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Causal inference using invariant prediction: identification and confidence intervals

Jonas Peters, Peter Bühlmann, Nicolai Meinshausen

arXiv:1501.01332v3stat.MEcs.LG

TL;DR

Causal discovery needs confidence statements for selected causal predictors, but post-selection inference is difficult. This paper uses invariance of predictive relationships across environments to identify causal predictors and construct confidence sets, establishing identifiability under assumptions for Gaussian structural equation models with interventions.

  • Problem

    Existing causal discovery methods often lack tractable confidence statements after searching and estimating a graphical structure.

    Source: §1.2 · p. 4

  • Method

    The method identifies predictor sets whose conditional target distribution remains invariant across environments or interventions, using this invariance to construct confidence sets and intervals.

    Source: §1 · p. 2, §1.1 · p. 3, §8 · p. 36

  • Results

    Under sufficient assumptions, the causal predictors are identifiable in linear Gaussian structural equation models with interventions, and the approach yields potentially conservative confidence statements.

    Source: §4.3 · p. 19, §8 · p. 36

  • Takeaways & Limitations

    Invariant prediction provides a framework for causal inference that links environmental stability to identifying causal predictors and quantifying uncertainty.

    Source: §8 · pp. 36–37

    The procedure assumes Gaussian errors and is not straightforward to extend to classification and nonlinear models.

    Source: §3.1 · p. 13

Abstract

from arXiv · show

What is the difference of a prediction that is made with a causal model and a non-causal model? Suppose we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables. Here, we propose to exploit this invariance of a prediction under a causal model for causal inference: given different experimental settings (for example various interventions) we collect all models that do show invariance in their predictive accuracy across settings and interventions. The causal model will be a member of this set of models with high probability. This approach yields valid confidence intervals for the causal relationships in quite general scenarios. We examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable. We further investigate robustness properties of our approach under model misspecification and discuss possible extensions. The empirical properties are studied for various data sets, including large-scale gene perturbation experiments.

1 Introduction

The paper introduces invariant prediction for causal discovery: direct-cause predictors yield conditional distributions that remain unchanged across suitable experimental settings, enabling statistically controlled causal inference. It develops confidence statements and intervals without requiring known intervention mechanisms, causal ordering, faithfulness, or other identifiability assumptions.

1 Introduction

  • Causal inference is framed through potential outcomes, structural equation models, and graphical models, with existing discovery methods often identifying Markov equivalence classes before estimating causal effects.The introduction situates the method within established causal-inference frameworks and related causal-discovery approaches.Source: §1 · pp. 1–2
  • Across environments, the conditional distribution of a target given all its direct causes remains invariant when interventions affect other variables but not the target.Omitting direct causes can destroy invariance, whereas the complete direct-cause set is an invariant submodel with controlled high probability.Source: §1 · p. 2, Figure 1 · p. 3, §1.1 · p. 4

1.1 Data from multiple environments or experimental settings

  • The framework uses i.i.d. samples from multiple environments and assumes a common prediction function and error distribution, equivalently identical target conditionals given the causal predictors.The environments may arise from unknown or uncontrolled interventions, and intervention targets need not be known.Source: §1 · p. 2, §1.1 · p. 3

1.2 New contribution

  • The proposed method exploits invariance for causal discovery, providing confidence statements for causal predictors and intervals for effects without assuming causal ordering, faithfulness, or other identifiability conditions.If an effect is not identifiable from the data, the procedure is designed to detect this rather than make false causal discoveries.Source: §1.2 · p. 4

1.3 Organization

  • The paper develops invariance assumptions, confidence procedures, structural-equation identifiability results, hidden-variable and instrumental-variable analyses, nonlinear and robustness extensions, and simulations and applications.Its software is released in the R package InvariantCausalPrediction.Source: §1.3 · p. 5, §1.4 · p. 5

2 Assumed invariance of causal prediction

The section formalizes invariant prediction across experimental settings and connects it to causal predictors, while distinguishing plausible from identifiable predictors and coefficients. Identifiability strengthens with additional or stronger interventions, but observational data alone cannot identify causal variables.

