Foundations of Structural Causal Models with Cycles and Latent Variables

Scope and contributions

• The work establishes foundational theory rather than detailed causal reasoning, discovery, or prediction, while noting that several existing results and algorithms extend to simple SCMs [44].Its scope covers definitions, equivalences, solvability, marginalization, Markov properties, graph interpretation, and simple SCMs.Source: §1 · pp. 2, 4, 5

Solvability

• Cyclic SCMs may lack solutions or unique observational, interventional, and counterfactual distributions, so the paper establishes sufficient solvability and measurability conditions.Interventions can destroy unique solvability even when the original SCM is uniquely solvable.Source: §1 · p. 2

Equivalence

• The paper defines observational, interventional, counterfactual, and strong equivalence relations that apply even when cyclic SCMs have no solution or multiple induced distributions.Counterfactual equivalence uses distributions induced by the twin SCM via the twin network method.Source: §1 · pp. 2–3

Marginalization

• Under local unique solvability, cyclic SCM marginalization preserves observational, interventional, and counterfactual semantics; under local ancestral unique solvability, it also respects latent projection.The paper shows that marginalization need not exist generally, and establishes these conditions for its existence and graph-level compatibility.Source: §1 · p. 3

Markov properties and causal interpretation

• For cyclic SCMs, usual Markov properties and graph-based causal interpretations can fail, but unique solvability conditions recover σ-separation-based Markov properties and graph-consistent causality.Under ancestral unique solvability, interventions are compatible with the causal structure encoded by the graph.Source: §1 · p. 3

Simple SCMs

• Simple SCMs are uniquely solvable with respect to every variable subset and consequently retain unique distributions, intervention and marginalization closure, latent-projection compatibility, Markov properties, and intuitive causal graphs.They include acyclic SCMs and have stronger directed global Markov properties in special cases including linear and discrete models.Source: §1 · pp. 3–4

2.1. Structural causal models and their solutions

• SCMs comprise endogenous and exogenous variables, a measurable causal mechanism, and a product exogenous distribution, with solutions satisfying X = f(X,E) almost surely.The definition is independent of the random variables solving the model, supporting technical analysis of cycles.Source: §2.1 · pp. 5–6, §2 · p. 5
• Cyclic SCMs may have multiple observational distributions or none: mutual-copy equations admit arbitrary distributions on the diagonal, whereas adding one to an equation eliminates all solutions.Unlike acyclic models, cycles also permit self-cycles and make observational distributions generally nonunique.Source: §2.1 · p. 6

2.2. The (augmented) graph

• The augmented graph records endogenous and exogenous parent relations, while the graph replaces shared exogenous parents with bidirected edges; acyclicity is defined on the augmented graph.These graph mappings are invariant under SCM equivalence, and the augmented graph is the more detailed standard representation.Source: §2.2 · pp. 8–9

2.3. Structurally minimal representations

• Every SCM has an equivalent structurally minimal representative whose mechanisms depend only on their parents, making the representation compatible with its augmented graph.This representative exists even when the original mechanism depends on nonparents.Source: §2.3 · p. 9

2.4. Interventions

• Perfect interventions replace selected mechanisms with fixed values and remove their incoming graph edges; disjoint interventions commute and preserve acyclicity.In cyclic models, intervention can destroy or restore solvability, so a solution before intervention does not guarantee one afterward.Source: §2.4 · pp. 9–10

2.5. Counterfactuals

• Counterfactual semantics are constructed by intervening on a twin SCM, which duplicates endogenous variables while sharing the exogenous variables.This framework builds on the twin-SCM idea introduced in.Source: §2.5 · p. 11
• Twin operations preserve SCM equivalence, are compatible with augmented graph mappings, and preserve acyclicity.The SCM and graph constructions therefore maintain acyclicity under the twin operation.Source: §2.5 · p. 11
• Perfect intervention commutes with the twin operation for both SCMs and directed graphs.For SCMs, interventions apply to both original and copied variables; the analogous graph operation also commutes under the stated exogeneity condition.Source: §2.5 · p. 11
• A counterfactual distribution is defined as P(X,X′) whenever the intervened twin SCM has a solution (X,X′).This formulation also applies to cyclic SCMs when the required solution exists.Source: §2.5 · p. 12
• SCMs can share observational and interventional distributions yet differ in their counterfactual statements, making counterfactual identification from data nontrivial.The subsection situates this issue within broader work on counterfactual interpretation.Source: §2.5 · p. 12

