noise mechanisms restrict each causal mechanism fi the SCM the form (((pa())) + (V)(t) Ug where 3 is required be strictly positive function. Consequently, even for vector-valued cases, the causal mechanism .)) for each I and any E By is always TMI mapping. This follows from the Jacobian matrix being diagonal, with each diagonal element (B (V(Vpu()))) >> 0 due to the constraints on B. Relevant works include LiNGAM (Shimizu et al., 2006), ANM (Hoyer et al., 2008), and LSNM (Immer et al., 2023). These studies have inspired the construction of DNME-type neural TM-SCMs.
Univariate Monotonic Mechanism (UMM): In the univariate case, functions are necessarily strictly Monotonicity they are invertible. Thus, it suffices to further constrain the monotonicity signature each causal mechanism fi(,,)) to be consistent for every E I and any Ωy. This consistency can be ensured by various approaches: (1) enforeing the function to be s.m.i s.m.d through prior construction, (2) indirectly composing an additive noise mechanism with an invertible function, (3) deriving monotonicity consistency from C smoothness. These approaches correspond to BGM ENasr-Esfahany et al., 2023). PNL (Zhang & Hyvärinen, 2009), and FiP (Scetbon et al., 2024), respectively. These works have inspired the construction of TNME-type neural TM-SCMs, extending from the univariate to the multivariate case. Discrete Causal Flow (DCF): Certain discrete-time transformations in normalizing flows, such as affine transformations (Dinh et al., 2017), unconstrained monotonic neural networks (Wehenkel & Louppe, 2019), and monotonic rational quadratic splines (Durkan et al., 2019), can be shown composed, the entire normalizing flow also remains TMI mapping. Several works establish a connection between r and normalizing flows, such CAREFL (Khemakhem et al., 2021), StrAF (Chen et al., 2023), CausalNF (Javaloy et al., 2023), and CCNF (Zhou et al., 2025), each employing different transformation blocks. These works can be classified as the CMSM-type neural TM-SCMs. Continuous Causal Flow (CCF): In continuous time settings where ODEs are used to construct SCMs, structural constraints are introduced into the velocity field, providing a novel pathway to induce TM mappings. Specifically, when the velocity field is triangular, the solution to the ODE always a T'I mapping, as guaranteed by the
TMI mappings. When these transformation blocks are
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Exogenous Isomorphism for Counterfactual Identifiability
condition (see Lemma A.21). Some works establish equivalence between the ODE solution and the solution mapping I of an SCM, such CKM (Peters et al., 2022) and CFM (Khoa Le al., 2025). These studies lay the foundation for constructing TVSM-type neural TM-SCMs.
Corollary 5.4 establishes that, under the additional assumptions of Markov property and identical causal ordering, and provided they induce the same observational distribution, the constructed models are identifiable up to the ~gu-equivalence class. Furthermore, by Theorem 3.2, these models are mutually Cg-consistent, implying that they are indistinguishable at the counterfactual level. Therefore, when these models are claimed be counterfactually consistent or counterfactually identifiable, Corollary 5.4 and Theorem 3.2 theoretically justify such assertions.
B.3. Theories
We now elaborate on how other related theorems mentioned in the main text can be interpreted as special cases of Theorem 3.2 and Corollary 5.4, starting with restatement of these theorems using the notation of this paper.
Special cases of Theorem 3.2 Several prior works have introduced notions similar to exogenous isomorphism and shown that SCMs satisfying these equivalence relations enjoy counterfactual consistency, mirroring the result of Theorem 3.2.
For instance, (Peters et al., 2017) formalizes the concept of counterfactual equivalence and proves that, when two SCMs share the same exogenous distribution, counterfactual equivalence is guaranteed by the following theorem:
Theorem B.1 (restated from Peters et al. 2017, Proposition 6.49). For two recursive SCMs M and M if their exogenous distributions satisfy P(U P( Pu, and for each i I, for almost every u4 Su and all Gvv,
(V u:) (V u.),
then M and (2) are counterfactually equivalent.
The original purpose of this theorem was to establish the minimality property of SCMs: namely, a mechanism in an SCM does not depend one of its parent endogenous variables, one can always select an SCM with a sparser structure. However, as elaborated in the main text, the counterfactual equivalence defined in this way mandates fixed latent representation.
(Nasr-Este Eussr-Esahhnn et al., 2023) relaxes counterfactual equivalence to non-fixed latent encodings: rather than requiring the exogenous distributions to coincide exactly, it only demands a component-wiss bijection.
Theorem B.2 (restated from Easr-Esfahany et al. 2023, Proposition 6.2). Suppose f(V (Vpp())) U.) is a BGM, meaning that for each realization V ppa(i) of V pat()) ( (Vpo()): -) is invertible. For two BGMs fi and f2. for almost every u E 22) and all v E Sby.
((2) (ap g(u (() = (V (op u ((").
(2) where g D.1 + D.. is a bijection, then we say that fi and f2 are equivalent. Two BGMs are equivalent if and only if they produce the same counterfactual outcomes.
Although this result is weaker than that of (Peters et al., 2017), it is confined to BSCMs. Moreover, the original paper considers only single causal mechanism within a BSCM, referred to as a BGM. In addition, the theorem addresses counterfactual outcomes rather than counterfactual distributions, a gap filled later by Theorem B.6. In contrast, although conceptually similar, Theorem 3.2 is not limited to BSCMs; it applies to any recursive SCM.
Recent advances in causal representation learning yield analogous results. Specifically, (Brehmer al., 2022) investigates class of models termed Latent Causal Models (LCMs). Because the causal mechanisms in an LCM are assumed to be point-wise diffeomorphic, LCM can be viewed BSCM augmented with latent endogenous variables. Using an isomorphism-base criterion to characterize equivalence between LCMs, the authors show that a component-wise diffeomorphic latent representation aligns the counterfactual distributions of two LCMs (counterfactual distributions are referred to in their paper as weakly supervised distributions).
Theorem B.3 (restated from Brehmer et al. 2022, Theorem 1). For LCMs M and M'. If they can be reparameterized through the graph isomorphism and elementwise diffeomorphisms, we say there is an LCM isomrrphism between them, or LCM equivalence, and we write M LLM M'. Assume that (i) the two models share the same observation space; (ii) the domains of every endogenous variable and its corresponding exogenous noise are R;; (iii) the intervention set contains all
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Exogenous Isomorphism for Counterfactual Identifiability
atomic, perfect interventions; and (iv) the intervention distributions have full support. Under these conditions, M LCM M' if and only if M and M' entail equal weakly supervised distributions.
(Zhou et al., 2024) also investigates domain counterfactual equivalence between Invertible Latent Domain Causal Models (ILDs). An ILD incorporates latent endogenous variables together with a mixing function g. Under fully connected DAG with known causal ordering, the causal mechanism fa in domain d is autoregressive. Because ILD assumes invertibility, the latent portion of its SCM can be regarded as a BSCM, and the domain counterfactual can be interpreted as a special instance of the generalized counterfactual in which the domain itself serves as the intervention. Accordingly, a domain -ppecfi mechanism fa corresponds the mechanism fid] in the submodel M[4]-
Theorem B.4 (restated from Zhou al. 2024, Theorem 1). For two ILDs M and M' whose mixing functions are g and g' and whose causal mechanisms are and f', if for any domains d, d' it holds that
go fa o fa -g--== =g f'v
o
then M and M' are said to be domain counterfactual equivalent, denoted M <DC M' The relation M <DC M' holds if and only if there exist invertible maps h1, h2 such that g' = g o h is invertible and f/a = h1 o fa h2 is autoregressive.
In summary, Theorem B.3 and Theorem B 4 achieve results comparable to Theorem B.2. Their main contribution is to extend counterfactual equivalence the setting of invertible latent causal models, where, in addition to the standard counterfactual reasoning, an unknown mixing function maps latent causal variables the observable space, thereby advancing the development of causal representation learning.
Spectial cases of Corollary 5.4
Several existing results on counterfactual identifiability for SCMs under triangular monotonicity can be regarded as special cases of Corollary 5.4.
(Lu et al., 2020) applied counterfactual reasoning in reinforeement learning, where Theorem states that counterfactual outcomes are identifiable when noise independent of states and actions, and the state transition function is strictly monotonically increasing:
Theorem B.5 (restated from Lu et al. 2020, Theorem 1). Assume S [(S,, Ar, U1+1), where U4t1 IL (St, At). and Sr, Av, U. denote the state, action, and noise time step in reinforcement learning, respectively. Suppose f is smooth and strictly monotonically increasing for fixed St and Aq. Given the observed values S. = st. A = a, and S1+1 St+1, the counterfactual outcome S141,A,=a' S = $t At = S1+1 = Si+1 is identifiable for a counterfactual action A₁ = a'.
Here, the independence of noise from states and actions corresponds to the Markovianity assumption. The function f. being smooth and strictly monotonically increasing, represents a univariate TMM mapping. The causal ordering assumption is implicitly given by time step f, and the observational distribution assumption is implicitly encoded in the observed values S. = St. A₁= = a, and S,+1 St+1. Thus, all four assumptions in Corollary 5.4 are satisfied, making the SCM described in Theorem B.5 ~Vr---iieeene -erriedntiiiable.. By Theorem 3.2, this SCM is also ~ -identifiablle thereby ensuring the identifiability of counterfactual outcomes.
ENasr-Esfahany et al., 2023) further distilled the properties described in Theorem B.5 to uncover results more aligned with general SCM formalism. Specifically, they examined a special causal mechanism known as BGM:
Theorem B.6 (restated from Nast-Es Esfahaaay et al. 2023, Theorem 5.1). Suppose f(Vpots), U.) is BGM, meaning that for each realization Vop(s)) V po( (Vpatt), -) is invertible. Then o is counterfactually identifiable given Pa if: (i) the Markovianity assumption holds, i.e., U; 1L V pa(s); and (ū)) for all ppatiy [VV )) is either strictly monotonically increasing or strictly monotonically decreasing.
pa(s),
Compared to Theorem B.5, Theorem B.6 extends the analysis strictly monotonically decreasing functions, which still conform the definition of TM mappings. The assumption of causal ordering is implicitly provided by the known pa(i). Lastly, while the original theorem is restricted to the univariate case, Corollary 5.4 does not impose such restriction. Thus, Corollary 5.4 accompanied by Theorem 3.2 fully subsumes Theorem B.6.
(Scetbon et al., 2024) adopted an alternative fixed- fixed-point- point-based representation of SCMs, referring to the identifiability of counterfactual outcomes as weakly partial recovery of fixed-point SCMs, and proposed the following theorem:
Theorem B.7 (restated from Scetbon et al. 2024, Theorem 2. 14). Assume that Qy and Qu: are subsets of for each i E I. Pu is absolutely continuous with a continuous density, and each U, is mutually independent. Let [M]N denote the set of
