The Causal-Neural Connection: Expressiveness, Learnability, and Inference

1 Introduction

• Causal analysis uses SCMs to represent mechanisms and exogenous variation, while the Pearl Causal Hierarchy organizes inference into strictly nested layers for seeing, doing, and imagining [5, Thm. 1].A fully specified SCM induces the hierarchy, whose layers correspond to distinct formal query languages [5, Def. 8; 5, Def. 9].Source: §1 · pp. 1–2
• The core inferential challenge is recovering higher-layer causal effects when the true SCM is only partially observed, especially when interventions must be inferred from passive observational data.The paper focuses on causal identification—whether a query is uniquely determined—and causal estimation after identifiability, extending beyond backdoor-form effects.Source: §1 · p. 2
• Neural universality does not ensure causal validity: models matching observational layer L1 can disagree on intervention layer L2 or counterfactual layer L3.Multiple neural models may fit the same observational distribution while inducing different higher-layer distributions, even when they are expressive enough to represent the true SCM.Source: §1 · p. 2, Figure 1 · p. 2
• The paper introduces neural causal models (NCMs) that are universally expressive, encode structural invariances as inductive biases, and support gradient-based causal inference.The framework targets both causal identification and estimation while preserving the Causal Hierarchy Theorem; the paper reports feasibility experiments for one implementation.Source: §1 · p. 3

1.1 Preliminaries

• An SCM is a tuple of exogenous variables, endogenous variables, structural functions, and an exogenous distribution; interventions replace mechanisms using the do-operator, defining layer L2 distributions.The preliminaries assume recursive SCMs with discrete finite-domain endogenous variables, while exogenous variables need not be independent.Source: §1.1 · pp. 3–4

2 Neural Causal Models and the Causal Hierarchy Theorem

• NCMs can perfectly represent the observational, interventional, and counterfactual distributions of any SCM, establishing equal expressive power despite their neural and uniform-noise restrictions.An NCM uses feedforward neural mechanisms, uniform exogenous sources, and shared sources to permit unobserved confounding; feedforward universality and the probability integral transform support this representation.Source: §2 · p. 4
• The Neural Causal Hierarchy Theorem shows that, except on a measure-zero subset, matching a lower causal-hierarchy layer does not determine any higher layer for NCMs.Thus, expressiveness alone does not make an NCM trained on observational data a valid proxy for the true SCM or enable cross-layer causal inference.Source: §2 · pp. 4–5

2.1 A Family of Neural-Interventional Constraints (Inductive Bias)

• A known causal diagram G supplies an inductive bias by constraining NCM confounding components to C2-components and neural parent sets to G’s directed edges.G-consistency means the candidate NCM imposes the same constraints on interventional distributions as the true SCM.Source: §2.1 · p. 5
• G-constrained NCMs form a parameterized family rather than a unique model, because the construction fixes function scopes and source independences but leaves neural parameters θ unspecified.The remaining degrees of freedom lie in θ, unlike general SCMs where both mechanisms and exogenous distributions can vary.Source: §2.1 · p. 5
• Every G-constrained NCM is G-consistent, and for any SCM inducing G, some G-constrained NCM is L2-consistent with it.Constraining the NCM family therefore preserves enough expressiveness to represent the true SCM’s entire Layer 2 while encoding G’s structural constraints.Source: §2.1 · p. 6, §A.2 · p. 22

Graphical-Neural Equivalence

• Theorem 4 shows that P(y | do(x)) is neural-identifiable from G-constrained NCMs and P(V) if and only if it is identifiable from G and P(V).Thus, neural models preserve the identification status of graphical causal queries while enabling inference without symbolic do-calculus.Source: §3 · p. 7

Operational Identification

• Under L1-consistency, G-constraint, and graphical identifiability, mutilating a proxy NCM computes the true intervention effect even when the true SCM is unknown.The proxy’s mechanisms and exogenous distribution need not match the true SCM, but violating these conditions generally invalidates proxy-based inference.Source: §3 · p. 7