3. Solvability

• Solvability is equivalent to existence of a solution, while unique solvability guarantees a unique induced observational distribution.These equivalences extend key acyclic-SCM properties to cyclic models under appropriate solvability conditions.Source: §3 · p. 12, §3.1 · p. 12, §3.2 · p. 14

3.1. Definition of solvability

• Solvability assigns measurable solution functions to endogenous-variable subsets, but strict-subset solvability need not imply global solvability.Measurable solution functions may fail to exist or be unique; proving existence in cyclic models requires a strong measurable selection theorem.Source: §3.1 · pp. 12–13

3.2. Unique solvability

• Unique solvability is equivalent to almost-everywhere uniqueness of subsystem solutions; acyclic SCMs satisfy it for every subset, and some cyclic SCMs do too.In general, unique solvability is not preserved under strict subset or superset operations, unions, or intersections.Source: §3.2 · pp. 13–14

3.3. Self-cycles

• An endogenous variable is uniquely solvable precisely when its graph has no self-cycle; in linear SCMs, a self-cycle occurs exactly when B_ii = 1.A self-cycle can still permit solvability without uniqueness, or prevent solvability entirely, depending on the structural equations.Source: §3.3 · p. 14

3.4. Interventions

• Intervention preserves solvability with respect to the enlarged subsystem when the original model is solvable with respect to the nonintervened variables.Without this condition, intervention can destroy unique solvability, eliminate solutions, or produce multiple induced distributions.Source: §3.4 · p. 14

3.5. Ancestral (unique) solvability

• Solvability with respect to a subset is equivalent to ancestral solvability, but unique solvability need not extend ancestrally except for linear SCMs.Ancestral unique solvability implies unique solvability, whereas the converse fails in general.Source: §3.5 · p. 15

4. Equivalences

• The section introduces these three coarser equivalence relations because the original SCM equivalence preserves graphs, solutions, and observational, interventional, and counterfactual distributions.The new relations compare models at different levels of observable and causal information.Source: §4 · p. 16

4.1. Observational equivalence

• Observational equivalence means that two SCMs induce the same observational distributions, but observationally equivalent models may have different augmented graphs and need not be equivalent.Equivalent SCMs are observationally equivalent, but the converse fails.Source: §4.1 · p. 16

4.2. Interventional equivalence

• Interventional equivalence requires observational equivalence after every perfect intervention and is finer than observational equivalence, yet interventionally equivalent SCMs may still have different augmented graphs.Equivalence with respect to O implies equivalence for strict subsets and implies observational equivalence on O, but not conversely.Source: §4.2 · pp. 16–17

4.3. Counterfactual equivalence

• Counterfactual equivalence compares twin SCMs under every perfect intervention, is finer than interventional equivalence, and can fail even when models share mechanisms and interventional distributions.Equivalence implies counterfactual equivalence, which implies interventional equivalence; counterfactually equivalent models may nevertheless have different augmented graphs.Source: §4.3 · p. 17

4.4. Relations between equivalences

• The equivalence relations form the hierarchy equivalence ⇒ counterfactual equivalence ⇒ interventional equivalence ⇒ observational equivalence, establishing a formal ladder of causal abstraction.Equivalence with respect to O requires matching endogenous-variable domains on O, while other model components may differ.Source: §4.4 · p. 17

5. Marginalization of structural causal models

• Under local unique solvability, substituting measurable subsystem solutions yields a marginal SCM observationally, interventionally, and counterfactually equivalent on the remaining variables.This extends marginalization beyond modular SCMs, although the operation is defined only up to equivalence because solution functions are unique only almost everywhere.Source: §5 · p. 18, §5.1 · pp. 19–20