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Exogenous Isomorphism for Counterfactual Identifiability
SCMs satisfying the observational distribution Py, causal ordering S, and causal mechanisms that are Ct and such that p(opot))))) -) is bijective for all i E I. Then [M]N + 0. and all models in this set are consistent (f(x) f f for any intervention E Sxx, where X V.
Here, (fxx o f P)PP represents the pushforward form of counterfactual outcomes defined via the abduction-action prediction framework, indicating that the theorem essentially discusses counterfactual identifiability. The mutual indepen- dence of U, corresponds to the Markovianity assumption. The causal ordering represented by a permutation matrix P in the original paper, and the observational distribution is described as a pushforward of the exogenous distribution.
Compared to Theorem B.6, the key difference lies in the lack of an explicit assumption that (Vpaa()) -) is strictly monotonic. Instead, this property is implicitly ensured through C' smoothness. Specifically, leveraging C" smoothness and (Scetbon et al., 2024, Lemma G.8), they indirectly proved that the partial derivative of f(ypa(s).) :) with respect to u; is always strictly positive or strictly negative, which implies consistency of the strict monotonicity of f(Vpa(o).))
There are other generalizations of the monotonicity assumption for counterfactual identifiability. (Wu et al., 2025) employs the rank preservation assumption to identify the counterfactual outcome, which relaxes the strict monotonicity assumption. As discussed in the main text, strict monotonicity essentially corresponds to lexicographical order; rank preservation can therefore be viewed as a redefinition of this lexicographical order, thereby generalizing the monotonicity assumption.
B.4. Neural SCMs
As for the related work neural TM-SCM, we focus here methods designed for counterfactual inference tasks that leverage proxy models constructed by mimicking the structure of SCMs and parameterizing them with neural networks, which we refer to as neural SCMs.
Neural SCMs without proven identifiability A series of works connect SCMs with various deep generative models, utilizing the inverse processes of these models to address the tractability issues in counterfactual inference. DSCM (Pawlowski et al., 2020) employs normalizing flows and variational inference to propose a general standardized framework for constructing SCMs using neural networks, enabling tractable inference of exogenous variables for counterfactual reasoning. Diff-SCM (Sanchez. & Tsaftaris, 2022) introduces diffusion models tailored for counterfactual inference on high-dimensional data, where latent variables are inferred through the diffusion process, and gradients are intervened upon during the reverse diffusion process, ultimately applied to counterfactual image generation. VACA (Sánchez-Martin et al., 2022) parameterizes SCMs using a variational graph autoencoder ( (VGAE), distinguishing itself by directly modeling the global causal mechanism rather than individual conditional mechanisms and encoding variable dependencies within the SCM using graph neural networks. While these models introduce the concept of mimicking SCMs to construct proxy models and enable counterfactual reasoning, they do not explicitly demonstrate the reliability of their inference results, i.e., whether the counterfactual outcomes are identifiable.
Neural SCMs proven to be identifiable for counterfactual effects A series of studies address the identifiability of counterfactual effects in proxy SCMs by leveraging deep generative models. CVAE- (Karimi et al., 2020) employs conditional variational autoencoders (CVAE) to model conditional mechanisms and proves that under the Markov assumption, certain types of counterfactual queries are identifiable solely from the observational distribution. Consequently, the constructed model ensures consistency for these counterfactual queries. MLEENCM (Xia et al., 2021) systematically explores the identifiability of connecting neural networks to SCMs and demonstrates that, in discrete cases and with sufficiently expressive neural networks, proxy models constructed from causal graphs and observational distributions exhibit duality in L2 causal queries. That is, a causal query is identifiable on the proxy model if and only if it is identifiable on the causal graph. iVGAE (Zečević et al., 2021) extends MLE-NCM by showing that such duality also holds when using VGAE. GAN-NCM (Xia et al., 2023) further generalizes MLE-NCM to L3 causal queries, proving that proxy models constructed from causal graphs and interventional distributions exhibit duality in the identifiability of counterfactual queries. The model is trained using generative adversarial networks (GANs).
Neural SCMs with empirically identifiable counterfactual outcomes A series of studies primarily focus on proving model identifiability and empirically demonstrate the identifiability of counterfactual outcomes through experiments. CAREFL (Khemakhem et al., 2021) connects causal inference with autoregressive normalizing flows (ANF), utilizing affine transformations primarily for cause-effect identification tasks in causal discovery and proposing methods for counterfactual reasoning. CausalNF (Javaloy et al., 2023) extends this approach to other types of ANF transformations and proves model
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Exogenous Isomorphism for Counterfactual Identifiability
identifiability in latent encodings. CCNF (Zhou al., 2025) further introduces partial causal transformations enhance inference performance. CFM (Khoa Le et al., 2025) generalizes this line of work from discrete normalizing flows to continuous normalizing flows, employing flow matching for modeling. While these studies demonstrate the identifiability of counterfactual outcomes on synthetic datasets, the connection between latent encoding identifiability (a problem in representation learning) and counterfactual identifiability (a problem in causal quantities) remains unclear. Proving under which conditions the former implies the latter is one of the key contributions of this work.
Neural SCMs proven to identifiable for counterfactual outcomes series of studies have established the identifiabil- ity of counterfactual outcomes under specified assumptions. BGM (Nast-Estahany et al., 2023) investigates the identifiability of causal mechanisms for counterfactual outcomes under the bijection assumption and proposes three identification methods, ultimately using conditional normalizing flows to construct proxy SCM to empirically support their findings. FiP (Scetbon et al., 2024) formalizes SCMs and counterfactual distributions using fixed points, framing the identifiability problem for counterfactual outcomes as the weak partial recovery problem. They identify a condition for recovering counterfactual distributions, though their experiments return to additive noise models based on attention mechanisms while employing optimal transport to model noise distributions. DCM (Chao et al., 2024) assumes that the exogenous distribution of the true latent SCM is uniform, focusing on encoder-decoder models, and proposes two identification methods as extensions of (Nasr-Esfahany al., 2023, Proposition 6.2). These methods rely on diffusion models to construct proxy SCMs. As shown in Appendix B.3, the first two studies are special cases of TM-SCM, while the last study leverages et al., 2023, Proposition 6.2), which corresponds to a special instance of Theorem 3.2 in this work that considers only one single causal mechanism. Although these studies demonstrate the identifiability of counterfactual outcomes, these results represent only a specific subset of counterfactual statements. In contrast, a key contribution of this work is proving that the theoretical foundations extend beyond the identifiability of counterfactual outcomes to encompass the more general and stronger ~c -identifiability.
C. Neural TM-SCM
In this section, we present the implementation details of various neural TM-SCMs used in the experiments.
C.1. Vectorization
In this subsection, we will specifically address the implementation issues related to vectorization and de-vectorization, particularly how to switch between the symbolic form of endogenous or exogenous values (primarily used for reasoning on M) and the vectorized form (primarily used for reasoning on the solution mapping 1) when a TM-SCM is given.
Vectorization Suppose we aim to vectorize an endogenous or exogenous value x, resulting in its vectorized form (x(x). Since these values originate from TM-SCM, their indices are fully aligned, and thus we do not distinguish between them here. Vectorization requires a predefined re-indexing map L. According to the definition provided in the main text, this only additionally requires a causal order S in the TM-SCM. The causal order can be assumed to be directly provided, as in the neural TM-SCM problem setting described in the main text. For the ground truth TM-SCM, any causal order can be obtained through topological sorting. Subsequently, we can construct such a map based on Lemma A,,9, defined as
(i,j) = - Σ dg.
kEPr(S)
From implementation perspective, the collection of values (X;)iεT, representing the symbolic form xf be expressed as dictionary where keys are I and values are vectors x,. Vectorization in this context involves concatenating the vectors x; corresponding to different causal variables in the causal order S. resulting in
k= nt
P2(x) =
X
|r(()/=k
where pr(i) denotes the causal prefix of i E I under S, and @ represents vector concatenation.
De-vectorization Suppose we aim to de-vectorize an endogenous or exogenous value P,(x) to obtain the symbolic form of x. This operation requires knowledge of the vectorization mapping 4, which is provided during the vectorization process.
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Exogenous Isomorphism for Counterfactual Identifiability
According to Lemma A.19 and (P.)-1 each causal variable is given by
x; (P. jeadt
where 4 Ekep(()) dg. From an implementation perspective, this is equivalent selecting the slice corresponding to 4:4 + de from the vector P. (x).
C.2. Exogenous Distribution
In the main text, we design the use of normalizing flows to model the exogenous distribution. However, in practical experiments, to demonstrate the insensitivity our theory to the type of exogenous distribution, we additionally consider other types of exogenous distributions. Recall that, according to the Markovianity assumption, it is required that Pt. = Ilizz PU... so the following discussion focuses on each Pú...-
Standard normal distribution A simple modeling approach is to standard normal distribution, such that
log PU., (u,) log (u, 00 I);
where N denotes the density of Gaussian distribution. However, this modeling approach may provide sufficient expressive power. For example, in the causal mechanisms of DNME, the diagonal structure results in non-interacting dimensions, and using standard normal distribution in such cases would enforce independence between these dimensions. This limitation is reflected in the experiments presented in the Appendix D.
Normalizing flow
As introduced the main text, the log-likelihood for each PU.o is given by
log . (2,) = log Pz,, (T, (u:) + log det J T., (..
where Ti 2, Ue is masked autoregressive flow (MAF), and det. J TTa (us) denotes the Jacobian determinant at u.
Gaussian mixture model
The log-likelihood for each Pu.,o is given by