Markovian Identification

• For Markovian G-constrained NCMs, every interventional distribution is identifiable through proxy-model mutilation.This follows because independent exogenous variables affect only one endogenous variable in the Markovian setting.Source: §3 · p. 7

Identification and Estimation

• Algorithm 1 minimizes and maximizes the intervention query over L1-consistent models; equal extrema identify and return the effect, whereas unequal values yield FAIL.The procedure is both necessary and sufficient for identification, making optimization-based neural inference theoretically as powerful as do-calculus for this decision.Source: §3 · p. 7

Model construction

• The proposed G-constrained NCM represents binary-variable causal systems with component-specific latent variables, Gumbel noise, and parent-conditioned neural mechanisms.The construction extends naturally to categorical and continuous variables, while its parameters must be learned to enforce L1-consistency.Source: §4 · p. 8

Probability estimation

• Causal and observational probabilities are estimated by computing intervention-consistent datapoint probabilities, using log-space evaluation and the log-sum-exp trick for numerical stability.Datapoints inconsistent with the intervention receive probability zero.Source: §4 · p. 8

Training objective

• With finite samples, the model parameters are learned by minimizing negative log-likelihood while adding a weighted term that maximizes the estimated interventional probability of the target outcome.The regularization weight λ is initially high and decreases during training.Source: §4 · p. 8

Optimization

• Algorithm 2 optimizes the training objective with standard gradient descent, maintaining parameterizations that minimize and maximize the interventional distribution.The procedure uses Monte Carlo samples and updates separate θmin and θmax parameterizations.Source: §4 · pp. 8–9

Experimental evaluation

• Figure 4 evaluates NCM-based identifiability decisions on four identifiable and four non-identifiable graph cases across training epochs and hypothesis-test thresholds.It reports classification accuracy and percentile summaries of max-min gaps over 3,000 training epochs.Source: Figure 4 · p. 9

Identifiability testing

• After optimization, comparing the corresponding P(y | do(x)) values determines identifiability: equality follows with perfect optimization and unbounded resources, whereas practice uses a thresholded hypothesis test.The threshold τ controls certain error types and can be selected empirically.Source: §4 · p. 9

Causal identification

• Across 8 SCMs, Alg. 2 determines whether P(Y | do(X)) is identifiable using the causal diagram and observational datasets, evaluating ATE identification.The target is ATEM(X, Y) = EM[Y | do(X = 1)] − EM[Y | do(X = 0)]; left-side Fig. 4 settings are identifiable and right-side settings are not.Source: §5 · p. 9
• Over 3000 epochs, max-min gaps shrink toward 0 for identifiable cases but rapidly grow and remain high for unidentifiable cases, enabling ID/non-ID classification.ID accuracy gradually increases during training, while non-ID accuracy remains high throughout and is perfect in the bow and IV cases.Source: §5 · pp. 9–10

Model extensions

• NCM training can use alternative discrepancy methods, including maximum mean discrepancy, kernelized Stein discrepancy, variational inference [9], or adversarial optimization, and can extend to continuous variables with normalizing flows [62] or VAEs [39].The continuous extension replaces the Gumbel-max trick with a model that directly computes a probability density.Source: §5 · p. 9

Causal estimation

• NCMs estimate the correct ATE in identifiable settings, whereas a naïve likelihood model fails except when P(y | do(x)) = P(y | x), while NCMs remain competitive with WERM.Both models improve at fitting P(V) as sample size increases, but only the NCM uses the causal inductive bias to generate identifiable interventional distributions.Source: §5 · p. 10