5.2. Marginalization of a graph

• General SCM marginalization need not respect latent projection: the marginalized SCM graph can fall outside the latent projection even when causal semantics are preserved.Under local ancestral unique solvability, the marginalized augmented graph is a subgraph of the latent projection; acyclic SCMs and linear SCMs therefore recover closure under marginalization.Source: §5 · p. 18, §5.2 · p. 21

5.1. Marginalization of a structural causal model

• Without solvability with respect to the hidden subset, no SCM on the margin can generally preserve even interventional semantics; unique solvability is sufficient but not necessary for counterfactual preservation.Marginalization may also be unavailable for individual variables with self-cycles, while marginalizing a larger subset can be possible.Source: §5.1 · pp. 18–20
• Sequential marginalization over disjoint subsets is equivalent to marginalizing their union, both for SCMs and for directed mixed graphs.For SCMs, this equivalence holds exactly when the intermediate model is uniquely solvable over the second subset.Source: §5.1 · p. 19, §5.2 · p. 21

• Cyclic SCMs can violate the directed global Markov property: d-separation may imply conditional independence that fails for every solution.In Example 6.1, X1 and X2 are d-separated given {X3,X4}, yet are dependent under every solution.Source: §6 · p. 22
• Acyclicification replaces mechanisms of strongly connected components with measurable solution functions, preserving solutions and observational distributions while translating d-separation into σ-separation [18].This construction requires unique solvability with respect to each strongly connected component.Source: §6 · p. 22
• For uniquely solvable SCMs, the observational distribution exists uniquely and satisfies the σ-separation criterion when the model is uniquely solvable within every strongly connected component.The general criterion is typically weaker than d-separation because σ-separation implies d-separation.Source: §6 · pp. 22–23
• The stronger d-separation criterion holds for acyclic SCMs, ancestrally uniquely solvable discrete SCMs, and suitable linear SCMs with nontrivial exogenous dependence and a Lebesgue-density exogenous distribution.The linear condition allows bidirected edges and need not assume Gaussianity.Source: §6 · pp. 22–23
• σ- or d-faithfulness is commonly assumed for constraint-based causal discovery, but general parametric and nonparametric guarantees remain unavailable beyond restricted DAG settings.Meek’s measure-zero result applies to multinomial and linear-Gaussian DAG models.Source: §6 · p. 23

7.2. Bidirected edges

• Neither directed nor bidirected edges are generally identifiable from observational, interventional, or counterfactual distributions, and no simple bidirected-edge criterion is known when the variables lie in a cycle.A bidirected edge may exist even when p(x2 |do(X1 = x1)) = p(x2 |X1 = x1) for every x1.Source: §7 · p. 23, §7.2 · p. 24

7.1. Directed paths and edges

• Under a specific unique-solvability condition, interventions can certify that a directed edge or path is present in the latent projection, despite cyclic SCMs sometimes violating graph-based causal interpretation.Without this condition, intervening on a nonancestor can change a variable’s distribution, and weaker componentwise solvability does not ensure the directed global Markov property.Source: §7.1 · pp. 23–24
• Proposition 7.1 detects i → j when two interventions on i, after fixing other observed variables, induce distinct unique marginal distributions for X_j; this covers edges and directed paths.Setting O = I targets a graph edge, whereas setting O = {i,j} targets a directed path, but the condition is not universally applicable.Source: §7.1 · p. 24

7.2. Bidirected edges

• Proposition 7.2 generalizes the common-cause principle to cyclic SCMs: a suitable mismatch between conditional and interventional distributions certifies a bidirected edge in the latent projection.The criterion requires unique solvability and marginal distributions under relevant interventions, with j not an ancestor of i in the intervened graph.Source: §7.2 · p. 24