log Pu:,o (u:) log Σ Wh NN IuR,o Ihiod
where A the density of Gaussian distribution with mean PR,i,0 and covariance matrix Ex,,, both parameterized by neural networks. Additionally, Σ Wk.i. = for any I.
In the implementation, these parameterized distribution models are provided by the normalizing flow library Zuko (Rozet al., 2024).
C.3. DNME
We begin with the simplest DNME prototype to demonstrate how construct the model while enabling the derivation of inverses and log absolute Jacobian determinants. Specifically, leverage the normalizing flow library Zuko, which provides unified framework and standard for building normalizing flow models. Thus, the constructed neural TM-SCM is effectively transformed into normalizing flow.
Recall that the symbolic form of DNME is given by
fr, (Vpa()) u:) = by (Vp()))))) + a. C u,
To transform DNME-type neural TM-SCM into a normalizing flow, we interpret the causal mechanism fú,0 as a discrete transformation within a normalizing flow. This can be represented using a coupling transformation e combined with an affine transformation a:
feo =
= c(vz(()), uy aa: (v))))()))))))) c(x,y:f) = (x, f()), a(x;a, b exp(a) ox,
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Exogenous Isomorphism for Counterfactual Identifiability
where the affine transformation satisfies the strict positivity requirement of a (Yp(()))))) and the coupling transformation ensures that Si,0 depends only ypa(s) and u, thereby constraining the causal structure. The function mg is multi-layer perceptron (MLP) that provides the parameters required for the affine transformation.
Correspondingly, the inverse of the causal mechanism, (V1,((z((())) (vi), can be expressed as:
foo(V240 v.) = (V2rtt))) ve:
c (x,,:)) = (x. (y)). a a' = (x- 2 exp(a),
where 2 denotes component-wise division. The log absolute Jacobian determinants for the coupling and affine transforma- tions are, respectively:
log det Je(x, y: f)] log det J (y)l,
log |dct Ja (x; b)l Σ aj
jEl:d.
where is the j-th component of a.
The solution mapping Ig then composed according to the causal order S as Se,e .... Sr.e, where pr(s) and pr (r) = J This forms a discrete normalizing flow, and since each transformation supports forward, inverse derivations, and log absolute Jacobian determinant calculations, Te inherits these properties. Consequently, the log-likelihood of endogenous value (t)) is computed as:
pv. pv. (v())) loggu, (v())) log det (r.-)((00)))
supporting maximum likelihood optimization process. Here, loguu, is the log-likelihood function of the exogenous distribution, discussed in Appendix C.2.
C.4. TNME
The TNME improves upon the DNME, with its formal representation defined
fv u,) = bb.o(Vpm(s)) + (A,
u..
Specifically, the component-wise affine transformation a is replaced by more sophisticated lower triangular affine transformation A, such that
S u:) c(vzy1), u; A(;moo(yui)))
A(x;A,b) b) = texp( + xT,
where the lower triangular affine transformation takes matrix A and a vector b parameters. The exp function constrains the matrix to be positive, ensuring monotonicity, while the tril function and a small positive scalar enforce the matrix to be a full-rank lower triangular matrix, guaranteeing invertibility. The MLP ma provides the parameters required for the lower triangular affine transformation.
Similar to DNME, the inverse of the lower triangular affine transformation is given by
A A (x; b) (exil( A + +x)))
where (tril(exp(A A + -1 is the inverse of tril(exp(AA A + c)) During implementation, instead of explicitly computing the matrix inverse, the inverse operation is treated as solving the linear system
(e c)) A (x a. b) =(x- (x-b)T,
where -- (x;; b) treated as the unknown.
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Exogenous Isomorphism for Counterfactual Identifiability
Due the properties of lower triangular matrices, the log-determinant of the Jacobian of the lower triangular affine transformation is straightforward to compute:
log det Ja (x; a. b)l=
Σ
Eass +c),
JEl:d,
where as is the (j. j-th element of the matrix A. Subsequently, the construction of the mapping T3 follows the same principles as DNME.
C.5. CMSM
The CMSM models the re-indexed solution mapping P, Te as sequence of discrete transformations,
P. PooTo = Tı,0 0.......0 Tn,0,
where n is required to be no less than the diameter of the causal graph, as stated in (Javaloy et al., 2023). Each T5,0 is modeled as an autoregressive transformation L, with an affine transformation a as the univariate transformation, such that
xj = f(x)-1;0),
where X (u( represents the vectorized exogenous values, and x,, Pv(v) represents the vectorized endogenous values. The autoregressive transformation t ensures that each component Zi,. of xj is expressed as
Xj.i =(f=--15j1,,,0033-11--111
where the MLP my,o provides the parameters required for the affine transformation. The autoregressive transformation t triangular transformation, and its inverse is computed in the same manner as triangular transformations, detailed in Lemma A.15. Furthermore, its log absolute Jacobian determinant is given by
log |det Jx(x.)) =
Σ
log J
iE &E d
as the Jacobian matrix of triangular transformation is lower triangular. Consequently, the construction of the re-indexed solution mapping P. o Ia follows the same principles as DNME.
C.6. TVSM
For the ODE
dx((t) = v(x(t), t) dt,
x00) = Xo.
denote the flow T(xp, t) at),, where Rd [0, 1] + Rd is time-dependent velocity field, and [0, 1] - Rd is the solution the ODE. A flow T(,,t) pushes forward the distribution at t that at time t, such that Pt,x (T(: ());Poos According to Lemma A.21, if the velocity field v is Lipschitz continuous triangular mapping for any 4 then the flow (-, t) is always TMI mapping.
In implementation, we adopt a method similar to (Chen et al., 2018) for modeling velocity fields, introducing a mask to ensure that is a triangular mapping:
v(x,)) f) = tanh(( @ M)x + (U @ sigmoid( G) @ M)T,
where
(W, U, = m²
W, U, G are matrices, and b is a vector, all parameterized by an MLP with as input. M is lower triangular mask.
Unlike previous approaches that model the re-indexed solution mapping using composite discrete transforms, we model P. To = odeint(v, 0, 1), and its inverse ro (P.) odeint( 1, 0), where odeint represents an IVP solver from the
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Exogenous Isomorphism for Counterfactual Identifiability
Algorithm 1 Pseudo Potential Response for TM-SCM
Input: a TM-SCM M, exogenous value u, intervened value x, vectorization i under a causal order S of M. Output: potential response VMM (u) u' u1 v T(u), D Eiez Ziez di
(Initialize potential response)
for k = 1 D do
(i.j) 6- (k), v
if Ix then
(v')., + (x),.j
(Do-intervention)
end if
(11))=11111] (T- (v))()))
{Find exogenous value for fully explaining the prefix part) [Assume the suffix part is not intervened, and update the potential response)
(v [1= f (((u))).'(==)
end for return v
torchdiffeq library (Chen, 2018). For log-likelihood computation, we employ the unbiased bog-likelihood estimation method in (Grathwohl et al., 2019):
logpv. (v lu, (T o (P.))+)(v((0) Ee [ (0000//00 eds
where r7 (P.) corresponds to odeint(v, 1, s),, and P. the standard normal distribution.
C.7. Inference
In the experiments, the trained neural TM-SCM infers counterfactual outcomes through process referred to as pseudo potential response (Algorithm 1). Direct intervention on causal mechanisms is challenging for TM-SCMs constructed from solution mappings (e.g., CMSM and TVSM), as they lack explicit causal mechanisms and thus cannot directly fix the values of causal parent variables.
The term pseudo potential response arises from the characteristic of Algorithm 1 performing indirect interventions. TM- SCM, as a BSCM, possesses fully supported exogenous variable space. Combined with TM-SCM's well-defined iteration order, the algorithm aims to inversely identify an exogenous value that perfectly explains the intervention and the ancestors already determined, rather than directly intervening on endogenous variables. This extends the intervention algorithm for single intervened variables in (Javaloy al., 2023) to support arbitrary intervened variables and random vectors.
To ensure correctness, prove that the output of the algorithm is consistent with the potential response V M. (u):
Theorem C.1 (Correctness of Pseudo Potential Response). The result returned by the pseudo potential response algorithm, v", equuls V Mix (u).
= Eier di, suppose E 4::4 + d;. By the definition of the potential response, we have Proof. For any / E 1: where D=
(V (u)),-(0) (T(xx(u)))-10
(Fipx (Man-
( (M) Mz) me))e i $ Ix, (xx)
Definition A.I.
E Ix
(fipe (unr - Consider the t-th iteration. Let v' v initially. If Ix then (v')--(0) xx-1(0)- By the property of T TM mapping, the component index d remains unchanged:
Suppose
(v),- (v)) ( = =x-1(0) = (Fixx) (Mm ()))-1)))
i Ix. then by the (t- 1)- iteration:
(v')--10) (T(u")).-2())
where for each is E 1:t
( ())) 1(x) = (f)b) (Ma (1) =(K))
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Exogenous Isomorphism for Counterfactual Identifiability
assuming k el : dz. Thus,
(v)) = (f(((fi (uin REp)) u:)),.. = (fi((fapo) (Man (4)) ke))))pp(() us)),,,, = (Fixel Uua (mm)())))--100)) Hence, after the 1-th iteration, (v=))-1(0) = (filx)
(1ix)(uan (Mon (1))) holds universally. Since Vi[1a-1) remains unchanged,
(v"), -((10))) = (Fits) (uan (6))) [1:0]-
This holds for any & E I.D. Thus, at D,
v' filx](unn (unn
where (1) = I, implying v = (u)
D. Experiments
D.1. Datasets
TM-SCM-SYM The dataset collection includes four small synthetic datasets: BARBELL, STAIR, FORK, and BACKDOOR. Each causal variable is represented as a random vector with dimensionalitty ranging from 1 to 8, with up to four causal variables in total. The exogenous distributions are either mutually independent standard normal distributions or multivariate normal distributions with a Markov structure. The causal mechanisms between variables are defined as manually designed symbolic TM mappings. The specific generation mechanisms for these four synthetic datasets are detailed as follows.
BARBELL: Includes causal variables, x and y, each with 8 dimensions. Their causal relationship follows the simplest binary cause-effect structure, x y, with the causal mechanisms represented as:
s(1 (1188tl xx
i 1,
EE=
1-1
Ijz Uzzii
1 1<i 5, 5<is7,
2. === = (fa(u)))
0.3 Uz +
Σ 2===1 Itj + -1
CDF
1.3 (2 x3)
+ +E x3) 13+11 1) + 2, 0.6, uz,s)
i-
(s((8. - x;:)
i= 1,
1 ( ((() [1 Vj: Un,
1 1<is5,
9: = (Jy(x, Un)):
CDF-1 1 (-s 1.3 ΣØ=1
+1 +1 i1 [E Uj :) / 3 + 2,06.6 uyy)) E リソ +1) -
5<is7,
0.3 Myyi 0.5
Ej= wb
+
i=8.
The function represents the softplus function defined as s(x) log(1 (1 + exp(x)). The binary function 7 is defined as 1(x, v) = s(x + 1) + .0.55 -3. The function CDF b, x) denotes the quantile function of the Laplace distribution at x, with location and scale b. The exogenous distribution follows a standard normal distribution.
STAIR Consider 3 causal variables x, y, and z with dimensions 1, 2, and 3, respectively. Their causal structure follows a chain: x + y + z. All causal mechanisms are linear and are represented as:
2 = (f )), = 0.75 Ugrs.
10 + Uyy
= 1, +2 = 2.
D1 (f,( uy)),
0.25 0.5 uy.i
-0.5 + Uz,i 4
i= 1, i = 2,
2 = (fz(x,
5 - 1.5
V/-1 + 2 Ug 0.5 i 3.
and the exogenous distribution follows a standard normal distribution.
FORK Consider 4 causal variables x, y, z, and w, each of dimension 2. The causal structure among them includes a collider structure x -> f- y as well as a direct causal relationship z - w. All causal mechanisms are additive and can
36