6 Conclusions

• NCMs are trainable SCMs that retain SCM-level expressivity but cannot generally perform cross-layer inferences without additional inductive bias.These results are stated in Def. 3, Thm. 1, and Corol. 1.Source: §6 · p. 10
• The proposed inductive bias uses non-parametric structural properties of the generating SCM to constrain interventional distributions while preserving full expressivity.A constructive procedure represents constraints analogous to causal diagrams, as established by Thms. 2 and 3.Source: §6 · p. 10
• With this bias, NCMs address canonical causal-inference tasks including causal identification and estimation.The capability is formalized in Thm. 4.Source: §6 · p. 10
• Algorithms 1 and 2, together with Cor. 4, provide a training procedure that is both sound and complete for the proposed framework.The framework supports different neural implementations combining architectures, training algorithms, and loss functions.Source: §6 · p. 10
• A proof-of-concept implementation produced experimental results supporting the feasibility of the proposed neural-causal approach.Figure 5 reports NCM estimation results for identifiable cases, comparing induced-distribution KL divergence and ATE MAE for naïve models, NCMs, and WERM.Source: §6 · p. 10, Figure 5 · p. 10
• The framework is intended to combine causal structural invariances for more explainable and generalizable decisions with deep-learning scalability for high-dimensional inference.This is presented as the broader motivation for future causal-neural AI systems.Source: §6 · p. 10

A Proofs

• The proofs section supplies proofs of the statements presented in the paper’s main body.It serves as the formal supporting material for the paper’s theoretical claims.Source: A Proofs · p. 15

• Every SCM has an L3-consistent canonical SCM, which represents SCM equivalence classes and remaps discrete exogenous variables to a discrete space.The canonical construction is the bridge from arbitrary SCMs to neural implementations.Source: §A.1 · pp. 15–16
• Binary-step MLPs express any function over finite numerical domains by encoding inputs and outputs as one-hot binary vectors and composing neural components.The construction first handles binary functions, then extends them to finite domains through encoding, computation, and decoding.Source: §A.1 · pp. 17–18
• An MLP can map Unif(0, 1) noise to any discrete probability mass function, supplying neural mechanisms for exogenous randomness.Threshold units partition the uniform input according to cumulative probabilities, after which a neural output function selects the corresponding support value.Source: §A.1 · pp. 18–19
• For any SCM M∗, there exists an NCM that is L3-consistent with M∗, because neural functions reproduce the canonical SCM’s exogenous mappings and structural mechanisms.The construction combines the discrete neural inverse probability integral transform with finite-domain function representation and matches the canonical model on all outputs.Source: §A.1 · pp. 19–20
• The Neural Causal Hierarchy Theorem shows that, except on a measure-zero subset, NCM layers do not collapse to lower layers, preserving the Causal Hierarchy Theorem’s limitations.Because every NCM is an SCM and every relevant SCM counterexample has an L3-consistent NCM representation, the collapse sets coincide.Source: §A.1 · p. 20

A.2 Proof of Theorem 2

• An SCM’s induced causal diagram and interventional distributions form a causal Bayesian network whose qualitative constraints are independent of the SCM’s specific mechanisms and exogenous distribution [5, Thm. 4].The graphical treatment incorporates structural constraints and extends them to latent-variable models [57, 5, Sec.1. 4].Source: §A.2 · pp. 20–21
• For any structural causal model M, its induced causal diagram and interventional distributions form a causal Bayesian network, with G serving as a CBN for L2(M).Source: §A.2 · p. 21
• Lemma 6 proves that any G-constrained NCM induces exactly G: directed edges match encoded parent sets, while bidirected edges match shared exogenous variables for each confounded component.Independent exogenous variables prevent bidirected edges except those created by shared exogenous variables, which are constructed according to G’s C-components.Source: §A.2 · pp. 21–22

Reduction

• Using Zhang et al., the proof replaces any SCM with a G-consistent discrete SCM having finite exogenous domains and matching it through L3-consistency, hence L2-consistency.The construction assumes independent exogenous variables, with unobserved confounding represented by shared exogenous variables.Source: §A.3 · p. 22

Theorem 3

• Theorem 3 states that every SCM inducing G has a G-constrained NCM that is L2-consistent with it.This is the paper’s counterfactual-level expressiveness result.Source: §A.3 · p. 22