• Simple SCMs are uniquely solvable with respect to every subset of variables, extending acyclic SCMs to cycles while excluding self-cycles.This local solvability condition means every subset of structural equations has a unique solution in terms of the remaining variables.Source: §8 · p. 25
• The class is closed under arbitrary-order marginalization, perfect intervention, and the twin operation, with marginalizations respecting latent projection and remaining simple.These closure properties preserve the model class across observational, interventional, and counterfactual constructions.Source: §8 · p. 25
• For simple SCMs, observational, interventional, and counterfactual distributions exist uniquely and satisfy general directed global Markov properties relative to the corresponding graphs.Under any one of Theorem 6.3’s conditions (1a), (1b), or (1c), the stronger directed global Markov property also holds.Source: §8 · p. 25
• Simple SCMs satisfy solvability conditions that allow direct causes, causes, and latent confounding to be defined from directed edges, directed paths, and bidirected edges.The paper gives sufficient distributional conditions for establishing each of these graph relationships.Source: §8 · pp. 25–26
• Not all causal and confoundedness relationships in a simple SCM can generally be identified from observational, interventional, or counterfactual distributions.This limitation already occurs for acyclic SCMs without additional assumptions.Source: §8 · p. 26
• Because all counterfactuals are defined even for cyclic simple SCMs, potential outcomes can be represented using the measurable solution function of the intervened model.For intervention do(I,ξ_I), the potential outcome is defined as X^ξ_I := g_Mdo(I,ξ_I)(E_pa(I)).Source: §8 · p. 26

9. Discussion

• Cyclic SCMs can lack solutions, unique distributions, valid marginalizations, Markov properties, and graphs consistent with causal semantics.These failures affect observational, interventional, and counterfactual distributions and contrast with acyclic SCM behavior.Source: §9 · p. 26
• Under appropriate solvability conditions, SCM operations, equivalence relations, marginalization, global Markov properties, and causal graph interpretations extend to cyclic models, though many conditions are only sufficient.The paper notes that some assumptions may be further relaxed.Source: §9 · pp. 26–27
• Simple SCMs preserve unique distributions, intervention and marginalization closure, latent-projection compatibility, global Markov properties, and intuitive causal interpretation, excluding self-cycles.They extend acyclic SCMs to cyclic settings while retaining many convenient properties.Source: §9 · p. 27
• Simple SCMs support σ-separation-based reasoning, extending causal discovery, adjustment, do-calculus, identification, and prediction methods from acyclic models.Examples include FCI, LCD, Y-structures, JCI, Tian’s ID algorithm, and neural-network construction methods.Source: §9 · p. 27
• The paper identifies future directions including exogenous-variable reduction, broader identifiability results, selection-bias graphs, and completeness for Markov properties in cyclic SCM subclasses.These directions aim to advance a general theory of statistical causal modeling.Source: §9 · p. 27

APPENDIX A: CAUSAL GRAPHICAL MODELS

• Appendix A organizes causal graphical-model foundations through directed mixed-graph terminology, cyclic graph concepts, modular SCMs, related graphical models, and supporting proofs.It introduces the appendix sections on graphs, Markov properties, modular SCMs, and related models.Source: Appendix A · p. 29

A.1. Directed (mixed) graphs

• Directed mixed graphs represent directed and bidirected edges, self-cycles, walks, paths, ancestors, descendants, strongly connected components, loops, and induced subgraphs.The graph of strongly connected components is always a DAG.Source: §A.1 · pp. 29–30

A.2. Markov properties

• For cyclic SCMs, ordinary d-separation-based global Markov properties fail generally, whereas σ-separation yields observational Markovness under component-wise unique solvability.Simplicity preserves solvability under interventions, extending the result to interventional and counterfactual distributions.Source: §A.2 · pp. 30–31

Definitions

• D-separation generalizes DAG d-separation and ADMG/mDAG m-separation by declaring paths blocked when colliders lack conditioned descendants or non-colliders are conditioned on.Endpoints are non-colliders, and d-separation requires every path between the queried node sets to be blocked.Source: §A.2.1 · p. 31

Path formulation

• In directed mixed graphs, a d-open walk exists exactly when a d-open path exists, so d-separation can be formulated using paths rather than walks.This lemma generalizes an earlier result to cyclic graphs.Source: §A.2.1 · p. 31