---

Exogenous Isomorphism for Counterfactual Identifiability
be expressed as:
Uzzi
i 1,
= (ux))
U2 0.22 Mge
i
Uy,i
= 1,
= (x, uy));
-0.5 My -0,50yy.i-
+ Uyy
2.
yi 0.5 Ug
i=
# (Ix(x,y,
+ exp( -2 - 1/2)
4 + 0.5( U2,1) i= 2.
1 + cxp(p11
1/1 x22
20
#wwi
i= 1,
w: (f))
cxp(0.5 =
22)
20
+ 0.25 Un
Uwy
i= 2.
+ exp(0.5 2 - 21)
and the exogenous distribution also follows standard normal distribution. BACKDOOR The causal graph involves 4 causal variables, y. 2, and each of dimension 4. The causal structure includes a direct causal link + w, an indirect causal path - z + w, and a backdoor path y + x + w. The causal mechanisms are represented as:
i 1, i 2, sigmoid(- (
tanh(ug.;)
T; =
Iz-1 ia))(
i 3, 1 + sigmoid(ux,. (uz,s)
i = 4.
tanh( + Uyys)
= 1, 2, = 3,
sigmoid Uy.i)
D1 = (x, uy))s
Vi-1 tnnh(
X14-1 (softmax(y sigmoid (uy,))
=
elu(y) - uz2s)
i = 1, i 2, i i= 3,
Seaky-relu(yr + Uz,s))
2 (f. uz)),
2 + clu(uz,))
V1s-1 (softmin(21:=11)) + leaky relu(ug,)) i 4.
(a,, Vt + 24) (softmax(1; ① y4 2))T + elu(2: - uw
1, (wi-1 ① Mw,(-1) (softmin(w; (1:1 @ (w.i- T + leaky_relu(1e + tmm
i = 2, i = 3, i 4.
w: (X, uw));
(xis ① Yiw @ Z (softmin (11 ① Yı. + Z ) T + x;-1 + (w
(Wi:i-1 1n,1: (softmax(wis. (Wi:-1 4 1 ) leaky_relu (uw,))
where tanh, sigmoid, clu, and maaky-relu widely used activation functions, ensuring monotonicity and continuity, The exogenous distribution follows multivariate normal distribution, with location 0 and covariance matrix:
0.8
0.3
0.2
0.8
1
0.4
0.1
0.3
0.4
1
0.6
0.2
0.1
0.6
ER--DIAG-50 and ER-TRIL-50 Each dataset collection consists of 50 randomly synthesized datasets, where each dataset contains 3-8 causal variables, and the dimensionality of each causal variable is randomly chosen between and 8. The causal structure among the variables modeled as an Erdös-Rényi graph, the exogenous distribution is randomly generated as a multivariate normal distribution, and the causal mechanisms are constructed using randomly generated TM mappings. The details for generating these components are described as follows.
37

---

Exogenous Isomorphism for Counterfactual Identifiability
Variables The number of causal variables, n,, is sampled from discrete uniform distribution Unif(3, 8), and these
variables are denoted X1,-.-,- Xn.
Dimensions The dimension d, of a causal variable is sampled from discrete uniform distribution Unif(1,8) with the
j-th component denoted as Tt.z-
Graphs An Frdōs-Rényi graph is generated over the n causal variables with a probability of 0.5 for each edge, ensuring the graph is directed and acyclic. Specifically, we construct an n x n adjacency matrix A, where each element ai,j is sampled from continuous uniform distribution On, 1). If az, > 0.5 and < j. the edge x; + Xj is included. Exogenous distributions For each causal variable, a multivariate normal distribution of dimension m is randomly generated to represent its corresponding exogenous distribution. By Markovianity, exogenous samples are formed by independently sampling from these distributions. Specifically, the parameters of the multivariate normal distribution are generated by sampling a mean vector μ and matrix C from the standard normal distribution. The covariance matrix is computed as Σ = CTC + I, ensuring positive definiteness, where I is the identity matrix.
Causal mechanisms The randomly generated causal mechanisms are ensured to TM mappings. For ER-DIAG-50, the causal mechanism represented as
(f((X)u())) u,)), = (9. Us.z);
for j E 1:d, For ER-TRIL-50, it is expressed as
(fi(X u:)); = ha.(( u:))s u4.5),
where for j = 1, (fi(Xpa))) u:));-1 (xp()())
The function 9i a randomly generated continuous function I Encnco de - To enhance its complexity, is expressed as combination of sinusoidal and segmented terms:
g(x) sin((ns, + Φ.) + +x)(()) ||x (x)l
segmented term
sinusoidal term
where jx(x) = arg minisisn ||x - Mاll, and c₁ Wi, Oi, Sj p2 are random parameters. The number of sinusoidal and segmented terms is = m 3.
The function h4. is randomly generated monotonic function R² + R, ensuring monotonicity with respect to the second dimension. It is represented in linear form:
h (2,( 81 21 82 322 b,
where the bias term b and scaling coefficients $1, $2 are random parameters. The signs of $1, $2 are not constrained be positive, indicating that the resulting mapping is not necessarily TMI mapping but rather more general TM mapping.
D.2. Metries
We employed three metrics evaluate the performance of the trained models.
OBSwD This metric measures the Wasserstein distance between the observational distribution derived from the model and the ground truth, evaluating how well the trained model fits the observational distribution. The Wasserstein distance computed using the geomloss library (Feydy et al., 2019), which calculates the un-biased Sinkhorn divergence with blur parameter of 0.05 and cost function of C(x, y) = 1 11 = v112.
This metric measures the error between the counterfactual results inferred by the model and the ground truth, assessing the accuracy of the trained model in counterfactual reasoning and reflecting the model's consistency in answering counterfactual queries. We use the root mean square error (RMSE), which retains the same unit as the original values, making it more interpretable for evaluating the accuracy of counterfactual reasoning.
CTFRMSE
CTFwD This is a metric unreported in the main text but included as a supplementary evaluation the appendix. measures the Wasserstein distance between the interventional distribution and the ground truth in an average sense. Since
38