Neural construction

• The constructed G-NCM assigns one Uniform(0,1) exogenous variable to each C2-component and uses MLPs to map these variables to the discrete SCM’s exogenous variables and then to endogenous variables.The neural functions are composed of smaller MLPs and respect the graph constraint.Source: §A.3 · pp. 22–23

Interventional matching

• For every interventional query Q = P(y | do(x)), the resulting NCM matches the original SCM’s interventional distribution, proving L2 consistency.The proof concludes that the NCM matches the SCM on the second layer of the causal hierarchy.Source: §A.3 · p. 23

Theorem 4

• Theorem 4 shows that Q is identifiable from G and P(V) if and only if it is neural identifiable from Ω(G) and P(V).The proof uses both inclusions: neural models cannot expand identifiable effects, while Theorem 3 constructs neural counterexamples whenever graphical identification fails.Source: §A.4 · p. 23

Corollary 2

• Corollary 2 shows that an identifiable causal query can be computed by mutilating any G-constrained NCM consistent with the observational distribution.The result replaces each intervention variable’s structural equation with the intervention constant and recovers the true effect.Source: §A.4 · p. 24

Corollary 4

• Corollary 4 proves Algorithm 1 is sound and complete: it returns FAIL exactly when Q is nonidentifiable, otherwise returning the true causal effect.The proof reduces identifiability to agreement among all observationally matching, graph-consistent NCMs and uses a disagreement as a counterexample.Source: §A.4 · p. 24

A.5 Proof of Corollary 3

• Markovianity requires independent, nonshared exogenous variables, corresponding to NCMs without bidirected edges or unobserved confounding.This is a strong constraint, and without it even two-variable queries can be non-identifiable.Source: §A.5 · pp. 24–25
• In Markovian models, every interventional distribution P(Y | do(x)) is identifiable via the backdoor adjustment P(Y | x, z)P(z), where Z contains X’s non-descendants [5].The result extends to NCMs through mutilation because neural and graph identification are connected.Source: §A.5 · p. 24
• Corollary 3 states that Markovian G-constrained NCMs always identify P(y | do(x)) by mutilating the proxy NCM via Corollary 2.The proof transfers graph-level identifiability from G and P(v) to the proxy representation Ω(G) and P(v).Source: §A.5 · pp. 24–25

A.6 Derivation of Results in Section 4

• The NCM architecture represents latent C2-component variables and Gumbel random variables, enabling computation of datapoint probability masses from samples of P(v).The architecture uses V as the observed-variable set and bU as the collection of latent component and Gumbel variables.Source: §A.6 · p. 25
• The derivation constructs Monte Carlo estimators for observational probabilities and interventional queries P(y|do(x)) using sampled latent variables.Interventional estimation uses a submodel of the NCM under do(X = x).Source: §A.6 · pp. 25–26
• Algorithm 3 implements Eq. 3’s estimator for P^cM(v | do(x)) from an NCM, topologically ordered variables, an intervention, and m Monte Carlo samples.It returns 0 when the queried assignment is inconsistent with the intervention.Source: §A.6 · p. 26

B Experimental Details

• The experiments use publicly released code, primarily PyTorch models, and PyTorch Lightning for training.The implementation is available at https://github.com/CausalAILab/NeuralCausalModels.Source: B Experimental Details · p. 26

B.1 Data Generation Process

• Randomly parameterized canonical SCMs generate expressive data distributions consistent with each graph’s constraints, with additional optimization enforcing challenging confounding cases for estimation.The canonical SCM is implemented in PyTorch, and estimation models are adjusted until the ATE–TV difference reaches at least 0.05 when needed.Source: §B.1 · p. 27

B.2 Identification Running Procedure

• Identification experiments evaluate eight graphs using repeated trials, 10,000 binary observational samples, and paired NCM minimization and maximization training to track max-min gaps.The models train for 3000 epochs while λ decreases logarithmically from 1 to 0.001; gaps are averaged over four runs and evaluated across three τ thresholds.Source: §B.2 · p. 27