Theorem A.7

• Under acyclicity, discrete ancestral unique solvability, or specified linearity and exogenous-density conditions, a uniquely solvable SCM has a unique observational distribution satisfying the directed global Markov property.The three sufficient conditions are: acyclicity; discrete endogenous spaces with ancestral unique solvability; or linear mechanisms with nontrivial exogenous dependence and an exogenous density.Source: §A.2.1 · p. 32

Counterexample

• A uniquely solvable, even simple, cyclic SCM can violate the property: X1 and X2 are dependent given {X3,X4} despite being d-separated by that set.The counterexample uses f1=e1, f2=e2, f3=x1x4+e3, and f4=x2x3+e4 with standard-normal exogenous variables.Source: §A.2.1 · p. 32

D-faithfulness

• D-faithfulness assumes that d-separation captures all observational conditional independencies, but general parametric and nonparametric guarantees remain unavailable beyond restricted DAG settings.For multinomial and linear-Gaussian DAGs, faithfulness holds for almost all parameter values under a natural parameterization.Source: §A.2.1 · p. 33

A.2.2. The general directed global Markov property

• Component-wise unique solvability guarantees an acyclic, observationally equivalent acyclification and a unique observational distribution satisfying the general directed global Markov property.The result is proved by translating σ-separation in the original graph into d-separation in the acyclified graph.Source: §A.2.2 · pp. 34, 36
• σ-separation extends d-separation by accounting for non-colliders in strongly connected components, reduces to d-separation in acyclic graphs, and is strictly weaker than the directed global Markov property.The general directed global Markov property is therefore applicable to cyclic SCMs, while σ-faithfulness remains an open completeness problem.Source: §A.2.2 · pp. 35–37
• For simple SCMs, observational, interventional, and counterfactual distributions exist uniquely and satisfy the general directed global Markov property on their corresponding graphs.The stronger directed global Markov property also holds when at least one of the three conditions in Theorem A.7 is satisfied; unrestricted solvability is insufficient because multiple solutions can induce dependence.Source: §A.2.2 · p. 36

A.3.1. Definition of a modular SCM

• A modular SCM is defined over a directed graph with hyperedges whose simplicial-complex structure is paired with product measurable spaces, product noise measures, and compatible solution functions for every loop.Compatibility requires solutions for nested loops to agree under restriction, enabling inductive construction of solutions; perfect interventions then transform the modular SCM by modifying its graph, hyperedges, noises, and functions.Source: §A.3.1 · pp. 37–38
• Perfect interventions on modular SCMs are directly defined on the underlying HEDG and depend on the choice of mapping φ.Source: §A.3.1 · p. 39

A.3.2. Relation between SCMs and modular SCMs

• A modular SCM’s underlying SCM is loop-wisely solvable and carries a compatible system of solution functions.Every modular-SCM solution also solves its underlying SCM, whose graph may be sparser than the modular SCM’s HEDG.Source: §A.3.2 · pp. 39–40

A.4. Overview of causal graphical models

• Modular SCMs add compatible solution-function structure to SCMs, while simple SCMs extend acyclic SCMs to cyclic settings and preserve many convenient properties.Modular SCMs include simple SCMs and are associated with properties such as marginalization and the general directed global Markov property.Source: §A.3.2 · p. 40, §A.4 · p. 41

A.3.2. Relation between SCMs and modular SCMs

• Simple SCMs are exactly the SCMs that are loop-wisely uniquely solvable, and their solution-function families are compatible.For simple SCMs, every family of measurable solution functions indexed by loops is compatible.Source: §A.3.2 · p. 40

APPENDIX B: (UNIQUE) SOLVABILITY PROPERTIES

• Appendix B develops additional solvability results, covering sufficient conditions for subsets and preservation under strict super- and subsets, unions, and intersections.The appendix states that proofs of these theoretical results are provided in Appendix E.Source: Appendix B · p. 41