---

Exogenous Isomorphism for Counterfactual Identifiability
the observed values in the ground truth of counterfactual datasets are sampled from the observational distribution, these samples, when considering only the counterfactual outcomes, correspond to the distribution expressed as
U Pvwivvl.vPPvdd Ex [Pvm]
ExxPx Px
where Px denotes the prior distribution of intervention values provided by the counterfactual dataset, and Pv represents the interventional distribution under intervention x.
D.3. Execution
This subsection details the preprocessing, training, and testing procedures of the experiments to ensure reproducibility.
Preprocessing During the initial execution, following the settings described in Appendix D.I, each synthetic dataset is divided into three splits: training, validation, and test datasets. The training dataset comprises observational data with a sample size of 20,000, directly sampled from the exogenous distribution, with exogenous samples propagated through the synthesized TM-SCM to derive endogenous observational values. The validation and test datasets, each containing 2,000 samples, consist of counterfactual data, providing observations, interventions, and counterfactual outcomes. These counterfactual datasets are synthesized in three steps: (i) Similar the observational dataset, exogenous values and their coressonning endogenous observations sampled. (ii) Intervened values sampled anew, where the intervened variables X satisfy certain conditions termed interventionally meaningful, and the intervened values adhere to the observational distribution. (iii) The counterfactual outcomes are derived using the exogenous values from (i) and the intervened values from (ii) to compute the potential responses.
The notion of interventionally meaningful refers to the following three conditions of an intervention set X: (i) For any X X, X must have least one child. (ii) If X X and X is parent of Y, then X and Y must have at least one common child. (iii) X is non-empty. In implementation. indices Ix sampled to satisfy these conditions, and mask is used to indicate whether variable has been intervened upon.
Finally, all dimensions in the datasets are standardized. To ensure consistency, the mean and variance from the observational dataset are used for standardization, including for the intervened values and counterfactual outcomes in the counterfactual datasets. Since standardization merely involves component-we wise scaling and shifting, the triangular monotonicity property of TM-SCM is preserved.
Training All neural network components in the models, as reported in Appendix C, are configured as MLPs with 2 hidden layers and width of 128. The outputs of these MLPs include an additional dimension representing the encoding length, and the parameters subsequently derived are averaged over this encoding. This design enhances the expressiveness of the MLPs while maintaining fairness in model parameters. Specifically, before each run, we perform binary search find an appropriate encoding length such that the total number of model parameters does not exceed IM. All models are constructed according to this standard, ensuring that the parameter counts are comparable across models.
For experiments on TM-SCM-SYM we train for 100 epochs with batch size of 64. Validation is performed every A training steps, where grows exponentially such that the interval after the t-th validation k = [r']. We set = 1.25. The validation results are used to plot the scatter relationship between OBSwD and CTFRMSE- All experiments under this configuration are conducted with 10 different random seeds.
For experiments on ER-DIAG-50 and ER-TRIL-50,, we train for 50 epochs with a batch size of 64, performing validation after every epoch. All experiments are once on their respective 50 pre- pre-generated random datasets.
For all the experiments mentioned above, the optimizer used is Adam with a learning rate of 0.001 and weight decay of 0.01 as a regularization term. The model weights corresponding to the epoch with the lowest CTFRMSE on the validation set are saved for testing.
Testing The test set is used to evaluate the model performance reported in the tables. Specifically, for OBSwD, we utilize the part of the test set labeled as observed values. According to the description the preprocessing section, these samples satisfy the ground truth observational distribution. For CTFRMSE and CTFwD. we use the parts of the test set labeled as observed values, intervened values, and counterfactual outcomes. The observed and intervened values are input into
39

---

Exogenous Isomorphism for Counterfactual Identifiability
Algorithm estimate the counterfactual outcomes, and the ground truth counterfactual outcomes used compute these counterfactual metries. During validation and testing, in where certain model configurations exhibit instability specific positions, leading to numerical overflow, the corresponding test cases are masked and ignored.
D.4. Full Results
Ablation TM-SCM-SYM We report below the complete ablation study results on the TM-SCM-SYM dataset as discussed in the main text. First, present the full version of the scatter plots shown in the main text, illustrated in Figure 4, which are obtained from the validation results of experiments conducted under 10 different random seeds.
TM-SCM
M
w/o
BARBELL
STAIR
FORK
BACKDOOR
DNME
TNME
CMSM
TVSM
C1PRNNS
OBSWD
Figure 4. Ablation results neural TM-SCMs on TM-SCM-SYM. Colored curves depict sliding- window regressions, with shaded areas showing 95% CI. To improve the readability of the plot scales, outliers helow the 0.01 quantile and above the 0.99 quantile were removed. Rows represent models (DNME, TNME, CMSM, TVSM), and columns represent datasets (BARBELL, STAIR, FORK, BACKDOOR).
Across all combinations of models and datasets, the non-ablated neural TM-SCM (blue) consistently achieves lower CTFRMSE. demonstrating superior performance counterfactual consistency. Specifically, as training progresses, the horizontal axis OBSwD gradually decreases, indicating an improved fit of the model to the observed distribution. can be observed that, except for certain settings where the model lacks sufficient expressiveness, OBSwp is reduced to a similar
40

---

Exogenous Isomorphism for Counterfactual Identifiability
Table Ablation results of neural TM-SCM on BARBELL and STAIR. The values shown the means of 10 experiments with different random seeds, with the subscript representing the 95% CL. The best pest-rerforming results are highlighted in bold.
BARBELL
STAIR
METHOD
OBSwD
CTFrmse
CTFwn
OBSwD
CTFrmse
CTFwD
#0.03
0.18
0.20
0.18 20.0
0.10
0.03
DNME
W/O O
2.44
0,94 0.94
2.8900.00
25.54
0.80
0.61
W/o M
1.
0.5020.02
1.53
0.17=0.01
0.47:1.03
0.48
0.39 +0.08
=.00
0.04
0.16_0.01
0. 0.16+0.01
0.07
TNME
W/O O
18,2611100
0.94:10.0
3.8910.00
32.65 =52.99
0.80 00
0.61
w/o M
7.50
0.52
1.55
0.18
0.39
0.
0.41
0.05
0.02+0.01
0.17=0.01
0.24
0.30
w/o O
0.39 0.04
0.04
3.89 1.000
15.05
0.83
0.60
CMSM
w/o M
1.01 1:046
0.07 0.001
0.04 0.001
0.17
0.13
0.05
w/o T
12212
0.98
25.38 $55.99
0.19
14.67 4:3.222. 22.28
0.51 40.70
0.37 #0.02
0.10
4..34
0.17
0.1640.01
0.08
W/O O
0.94
3.89 +0.0u
=00.7a
0.80
0.61
TVSM
W/o M
1.15
0.29
0,48
0.17=0.01
0.28
0.2120.02
w/o T
1.33
0.51
1.59
20.56
0.43
0.44.0.05
Table 4. Ablation results of neural TM-SCM on FORK and BACKDOOR. The values shown are the means of 10 experiments with different random seeds, with the subscript representing the 95% CI. The best-performing results are highlighted in bold.
FORK
BACKDOOR
METHOD
OBSwD
CTFRMSE
CTFwD
OBSwD
CTFRMSE
CTFwD
0.28
0.14
0.08
2.83
0.27
0.59
DNME
W/O O
2.15 2.11
4..00
0.57 20.00
4.79
0.46
1.45
w/o M
1.61 2:.88
0.62
0.47 10.01
4.36.10.12
0..4
0.88
0.27=0.01
0.14
0.07=0.o0
2.47
0.16=0.oo
0.20
TNME
w/O
=.88
0-0000000
+.000
9.20.+0.24
0.46
1.45
w/O M
15.31 4:32.44
0.58 +0.02
0.48
3.20
0.28 =0.01
0.63
0.30
0.08 20.01
0.02=0..0
2.03
20.00
0.19
w/O
1.56
00.0
0.71
6.35
20.00
1.50
CMSM
W/O M
0.00
0.08 40.00
0,02+4.00
2.68
0. 0.17
0.23
w/O T
0.37 0.
0.63
1.02
7.91 #11.0
0.37
1.04
0.29
0.16
0.08
2.29
0.12=0.01
0.11
W/O O
2. 2.10 10
4..0
0.57 20.00
5.79
0.46
1.45 _0.500.00
TVSM
W/O
0.32
0.48
0.3 0.31 10.01
2.53
0.21 20.01
0.36
W/O
0.46
0.61
0.43
2.6410.44
0.34
0.87
level across most configurations. However, the final performance on the vertical axis CTFRMSE varies across different ablation modes, highlighting the negative impact of the ablated components on counterfactual consistency.
In these ablation modes, w/o O (yellow) represents the case where the causal order reversed. Although OBSwD decreases in almost all settings, CTFRMSE remains nearly unchanged. This highlights the significant role causal order assumption Ag in ensuring counterfactual consistency. w/o M (green) represents the case where the exogenous distribution is non- Markovian. In most settings, OBSwD decreases, CTFRMSE instead increases. This indicates that counterfactual consistency cannot converge to a lower level expected under Mar-Markov conditions. underscoring the necessity of the Markovian assumption AM for counterfactual consistency. w/o T (red) represents the case where the solution mapping of the constructed SCM is not triangular (allowed only for CMSM and TVSM). In these settings, as OBSwD decreases, CTFRMSE diverges. This demonstrates the critical impact of the assumption that SCMs are TM-SCMs, ATM.SCM. on counterfactual consistency. Only in the no ablation mode (blue), where three assumptions in Corollary 5.4 AAMsSMM- AM. and Ag-are are satisfied, does counterfactual consistency improve (as evidenced by decreasing CTFRMSE) as Ary is gradually satisfied (indicated by