B.3 Estimation Running Procedure

• Estimation experiments compare NCMs, naïve likelihood maximization, and WERM across four identifiable graphs and fifteen logarithmically spaced sample sizes from 10^3 to 10^6.They report KL divergence and ATE means with 95%-confidence intervals over 25 trials; Fig. 5 places KL results in the top row and ATE results in the bottom row.Source: §B.3 · pp. 27–28

B.4 Identification Hypothesis Testing

• The identification test rejects non-identifiability when the estimated mean max-min gap is below τ at 95% confidence, while preliminary gap distributions motivate τ = 0.03 because ID and nonID cases separate around that threshold.Fig. 7 shows most ID gaps below 0.03 by the end of training, whereas most nonID gaps remain above it.Source: §B.4 · p. 29, Figure 7 · p. 29

B.5 Additional Results

• In higher-dimensional experiments, mapping non-X/Y binary covariates to recoverable 20-dimensional vectors preserves ATE and TV while making optimization harder, yet NCM performance remains superior or equal to WERM.The corresponding results are shown in Fig. 8 for one million observed samples.Source: §B.5 · p. 29, Figure 8 · p. 29

B.6 Other Possible Architectures

• The NCM framework is not tied to one architecture: divergence minimization, variational inference, adversarial optimization, and density-based continuous-domain models are possible alternatives, though optimization errors can impair identification and estimation.Perfect optimization guarantees correctness, but larger domains may require architectures that are more scalable and robust to optimization error.Source: §B.6 · p. 30

C Examples and Further Discussion

• The examples explain why expressiveness alone cannot support cross-layer inference, how causal diagrams encode structural constraints as inductive bias, how NCMs incorporate that bias for identification, and why neural ID may be preferred to do-calculus.This section complements the main paper through discussions in Secs. C.1–C.4.Source: C Examples and Further Discussion · p. 30

C.1 Expressiveness Examples

• An NCM can exactly match a complex SCM’s distributions across all three layers of the probability-of-causality hierarchy.The construction reproduces the SCM’s observational, interventional, and counterfactual behavior, demonstrating expressiveness beyond layer-1 queries.Source: §C.1 · p. 33
• Expressiveness alone cannot identify causal effects because multiple NCMs can fit the same observational distribution while disagreeing on interventions.The candidate models induce identical layer-1 data but yield different values for P(B = 1 | do(D = 1)), making arbitrary model selection potentially misleading.Source: §C.1 · p. 35
• The observationally equivalent models support sharply different substantive conclusions, ranging from beneficial to null or harmful effects of a high-vegetable diet.Models c M1 or c M4 imply benefit, c M2 implies no effect, and c M3 implies harm.Source: §C.1 · p. 35

C.2 Structural Constraints Embedded in Causal Bayesian Networks

• The ambiguity arises because the two SCMs in Example 3 induce the same layer-1 distributions despite differing causal mechanisms.Their structural functions and exogenous influences differ, so an intervention such as P(Y = 1 | do(X = 1)) can differ even when observational quantities match.Source: §C.2 · p. 36
• Causal diagrams encode equality constraints determined by functional arguments and shared exogenous influences, independently of specific functions or noise distributions.These structural constraints span observational and experimental distributions rather than only conditional independences.Source: §C.2 · p. 36
• A causal Bayesian network parsimoniously represents exponentially many such constraints, whose closure entails do-calculus and supports computation of all identifiable effects.The result follows from the stated completeness of do-calculus for identifiable effects.Source: §C.2 · p. 36
• Embedding diagram-derived constraints into an NCM supplies an inductive bias that automatically enforces the corresponding causal structure during inference.For Example 2, the diagram excludes c M2 because B cannot be an argument of fD.Source: §C.2 · pp. 36–37