B.1. Sufficient condition for solvability w.r.t. subsets

• A subset is solvable when the relevant solution space is nonempty and σ-compact for almost every exogenous setting and every fixed complement.This condition includes countable discrete spaces, real intervals, and Euclidean spaces, and supports a sufficient-and-necessary unique-solvability condition for subsets.Source: §B.1 · p. 42

B.2. (Unique) solvability w.r.t. strict super- and subsets

• Solvability for a subset generally does not imply solvability for its strict supersets or strict subsets.An example is solvable for {1,2} and {2,3}, but not for {1}, {3}, or {1,2,3}; unique solvability also fails for {2}.Source: §B.2 · p. 42
• Solvability with respect to O does imply solvability with respect to every ancestral subset of O in the SCM’s graph.This provides a structural preservation result despite the failure of preservation under arbitrary strict sub- and supersets.Source: §B.2 · p. 42

Intersections

• An SCM can be uniquely solvable with respect to two sets while failing to be solvable with respect to their intersection.The example is solvable with respect to {1,2} and {2,3}, but not their intersection.Source: §B.3 · p. 42

Unions

• Unique solvability is not preserved under unions, including unions of disjoint subsets.The section cites Examples B.2 and 2.4 for these failures.Source: §B.3 · p. 42

Union of ancestral subsets

• Unique solvability is preserved for the union of two ancestral subsets when solvability holds for each subset and their intersection.This is Proposition B.4’s condition for combining measurable solution functions.Source: §B.3 · p. 42

Nodewise characterization

• An SCM is ancestrally uniquely solvable with respect to O if and only if it is uniquely solvable with respect to the ancestral set of every node in O.Thus, ancestral unique solvability can be checked nodewise.Source: §B.3 · p. 43

APPENDIX C: LINEAR SCMS

• The appendix defines linear SCMs through linear combinations of endogenous and exogenous variables with coefficient matrices B and Γ and a product exogenous measure.It does not require the exogenous measure to be Gaussian.Source: Appendix C · p. 43
• For linear SCMs, solvability with respect to a subset is characterized by a necessary-and-sufficient pseudoinverse matrix condition.The condition uses A_LL = I_L − B_LL and must hold for almost every exogenous realization and every assignment outside the subset.Source: Appendix C · p. 43
• Unique solvability with respect to a subset holds exactly when A_LL = I_L − B_LL is invertible, yielding a measurable solution function.A sufficient condition is that the spectral radius of B_LL is less than one.Source: Appendix C · p. 44
• For linear SCMs, unique solvability, ancestral unique solvability, and unique solvability on every induced strongly connected component are equivalent.This positive equivalence contrasts with the general cyclic case, where unique solvability need not imply ancestral unique solvability.Source: Appendix C · p. 44
• When A_LL is invertible, marginalization over L exists by substitution, remains linear, and respects the latent projection.The marginal causal mechanism is defined using A_LL = I_L − B_LL.Source: Appendix C · p. 44
• Linear SCMs and their marginalizations are observationally, interventionally, and counterfactually equivalent on the retained variables.Unlike nonlinear SCMs, every marginalization in this class respects the latent projection, and simple linear SCMs are closed under marginalization.Source: Appendix C · p. 45

APPENDIX D: EXAMPLES

• Appendix D adds examples of SCMs describing equilibrium states of feedback systems governed by random differential equations and examples supporting the main text.These examples motivated the study of cyclic SCMs and extend the paper’s illustrative material.Source: Appendix D · p. 45