---

Exogenous Isomorphism for Counterfactual Identifiability
Table Ablation results of neural TM-SCM on TM-SCM-SVM for exogenous distributions. The values shown are the means of 10
experiments with different random seeds, with the subscript representing the 95% CL.
BARBELL
STAIR
FORK
BACKDOOR
METHOD
DIST
OBSwo
CTFRMSE
OBSwd
CTFRMSE
OBSwD
CTFRMSE
OBSwD
CTFRMSE
N
6.20+
0.19 0.19 +0.00
0.17 +0.00
0.11 20.00
1.14
0.29 t0.00
4.57+a.02
0.20
DNME
GMM
1.94
0, 00191
0. 0,1620.01
0. 0.12
0.37
0.17
3.41 23
0.22
NF
0.42
0. 180.00
0.18
0. 0.10
0.28+0.01
0. 0.14
2.8340.03
0.27
N
4.884
go.00
0. 0.17 +0.00
0. 0.10=g.00
1.12
0.26+0.00
3.12
0.15
TNME
GMM
3.654
0.0910.01
0.17
0.14
0.34.10.03
0.1610.01
2.82:10.22 22
0,15 .00
NF
0.39
0.07
0. 16.0000
0. 0.16220.0
0.27:10.01
0.14
2.47:1.04
0. 0.16...00 00
N
2.9642.68
0.06g0.01
0.17
0.12=0.022
0.32+4.002
0.07+0.01
2.8640.441
0.13
CMSM
GMM
1.991
0.00
0.19
0.1 0.11:00.00 0.002
0.30
0.08 1:0.0
15.93
0.1 00
NF
0.41 40.00
0.05 00.01
0.17 0..01
0.24 20.2
0.30
0.08
2.53
0.15 20.00
N
0.34
0,0940.00
0.14 0.16
0.31
0.17
2.29++0.01
0.10. 0.00
TVSM
GMM
0.
0.09 #0.00
0. 00.00
0.15 =0.01
0.29 +0.00
0.15 40.00
2.27+40.01
0.10=a.00
NF
0.37+0.02
0.10
+0.02
0.16
0.29#0.01
0.1640.00
2.29+0.u2
0. 0
decreasing OBSwD).
Furthermore, we can analyze the neural TM-SCM methods under different constructions (without ablations) row by row, It can be observed that DNME and TNME exhibit similar performance but perform slightly worse the BACKDOOR dataset. This may be attributed to their limited expressive capacity for the observed distribution, reflected by the OBSwD. which did not decrease as expected to the same level as other models. CMSM appears to be less stable during training, as evidenced by the widely dispersed scatter points across all settings. This could imply that the parameters of CMSM undergo significant changes during training, resulting in unstable performance. In contrast, TVSM demonstrates more concentrated scatter points, indicating higher stability. Moreover, it achieves a lower OBSwD on the BACKDOOR dataset compared to DNME and TNME, suggesting stronger expressive capacity. However, due to its reliance on IVP solvers, its inference speed is relatively slow.
We report the final performance of the above experiments on the test set in Table 3 and Table 4.
It can be observed that, apart from certain anomalies exhibited by CMSM under the non-Markovian ablation on the STAIR dataset (as also reflected in Figure 4), the non-ablation settings consistently achieve lower CTFRMSE- This further highlights the critical role of the three assumptions in Corollary 5.4, namely ATM.SCM. AM. and Ag, in ensuring counterfactual consistency.
Finally, we conducted an ablation study on exogenous distributions by constructing new models using different exogenous distributions as described in Appendix C.2. This analysis demonstrates that Corollary 5.4 and the proposed neural TM-SCM are insensitive to the type of exogenous distribution. The results, shown in Table 5, indicate that while different models may exhibit some variation in the fit of the observational distribution (i.e., OBSwD), their counterfactual consistency (i.e., CTFRMSe) remains largely similar. Furthermore, as asserted in Appendix C.2, using a standard normal distribution as the exogenous distribution may not provide sufficient expressive power, as evidenced by the higher OBSwD values in DNME and TNME models in Table 5.
Ablation on ER-DIAG-50 and ER-TRIL-50 We report below the complete results of the ablation study on the ER-DIAG-50 and ER-TRIL-50 datasets as presented the main text. The ER-DIAG-50 and ER-TRIL-50 datasets encompass diverse TM-SCM configurations, including various scales, structures, and parameter settings. Compared the four symbolically constructed datasets TM-SCM-SYM, they provide more comprehensive validation and testing for our framework.
The complete version of the ablation results table from the main text, as shown in Table 6, presents the final results on the test set. The table additionally includes the metrics OBSwD and CTFwD, which measure the average fit to the observational distribution and the prediction for the interventional distribution, respectively.
The results demonstrate that models without ablations achieve the best performance on the final counterfactual consistency
42

---

Exogenous Isomorphism for Counterfactual Identifiability
Table Ablation results neural TM-SCM on ER-DIAG-SU 50 and ER- -0. values shown are the means 50 experiments, with the subscript representing the 95% CL. The best pest-rrrformin results are highlighted in bold.
ER-DIAG-50
ER- ER-TRIL-550
METHOD
OBSwD
CTFRMsE
CTFwD
OBSwd
CTFRMsE
CTFwD
3.47 +o.03
0.53 #0.05
20.28
21.28 4:22.992.
+0.12
3.37+a.u
DNME
W/O O
1.0641.70
40.85
2.8844.43
25. 25.42-421.8 27
4.
4.7943.03
W/O M
4.35
0.6210.04
1.98
18.53
0.58
3.6013.01
2.654
0.47 20.06
+..2
3..4
0.55
3.49
TNME
W/O O
5.40.12.55
11.24:120.0
3.47:1108
23.75
6.4119.84
37.11 1:4.991
W/o M
3.88
0.62
2.08
5.57
0.7310.21
13.60
15.70423.20
0.37
0.96
9.78
0.42+0..1
3.17=3.07
CMSM
W/O O
41.743145.13
2.64.13377
2.91 1.044
36.65
4:30.28
2.12:12.9
4.69
W/o
8.00 9..888
1.69 1:2.0
0..05
10.39 10.58
0.7540.49
7.79 48.34
W/O T
3.74
0.64
2.47:40.5
23.36.436.32
1.25
1.33
3.66.41.00
1.46+0.05
0..28
3.40
0.50
3.35 422999
W/O O
2.25.42.a1
0.79
2.89
11.79+10.38
0.88
4.6843.03
TVSM
W/O M
3.54
0.53
1.63
4.25
0.53
3.33
W/O T
22.22:36.74
0.67
2.43
14.30
0.7810.12
6.6313.87
metric, CTFRMSE- This highlights the universality the three assumptions in Corollary 5.4, namely ATM.SCM. AM, and Ag, in ensuring counterfactual consistency. Furthermore, these findings empirically reinforce the validity of Corollary 5.4.
Beyond providing empirical support for counterfactual consistency, these results also reveal several characteristics of the four different models. For instance, DNME performs significantly better ER-DIAG-50 compared ER-TRIL-50, DNME is specifically designed based causal mechanisms where exogenous variables interact in a diagonal manner. The performance drop ER-TRA TE-------0 indicates that DNME lacks sufficient expressive power in more general settings. TNME addresses this limitation of DNME through its design. CMSM may exhibit instability due to its potentially poor fit to the observed distribution. Furthermore, as only numerically stable test results are recorded in the experiments, many omitted cases CMSM's results may contribute to its lower CTFwD values. contrast, TVSM demonstrates higher accuracy in counterfactual inference compared to DNME and TNME, while also achieving better to the observational distribution than CMSM.
43