C.3 Solving Identification through NCMs

• Algorithm 1 returns “FAIL” when compatible NCMs disagree on a causal effect and otherwise estimates the query, without requiring the NCM to match the true SCM’s functions.Identification requires only agreement on the effect across compatible models, not functional equivalence with the true SCM.Source: §C.3 · p. 37
• In the diet example, three G-compatible NCMs match the observational distribution but disagree on Q, whereas adding sodium makes every compatible NCM agree on Q.Without sodium, the query is non-identifiable; with the refined model, compatible NCMs recover the same interventional result.Source: §C.3 · pp. 37–38
• NCM expressivity enables correct identification even when the true model is any SCM, whereas less expressive classes can falsely declare effects identifiable.The NCM construction can match observational data while also recovering cross-layer effects through its structural constraints.Source: §C.3 · p. 40
• Markovian models infer 0.375 for P(B = 1 | do(D = 1)) instead of the true 0.46875 because their model class cannot represent the non-Markovian SCM.A second restricted SCM class likewise forces P(Y | do(X)) to fixed expressions despite the query being non-identifiable in the true model.Source: §C.3 · pp. 40–42

C.4 Symbolic versus Optimization-based Approaches for Identification

• Symbolic methods derive closed-form effects from diagram constraints, whereas optimization-based methods search structural models to maximize or minimize the query under data constraints; Theorem 2 establishes shared identification status.Symbolic approaches avoid directly searching the infinite-dimensional SCM space, while optimization-based approaches place that space at the center of analysis.Source: §C.4 · pp. 43–44
• Optimization-based identification searches SCMs matching the observational distribution and causal constraints, using agreement or disagreement in their interventional predictions to assess identifiability.For non-identifiable effects, distinct SCMs can generate identical observations but different interventions; approximate optimization makes the resulting statements probabilistic rather than deterministic.Source: §C.4 · p. 45, Figure 17 · p. 45
• Identification succeeds when assumptions determine the target interventional distribution; otherwise, stronger assumptions are required to make any statement about the true SCM.Symbolic methods use graph structure, whereas optimization-based methods use function-scope and exogenous-independence constraints.Source: §C.4 · p. 45
• Unlike symbolic pipelines that separate identification from estimation, the proposed optimization framework performs both tasks in an integrated manner but may require substantial samples and computation for accurate training.Its performance depends on both causal assumptions and optimization quality because the procedure is sampling-based.Source: §C.4 · p. 45
• The hybrid alternative first uses a deterministic symbolic algorithm for identification and then an NCM for estimation, combining probability-one correctness with neural scalability in complex settings.The procedure is specified as Alg. 4 in Fig. 18, contrasted with the more purely neural Alg. 1.Source: §C.4 · pp. 45–46, Figure 18 · p. 46
• The work claims that neural networks and proxy SCMs can perform causal reasoning for the ID problem, providing a complete neural-framework solution beyond their established use for estimation.In the hybrid approach, causal reasoning remains with symbolic identification while neural methods perform estimation.Source: §C.4 · p. 46

D Generalizations

• Proxy SCMs can generate effectively unlimited data, estimate multiple queries after mutilation without retraining, retain interpretable SCM semantics, and simplify extensions to other identification settings.Potential extensions include identifying other query types and fusing information across sources or experimental conditions.Source: §C.4 · p. 46

General NCM Definition

• NCMs can use alternative function and noise types when they satisfy the framework’s required properties, so the uniform-noise MLP construction is an implementation choice rather than a limitation.The general case requires each exogenous variable to have an absolutely continuous density and the function class to support the stated parameterization property.Source: §D.1 · p. 47

NCM Expressiveness

• For any SCM, there exists a general NCM that is L3-consistent with it, establishing expressiveness beyond the specific uniform-noise MLP instantiation.The proof uses the probability integral transform to convert admissible exogenous distributions to uniform noise and constructs functions matching a canonical SCM.Source: §D.1 · p. 47

G-Constrained NCM

• For any SCM inducing causal diagram G, a G-constrained NCM exists that is L2-consistent with the SCM.The construction assigns independent exogenous distributions to C2-components, shares each component’s noise across its variables, and uses exactly the directed parents in G.Source: §D.1 · p. 48