D.1. SCMs as equilibrium models

• SCMs can model the causal semantics of equilibrium states in systems with feedback loops, including economic markets and physical dynamical systems [6].The paper motivates this use through interacting masses attached to springs and market price–quantity feedback.Source: §D.1 · p. 45
• With positive friction, the coupled harmonic oscillator converges to a unique equilibrium, whose force-balance equations induce a linear SCM.The equilibrium positions are obtained by setting velocities and accelerations to zero and treating spring lengths, constants, and endpoint distance as fixed parameters.Source: §D.1 · pp. 45–46
• Intervening to fix one mass’s position makes the remaining equilibrium positions solutions of the intervened model, and the oscillator SCM is simple.Some physical parameters may instead be represented as exogenous variables.Source: §D.1 · p. 46
• The market equilibrium model is a uniquely solvable linear SCM in which a self-cycle for price implements the equilibrium condition that demand equals supply [65].Allowing self-cycles shows that this market model can be represented as an SCM, contrary to the cited objection that it cannot [65].Source: §D.1 · p. 46
• The examples use augmented, twin, and intervened-twin graphs to represent observational, interventional, and counterfactual semantics for cyclic equilibrium models.Figure 9 depicts these graph constructions for the price, supply, and demand examples.Source: Figure 9 · p. 47, §D.1 · p. 47
• For the market model, a counterfactual intervention changing supply from s to s′ shifts price deterministically to p + (s′ −s)/βD, conditional on observed price p.This query is formulated using the intervened twin model and represents a ceteris-paribus change from the factual intervention.Source: §D.1 · p. 47

D.2. Additional examples

• Additional examples are provided to support the main text’s discussion of SCMs with cycles, equilibrium models, and latent-variable causal semantics.The supplied passage identifies this subsection’s role but does not specify the additional examples’ contents.Source: §D.2 · p. 47

Section 2

• Section 2: Structural equations that differ on a probability-zero event can induce exactly the same random-variable solutions almost surely.In Example D.4, the two mechanisms differ only at e = 0, which has probability zero, so their solution sets agree for random variables.Source: Section 2 · p. 47
• Section 2: Latent confounding may not admit an interventionally equivalent SCM with separated exogenous components and strictly monotone mechanisms.Example D.6 derives a contradiction because the transformed second exogenous component would need both a standard-normal distribution and a distribution with nonzero mean.Source: Section 2 · pp. 48–49
• Section 2: Observational and interventional densities may identify neither a counterfactual distribution nor the dependence parameter governing it.For M_ρ, the counterfactual is N(ρc, 1 −ρ^2), while observational and interventional densities do not depend on ρ; thus interventionally equivalent SCMs need not be counterfactually equivalent.Source: Section 2 · p. 49

Section 3

• This SCM has a unique solution (0,1); a perfect intervention do({1},ξ1) for some ξ1̸ = 0 leads to an intervened model that is not solvable, while do({2},ξ2) for some ξ2 > 1 yields a nonuniquely solvable SCM with multiple induced distributions.Source: Section 3 · p. 50

Section 4

• Section 4: Counterfactually equivalent SCMs can have different causal graphs.The models in Example D.10 are counterfactually equivalent, although their graphs differ.Source: Section 4 · p. 50

Section 5

• Although the SCM is not uniquely solvable with respect to L, so Definition 5.3 cannot be applied, a marginal SCM on {1,3} remains counterfactually equivalent to M.For the marginal SCM Mmarg({3}), the augmented graph has no directed path, unlike the augmented graph of M.Source: Section 5 · p. 51

Section 7

• Section 7: Comparing conditional and interventional distributions can detect a bidirected edge in an SCM graph.For ˜M, their inequality at both intervention values identifies 1 ↔2; for M̄, their equality is consistent with the absence of that edge.Source: Section 7 · p. 51

APPENDIX E: PROOFS

• The appendix provides proofs for all theoretical results in Appendices A–C and the main text, with some arguments relying on Appendix F’s measure-theoretic results.Source: Appendix E · p. 51

Appendix A

• The proofs establish path-reduction lemmas, observational equivalence of acyclification, and solvability and distributional results under component-wise unique solvability.Additional arguments prove loop-wise solvability and compatibility of solution functions, including inheritance to induced subsets and preservation under simplicity-related operations.Source: Appendix A · pp. 52–54

Appendix B

• The proofs use measurable selection to derive solvability, combine solution functions on overlapping ancestral sets to obtain unique solvability on their union, and extend this through induction over ancestral subsets.Source: Appendix B · pp. 55–56

Appendix C

• For linear SCMs, unique solvability is equivalent to invertibility of A_LL, equivalently to unique solvability on every strongly connected component; under these conditions, marginalization respects the latent projection.Source: Appendix C · pp. 56–57