Reviews:
[{"review_id": "O3Yu1dc1Av", "reviewer": "Reviewer_BQ84", "summary": "The paper introduces the notation of exogenenous isomorphism (EI) between SCMs, a suffient relationship (Theorem 3.2) that ensuring counterfactual equivlance. In section 4 and section 5, the paper focus on two special types of SCMs: Bijective SCMs (BSCMs) and Triangular Monotonic SCMs (TM-SCMs). The paper first introduces the sufficient and necessary condition of achieving EI identification is that there exists a bijection $h_i$ between the exogenous variable $\\textbf{u}\\_i^{(1)}$ and  $\\textbf{u}\\_i^{(2)}$ of two SCMs $\\mathcal{M}^{(1)}$ and $\\mathcal{M}^{(2)}$ (Theorem 4.3). Further, the paper proves EI identifiability under assumption set $\\mathbf{\\mathcal{A}}$ from counterfactual transport (Theorem 4.6) and KR Transport (Theorem 4.8). Similar result applies to Triangular Monotonic SCM (Proposition 5.3).\n\nFinally, the paper shows how these theoretical results can inform practical modeling strategies. By designing “neural” versions of TM-SCMs—where monotonic and triangular structures are imposed on neural networks—the authors illustrate how to fit a model solely to observational data while still supporting consistent answers for counterfactual queries. Empirical evaluations on synthetic data demonstrate that the learned models reproduce ground-truth counterfactual outcomes accurately, underlining the paper’s main claim that consistent counterfactual inference can be achieved without recovering the full latent structure.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for pointing out the typo and have added three references relevant to our work:\n\n- (Brehmer et al., 2022): In this paper, Theorem 1 proves that the latent causal model is identifiable from weak supervision up to graph isomorphisms and elementwise diffeomorphisms, where the latter is a prerequisite for exogenous isomorphism. Our results further show that if such a model is $\\sim_{\\text{EI}}$-identifiable, then the counterfactual distribution is consistent. This answers the question raised in the paper regarding whether a redefinition of causal variables preserves the counterfactual distribution.\n\n- (Wu et al., 2025): This paper focuses on relaxing the strictly monotonicity assumption to a rank preservation assumption to prove the identifiability of counterfactual outcomes. The corresponding SCM can be regarded as a generalization of the TM-SCM, so the results of this paper indeed fall within the discussion framework of $\\sim_{\\text{EI}}$-identifiability.\n\n- (Zhou et al., 2023): This paper studies the invertible latent causal model, which is a version of the BSCM with latent causal variables, and is therefore related to the subject of our study. The paper mainly focuses on what is referred to as the domain counterfactual, defines the concept of domain counterfactual equivalence, and concentrates on deriving the counterfactual error bound.\n\nWe will refine these points in our revision.\n\n> **Question 1**: In real-world scenarios (where full knowledge of mechanisms is rarely available), do you envision practical approaches to test whether two learned SCMs might be exogenously isomorphic or close to it?\n\nIf we have full knowledge of the mechanisms (for example, if we have learned proxy SCMs and have access to their mechanisms, as discussed in (Zhou et al., 2023)), we can directly verify whether two SCMs are exogenously isomorphic according to the definition. However, in real-world scenarios, SCMs learned under incomplete knowledge are unlikely to happen to fall into the equivalence class of exogenous isomorphism, which corresponds to the identification task.\n\nIn identification tasks, rather than using statistical tests, we typically introduce assumptions to prove identifiability; this is a means to ensure favorable properties a priori, with the connection to reality mainly lying in the strength of these assumptions. Our paper later discusses sets of assumptions that guarantee exogenous isomorphism for two specific types of SCMs: BSCM and TM-SCM.\n\nOf course, there are also statistical methods that can indicate whether a learned SCM is close to being exogenously isomorphic, such as the counterfactual error bound presented in (Zhou et al., 2023) or the partially identifiable aspects that have recently received attention in counterfactual reasoning. However, this is somewhat beyond the scope of our discussion and can be explored in future work.", "claims_and_evidence": "The paper claims EI identification results on Bijection SCMs and Triangular Monotonic SCMs supported by various theorems. The paper further proposes Neural TM-SCM, constructing the neural network parameterized SCMs to satisfy the assumptions the theorem requires.", "methods_and_evaluation_criteria": "The methods and evaluation criteria reflects the problem.", "theoretical_claims": "I did not check the correctness of proofs.", "experimental_designs_or_analyses": "The experiments reflects the theorem, showing the constructrion constraints attemping to satisfy $\\mathbf{\\mathcal{A}}$ achieves lower coutnerfactual error (CTF RMSE).", "relation_to_broader_scientific_literature": "Thie paper extends and generalize the existed counterfacutal identification results. Like it extends [1]'s theroem and generalize [2] (bijective, proposition 6.2) )and [3] (bijective, markovian, Guassian exogenous noise in Theorem 1) to a more generalize scenario.", "essential_references_not_discussed": "Here are some paper talking about bijective or invertible SCM in the literature, and similar (partial) results in latent causal model counterfactual identification or causal representation learning. For example, some of the work here talks about TR transformation on identification (upto point-wise bijection and a re-ordering operator due to the nature of latent causal model) [3,4,5]. \n\n[1] Nasr-Esfahany, A., & Kiciman, E. (2023). Counterfactual (non-) identifiability of learned structural causal models. arXiv preprint arXiv:2301.09031.\n\n[2] Nasr-Esfahany, A., Alizadeh, M., & Shah, D. (2023, July). Counterfactual identifiability of bijective causal models. In International Conference on Machine Learning (pp. 25733-25754). PMLR. \n\n[3] Brehmer, J., De Haan, P., Lippe, P., & Cohen, T. S. (2022). Weakly supervised causal representation learning. Advances in Neural Information Processing Systems, 35, 38319-38331.\n\n[4] Wu, P., Li, H., Zheng, C., Zeng, Y., Chen, J., Liu, Y., ... & Zhang, K. (2025). Learning Counterfactual Outcomes Under Rank Preservation. arXiv preprint arXiv:2502.06398.\n\n[5] Zhou, Z., Bai, R., Kulinski, S., Kocaoglu, M., & Inouye, D. I. (2023). Towards characterizing domain counterfactuals for invertible latent causal models. arXiv preprint arXiv:2306.11281.", "other_strengths_and_weaknesses": "The paper writes clearly and well-structured. Though notation heavy, it is understandable since the paper contains heavy theorem contributions.", "other_comments_or_suggestions": "Typos: Title of Seciton 5, should be triangular.", "questions_for_authors": "1. You show that two SCMs are exogenously isomorphic if their causal mechanisms align via component-wise bijections and they induce the same L3 distributions. In real-world scenarios (where full knowledge of mechanisms is rarely available), do you envision practical approaches to test whether two learned SCMs might be exogenously isomorphic or close to it?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kJuqGy5hHf", "reviewer": "Reviewer_PYFF", "summary": "This paper explores ∼L3-identifiability, aiming to ensure that all Structural Causal Models (SCMs) satisfying given assumptions provide consistent answers to causal questions. The authors introduce exogenous isomorphism and propose ∼EI-identifiability, showing that full recovery of exogenous variables is not needed for ∼L3-consistency. They explore this in two special SCM classes: Bijective SCMs (BSCMs) and Triangular Monotonic SCMs (TM-SCMs), offering new methods to achieve identifiability. The paper also leverages neural TM-SCMs for counterfactual reasoning, with experiments validating the effectiveness of the proposed approach. Key contributions include the novel concept of exogenous isomorphism and the unified framework for achieving counterfactual identifiability.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their careful reading and for pointing out some weaknesses and suggestions. Below are our responses to these points.\n\n> **Weakness 1**: Limitations of Assumptions  \n\nIt is acknowledged that the strength of the assumptions is a common drawback in identification tasks; however, these assumptions are necessary to guarantee identifiability. For instance, the Markovianity assumption is routinely employed in causal inference, causal discovery, and causal representation learning, especially in situations where only observational data is available. This is also the primary scenario studied in our work with BSCM and TM-SCM. \n\nAlthough the assumptions in our current results for BSCM and TM-SCM are relatively strong, we remain optimistic because the theory regarding exogenous isomorphism applies to **any recursive SCM** without imposing additional assumptions about distributions or mechanisms. This lays the groundwork for future analysis of $\\mathcal{L}_3$-identifiability in imperfect real-world scenarios.\n\n> **Weakness 2 & Suggestion 1**: Complexity When Dealing with Large-Scale Datasets  \n\nIn terms of model construction, TM-SCM represents a class of models that includes additive noise models, causal flows, and others (see Appendix B.2, page 25). These models are closely related to current deep generative models, which provides potential for future applications on high-dimensional and large-scale data. Since our paper mainly focuses on proof-of-concept experiments aimed at empirically validating the correctness of the related theory, we have not yet explored applications on large-scale datasets.\n\nRegarding inference, especially for counterfactual inference with invertible SCMs, we present the Pseudo Potential Response algorithm in detail in Appendix C.7 (see page 33). This algorithm requires $n$ iterations, where $n$ is the total dimensionality of the variables, with each iteration involving one inverse operation and one forward operation. Although the algorithm iterates over the dimensions, the algorithm extends the approach in (Javaloy et al., 2023) to handle vector spaces and multiple interventions. We also note that (Le et al., 2024) and (Almodóvar et al., 2025) have proposed approximate methods that learn a encoder fitting the inverse operation, which can accelerate counterfactual inference on invertible SCMs. We will clarify and expand on these points in the appendix.\n\n> **Question 1**: Real-World Applications   \n\nCounterfactual reasoning has already been widely applied in real-world tasks (see lines 38-42). This work focuses on the identifiability of counterfactual reasoning, and its impact lies in theoretically ensuring the reliability of related applications.\n\nAt present, this work remains in the early stages of theoretical development and synthetic data experiments. We primarily use proof-of-concept experiments to demonstrate the practicality of addressing counterfactual consistency issues. We have not yet ventured into real-world applications, and therefore have not focused on the complexities and noise involved therein. Nonetheless, the results from these synthetic experiments offer preliminary empirical support for the theoretical guarantees of counterfactual identifiability established in several related works.\n\n> **Suggestion 2 & Question 2**: Comparison with Other Methods \n\nThe theoretical comparisons have already been listed in the related work section in Appendix B (see page 25), and we will also include the literature mentioned by reviewer BQ84.\n\nIn practice, since we have not proposed a specific model, a direct comparison with other methods is not necessary. Instead, we have refined four prototypes from previous literature (simplified implementations of several state-of-the-art methods) to validate the theory in the ablation experiments. These prototypes were built and run in all ablation experiments, so the ablation studies inherently serve as a comparison among different methods, as detailed in Appendix D.4 (see page 37).