Proof Extensions

• The remaining NCM results, including the stated theorem and corollary extensions, require only minimal changes to their original proofs.The section explicitly includes Corollaries 1–4 and Theorems 2 and 4 among the results carried over.Source: §D.1 · p. 48

Expressiveness across the causal hierarchy

• NCMs are L3-expressive, but expressiveness beyond Layer 1 is nontrivial because higher causal-hierarchy layers contain substantially more possible SCM distinctions.Theorem 1 establishes NCM L3-expressiveness, while even constructing an NCM that matches another SCM across all three layers can be complicated.Source: §D.2 · pp. 48–49

Other neural model classes

• Many neural model classes are L1-expressive: variational models with normalizing flows [62] are generative and proven able to fit any data distribution, while GANs are believed to share this property.Source: §D.2 · p. 49

Causal models under restricted assumptions

• Causal neural models that estimate effects under backdoor or conditional ignorability conditions [70, 64, 58, Sec. 3.3.1] are not generally L2- or L3-expressive outside those settings.Much existing work estimates identifiable effects from P(V), leaving causal reasoning—whether Q = P(Y|do(X = x)) equals some function f(P(V))—outside the network.Source: §D.2 · p. 49

L2-expression wrapper

• An L2-expression wrapper is L2-expressive by fitting a generative model for every Layer 2 distribution, but the required number of distributions grows exponentially with variables and is infinite for continuous cases.For any SCM, the wrapper can assign a fitted model to each L2 distribution and answer an L2 query by returning the corresponding distribution.Source: §D.2 · p. 50

Structural constraints and practicality

• The wrapper does not automatically enforce causal-graph constraints, so jointly fitting all desired distributions can violate the true interventional distribution and become infeasible even with three binary variables.Maintaining the numerous constraints induced by the causal graph during optimization is the central practical difficulty, whereas NCMs incorporate structural constraints while achieving L3-expressiveness.Source: §D.2 · p. 50

Identifiability and learnability

• Observationally matching an SCM does not ensure correct interventional distributions, so universal approximators cannot generally recover causal effects from observational data alone.A learned model may reproduce observational data perfectly while differing from the true SCM on interventions, making causal validity unverifiable when the true SCM is unobserved.Source: E Frequently Asked Questions · p. 51

NCM interpretation

• NCMs are trainable proxy models for SCMs, but their higher-layer distributions remain underdetermined when trained only on layer 1 data and become useful only under appropriate consistency and identifiability conditions.NCMs are not causal diagrams; with G-consistency, they can generate interventional distributions contingent on L1-consistency and query identifiability.Source: E Frequently Asked Questions · p. 52

Assumptions

• The framework assumes a causal diagram because cross-layer inference and complete diagram recovery are generally impossible from observational data alone, although partial diagrams or additional structural assumptions may suffice.The work assumes recursive ground-truth SCMs, while the generalized NCM definition need not impose acyclicity a priori.Source: E Frequently Asked Questions · p. 52

Expressiveness and identifiability

• Theorem 3 shows G-constrained NCMs retain maximal expressiveness for SCMs compatible with G, despite enforcing causal-diagram constraints that generic universal approximators need not preserve.This expressiveness connects NCM-based identifiability to SCM-based identifiability and supports using NCMs as proxies for the true model.Source: E Frequently Asked Questions · pp. 52–53

Optimization

• The optimization objective balances learning P(V) against optimizing the target interventional query through λ, starting large for query optimization and decreasing so observational fitting is accurate by convergence.Too-small λ can impair optimization in non-identifiable cases, whereas too-large λ can compromise P(V) and produce incorrect identifiable-case inferences.Source: E Frequently Asked Questions · p. 53

Practical scope and limitations

• NCMs provide a unified optimization-based pipeline for identification and estimation, can generate layer 1 and identifiable layer 2 distributions, and are empirically intended to work when training error is sufficiently minimized.The paper makes no formal robustness claims for training error and leaves refined robustness analysis for future work.Source: E Frequently Asked Questions · pp. 53–54