Section 2

• The proof of structural minimality constructs mechanisms depending only on parent variables and parent disturbances, while perfect intervention removes the corresponding functional dependencies and incoming graph edges.Source: Section 2 · p. 57
• Section 2 shows that perfect interventions remove parental relations without creating cycles, while twin constructions remain acyclic under the stated graph assumptions.Intervening on both original and copied nodes after twinning yields the same graph as twinning after intervention.Source: Section 2 · pp. 57–58

Section 3

• Section 3 characterizes solution existence through almost-everywhere solvability of structural equations and measurable solution functions, with unique solvability yielding observational distributions as push-forwards.Acyclic augmented graphs guarantee unique solution functions via topological substitution, and solvability propagates to interventions and ancestral subsets.Source: Section 3 · pp. 59–60

Section 4

• Section 4 proves that twin SCMs are observationally equivalent and that SCM equivalence transfers to counterfactual equivalence through interventionally equivalent twin constructions.The twin operation duplicates variables while preserving the relevant observational behavior.Source: Section 4 · p. 61

Section 5

• Section 5 proves that sequential marginalization is valid exactly when joint and successive unique solvability conditions hold, and preserves observational and interventional equivalence.It also establishes compatibility with perfect interventions and twin operations under unique solvability.Source: Section 5 · pp. 63–64
• Section 5 proves that marginalizing a uniquely solvable subset preserves observational equivalence on the remaining variables and supports subsequent solvability and causal transformations.The marginal and original SCMs can reconstruct each other’s solutions through the measurable solution function.Source: Section 5 · p. 64
• Section 5 proves that marginalizing a solvable SCM yields a graph that is a subgraph of its latent projection.The proof bounds endogenous and exogenous parents via directed paths whose intermediate nodes lie in the marginalized set.Source: Section 5 · pp. 65–66

Section 6

• Section 6 derives its theorem directly from Theorems A.7 and A.21.Source: Section 6 · p. 66

Section 7

• Section 7 shows that absent directed edges in the latent projection force the corresponding intervened distributions to coincide, while absent bidirected edges imply Xi ⊥⊥ E_fpa(A\i(j)) for every solution.The independence conclusion supports the resulting conditional-probability argument for the intervened model.Source: Section 7 · pp. 66–67

Section 8

• Section 8 proves that simple SCMs are closed under marginalization, perfect intervention, and the twin operation, and therefore induce unique observational, interventional, and counterfactual distributions.The twin model is shown uniquely solvable, after which Theorem 3.6 supplies uniqueness of all three distribution types.Source: Section 8 · p. 67

Measure-theoretic foundations

• Standard measurable spaces are those isomorphic to Borel spaces of Polish spaces, with standard probability spaces additionally carrying a probability measure.Open or closed subsets of R^d and finite sets with their usual metric are examples.Source: Appendix F · p. 68
• Analytic sets are preserved under measurable images and preimages, are measurable under standard probability measures, and admit measurable inner and outer approximations.These properties support later measurable-selection arguments.Source: Appendix F · pp. 68–69

First measurable selection theorem

• The first selection theorem yields a measurable g:E→X selecting (e,g(e))∈S for P_E-almost every e when S is measurable and its projection misses only a null set.The proof combines analytic-set measurability, Jankov–von Neumann uniformization, and almost-everywhere replacement by a measurable map.Source: Appendix F · pp. 69–70

Second measurable selection theorem

• The first theorem can fail on product spaces without a probability measure on every factor, but σ-compact fibers restore selection uniformly over all x.The second theorem produces measurable g:X×E→Y for almost every e and every x under this fiber condition.Source: Appendix F · p. 70

Almost-everywhere quantifiers

• Additional lemmas transfer almost-everywhere properties through push-forward measures and formalize valid implications among almost-everywhere and universal quantifiers on product spaces.These include exchanging product coordinates, moving from almost-everywhere statements to existence claims, and deriving almost-everywhere uniformity over x.Source: Appendix F · pp. 71–72