\n\nIn the revised version, we will include a brief version of the related work in the main text to provide a more detailed comparison, which will highlight the advantages.", "claims_and_evidence": "Yes, the claims made in the paper are well-supported by theoretical proofs and empirical evidence: \nTheoretical Proofs: The authors provide rigorous mathematical proofs for their key claims, such as Theorem 3.2 (which shows that exogenous isomorphism implies ∼L3-identifiability) and Theorem 4.6 (which demonstrates ∼EI-identifiability for Bijective SCMs). These proofs are detailed and logically sound, establishing a strong theoretical foundation.2. \nEmpirical Validation: The paper includes experiments on synthetic datasets that demonstrate the effectiveness of the proposed methods. The results show that neural TM-SCMs achieve higher counterfactual consistency (measured by CTFRMSE) compared to ablations, validating the necessity of the assumptions made for ∼EI-identifiability.", "methods_and_evaluation_criteria": "Yes, the methods and evaluation criteria used in the paper are appropriate for the problem of counterfactual reasoning. The authors propose a novel approach to achieve∼EI-identifiability by focusing on specific classes of SCMs(BSCMs and TM-SCMs)and demonstrate its effectiveness through neural network parameterizations. The evaluation metrics, including OBSWD(Wasserstein distance for observational distribution)and CTFRMSE(root mean square error for counterfactual outcomes),are well-suited to assess the model's performance in fitting the observational data and generating consistent counterfactual results.", "theoretical_claims": "Yes, I checked the correctness of the proofs. The authors introduce several theorems and lemmas to establish the connection between exogenous isomorphism and∼L3-identifiability. For example, Theorem 3.2 shows that exogenous isomorphism implies∼L3-identifiability, providing a solid theoretical foundation for the proposed method. The proofs are detailed and logically sound, supporting the claims made in the paper.", "experimental_designs_or_analyses": "Yes, I checked the soundness of the experimental designs and analyses. The authors conducted experiments on synthetic datasets to validate the correctness of the theory and the effectiveness of the proposed method. The experimental design considered different model structures (such as DNME, TNME, CMSM, and TVSM) and various datasets (such as TM-SCM-SYM, ER-DIAG-50, and ER-TRIL-50), which are representative to some extent. In the experimental analysis, the authors provided a detailed discussion on the impact of each assumption on model performance and used ablation studies to verify the necessity of these assumptions.", "relation_to_broader_scientific_literature": "The paper situates its contributions within the broader literature on causal inference and machine learning,:\nIn the field of causal inference, counterfactual identifiability is an important research direction. This paper proposes a new identifiability target—achieving identifiability across the entire counterfactual hierarchy—which offers new ideas and methods for research in causal inference. Additionally, the concept of exogenous isomorphism introduced in the paper and the ∼EI-identifiability based on it provide a new perspective for the theoretical study of causal models.\nIn the field of machine learning, causal inference and causal representation learning are current research hotspots. The research findings of this paper not only provide theoretical support for research in these areas but also offer valuable references for practical applications.", "essential_references_not_discussed": "No, there are no essential related works that are missing from the citations or discussions in the paper.", "other_strengths_and_weaknesses": "Strengths:\n1. Theoretical Innovation: The paper introduces the concept of exogenous isomorphism and proposes ∼EI-identifiability based on this, providing new theoretical support for the problem of causal identifiability. \n2. Methodological Practicality: The paper not only presents theoretical insights but also applies the theory to practical problems through neural TM-SCMs, demonstrating its effectiveness in counterfactual reasoning. \n3. Experimental Validation: The paper validates the correctness of the theory and the effectiveness of the method through experiments on synthetic datasets, offering strong support for the reliability of both the theory and the approach.\nWeaknesses:\n1. Limitations of Assumptions: Although the assumptions proposed in the paper are theoretically sound, they may be difficult to satisfy in practical applications. For example, the assumption that the causal model is Markovian may not hold in some complex scenarios.\n2. Computational Complexity: The neural TM-SCMs proposed in the paper may have high computational complexity, especially when dealing with large-scale datasets. This could limit their scalability in practical applications.", "other_comments_or_suggestions": "1. Scalability Analysis: The authors should provide a detailed analysis of the computational complexity and scalability of the proposed neural TM-SCMs, especially for large-scale datasets. This would help address potential limitations in practical applications.\n2. Comparison with Other Methods: A more detailed comparison with other state-of-the-art methods for counterfactual reasoning would strengthen the paper. This could include both theoretical and empirical comparisons, highlighting the advantages and trade-offs of the proposed approach.", "questions_for_authors": "1. Real-World Applications: Can the authors provide examples of real-world applications where the proposed method could be particularly beneficial? How would the method handle the complexities and noise typically found in real-world data?\n2. Comparison with Other Methods: Could the authors provide a more detailed comparison with other causal representation learning methods, especially those that also aim to improve counterfactual consistency? What are the key differences and advantages of the proposed approach?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "94yShrJgco", "reviewer": "Reviewer_CJgk", "summary": "This paper analyses L3 identifiability, showing that full recovery of exogenous variables in SCMs is not required to achieve it. It also unifies the existing theories of bijective SCMs and TM-SCMs, implementing neural TM-SCM for the experimental section, thus showing the practical applicability of the developed ideas.\n\n## Update after rebuttal\nAuthors carefully addressed my concerns about novelty. I believe this work is sound and rigurous and, in light of the arguments in the response, not only makes valuable connections among existing theories but also extends them. I am now more confident in changing my initial score from 3: Weak accept to 4: Accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewers for raising questions regarding the novelty of this paper. Below are clarifications to address these misunderstandings:\n\n> **Question 1**: Specify the degree of novelty of the paper\n\nThe main result of this paper is that if causal mechanisms are aligned via **exogenous isomorphism**, then they yield **consistent $\\mathcal{L}_3$ distribution** (Theorem 3.2). The former serves as a target in representation learning, while the latter is formulated by PCH. In this way, our primary contribution resides in establishing a novel connection between these two distinct perspectives.\n\nTheorem 3.2 applies to **any recursive SCM**. BSCM and TM-SCM then introduce additional parametric assumptions, and they are special cases identified in our effort to relate our work to previous literature.\n\nFrom the perspective of BSCM or TM-SCM, our paper further extends existing counterfactual identification theories (e.g., Nasr-Esfahany et al., 2023) by:  \n1. Addressing not only counterfactual outcomes but all $\\mathcal{L}_3$ distributions defined by PCH;  \n2. Extending the framework to vector spaces, where each causal variable take values in $\\mathbb{R}^d$;  \n3. Presenting our derivations from a novel perspective of counterfactual transport.\n\nRegarding model design works (e.g., Javaloy et al., 2023), our contribution is to help formalize proofs that model indeed possess counterfactual identifiability (previously only demonstrated to achieve component-wise bijective identifiability in representation learning). This aspect has been rarely discussed or is only empirically illustrated in those studies.\n\nThese theoretical and practical connections are further detailed in the related work section in Appendix B (see page 25). In the revised version, we plan to add a dedicated related work section in the main text to briefly highlight the novelty of our contributions.\n\n> **Question 2**: Formalization of state-of-the-art ideas or bringing in some novel idea?\n\nAs suggested by the title, the novel perspective of exogenous isomorphism is the primary contribution of our paper. The reviewer’s concerns might arise for the following reasons:  \n- Experimentally and in terms of conclusions, our work may appear similar to previous studies—partly because we discuss extensive material on BSCM and TM-SCM. However, these are used as bridges to connect the idea of exogenous isomorphism with prior work and to design ablation experiments that validate our theoretical findings. Hence, referring to their results is necessary.  \n- In addition, the imprecise wording in the abstract may have led readers to mistakenly believe that proving that full recovery of exogenous variables in SCMs is not required to achieve counterfactual identifiability is our primary contribution. In fact, our focus is on clarifying the degree of representational identifiability that leads to complete counterfactual identifiability. We will update the relevant text accordingly.\n\nWe appreciate the reviewer’s valuable feedback, which will help us enhance the clarity and impact of our revised manuscript.", "claims_and_evidence": "The claims are well supported.", "methods_and_evaluation_criteria": "Benchmarks and datasets are quite complete. No external evaluation is performed, but this is not required due to the nature of the paper.", "theoretical_claims": "Theoretical claims are well supported.", "experimental_designs_or_analyses": "I did not see any important issues regarding experimental designs.", "relation_to_broader_scientific_literature": "All relevant related papers are properly cited.", "essential_references_not_discussed": "All relevant related papers are properly cited.", "other_strengths_and_weaknesses": "As a strength, the paper is generally well written.\n\nAs weakness, the paper does not seem to present great novelties. The fact that full recovery of exogenous variables in SCMs is not required to achieve counterfactual identifiability has been previously demonstrated in works like (Nasr-Esfahany et al., 2023) or (Javaloy et al., 2023). Neural TM-SCM are not new either. This paper has some value as it collects the state-of-the-art ideas about counterfactual identifiability and makes and effort in unifying the ideas of bijective SCMs and of TM-SCM, but I am not completely sure if it has an important enough novelty for ICML. I would appreciate if authors could address this issue.", "other_comments_or_suggestions": "-", "questions_for_authors": "Could the authors specify the degree of novelty of the paper? Is it a compilation and formalization of state-of-the-art ideas or does it bring some novel idea?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: All reviews are unanimous about this paper. Authors did clarify aspects relating to novelty and comparisons to existing works during the discussion which seems to have satisfied the reviewers. 

The paper proposes exogenous isomorphism - an equivalence relation between Structural Causal Models (SCMs) such that for all SCMs that are equivalent under this relation, all the counterfactual queries are answered uniquely. This identifies all queries in the three layers of Pearl's Causal Hierarchy Uniquely.  Authors then argue that if this is satisfied, it is not necessary to estimate the true exogenous variables. Authors also discuss sub-classes of SCMs where such an equivalence relation holds and also propose Neural architectures that can utilize the theory to answer counterfactual queries.

The paper addresses a fundamental problem and makes counterfactual identification possible for a class of SCMs satisfying some equivalence relation. Solid contribution. I recommend acceptance.
Decision: Accept (spotlight poster)