2 Assumed invariance of causal prediction

  • An invariant linear model requires a support S* and coefficients γ* such that Y^e = X^e_S*γ* + ε^e, with mean-zero noise having the same distribution across environments.The noise distribution need not be known, and an intercept is included by default.Source: §2 · p. 5
  • In a linear SEM, the parents of Y satisfy the invariance assumption under interventions on non-target variables, including do- and soft-interventions, provided the stated noise-independence condition holds.More general interventions are allowed when the conditional distribution of Y given the invariant predictors remains unchanged; interventions on Y are excluded.Source: §2 · pp. 7–8

2.1 Plausible causal predictors and identifiable causal predictors

  • The invariant set need not be unique: the method is valid for any set satisfying the assumption, but generally identifies only predictors common to all invariant solutions.With one observational environment, many supports can satisfy invariance, whereas complete identifiability requires additional conditions.Source: §2 · pp. 7–8, Definition 1 (Plausible causal… · p. 9

Definition 1 (Plausible causal predictors and coefficients)

  • Identifiable causal predictors are a subset of plausible predictors, expand monotonically as environments are added, and may coincide with all parents of Y under suitable SEM conditions.For a single environment, the identifiable set is empty, so no causal variables are identified.Source: Definition 1 (Plausible causal… · p. 9

2.2 Plausible causal coefficients

  • Plausible causal coefficients are regression vectors satisfying invariant residual behavior across environments; their global set shrinks when stronger or more numerous interventions are added.For a given predictor set, the plausible coefficient set is either empty or contains only the population regression vector.Source: §2.2 · pp. 9–10

3 Estimation of identifiable causal predictors

The procedure tests invariance across predictor subsets, pools accepted models to form causal-coefficient confidence sets, and guarantees conservative coverage under valid tests. Concrete tests, pooling choices, observational splitting, and computational shortcuts determine its applicability and efficiency.

3 Estimation of identifiable causal predictors

  • Valid invariant-prediction tests yield an estimator that excludes non-causal predictors with probability at least 1 −α and confidence sets with coverage at least 1 −2α.The guarantees hold for any experimental or intervention settings, although the estimated predictor set may be empty; in linear structural equation models they apply to causal coefficients and parent variables.Source: §3 · p. 11

3.1 Two concrete proposals

  • Method I tests invariant regression coefficients and residual variance across settings, whereas faster Method II fits one global model and compares residual means and variances.Method I assumes Gaussian errors and can be computationally slow; Method II ignores coefficient-estimation variability but extends more readily to nonlinear and logistic models.Source: §3.1 · pp. 12–14

3.2 Data pooling

  • Pooling is valid when the conditional distribution given the true causal predictors is invariant, but trades statistical power against identifiability as fewer settings remain.Richer experimental settings shrink the plausible coefficient set and enlarge the identifiable predictor set; pooling reduces the number of identifiable causal variables.Source: §3.2 · pp. 14–15, identifiable causal predictors · p. 15

3.3 Splitting purely observational data

  • With purely observational data, the method conservatively returns no causal predictors, while splitting by a non-descendant variable preserves confidence-interval validity but leaves predictor identifiability unresolved.The method does not exploit other directional information such as non-Gaussianity, nonlinearities, or conditional-independence structure.Source: §3.3 · pp. 15–16

3.4 Computational requirements

  • Brute-force search is super-exponential, but restricting causal-set size gives O(p^s) complexity, while screening to q variables reduces it to O(q^s) with coverage at least 1 −2α.Early stopping can terminate searches when the empty set is accepted or accepted sets have no common overlap.Source: §3.4 · p. 16

4 Identifiability results for structural equation models

For linear Gaussian structural equation models, invariant prediction identifies causal predictors under three intervention schemes, with sufficient conditions detailed in Theorem 2. A generic single intervention can also suffice, while counterexamples show why some conditions matter.

4.1 Linear Gaussian SEMs

  • Under linear Gaussian SEMs, the target is Y:=X1, all variables are observed, the graph is acyclic, and causal effects remain invariant across interventions.The observational block is a linear Gaussian SEM, while other blocks may change mechanisms for non-target variables.Source: §4.1 · p. 17

4.2 Interventions

  • Do-interventions fix selected predictors to specified values, whereas noise interventions alter their error distributions through scaling or shifting.These are contrasted as hard and softer intervention mechanisms.Source: §4.2.1 · p. 18, §4.2.2 · p. 18

4.2.3 Simultaneous noise interventions

  • Simultaneous noise interventions pool experiments into one interventional dataset, allowing error distributions to change at many variables and accommodating off-target effects.This setting uses two environments and includes gene knockout experiments as an example.Source: §4.2.3 · pp. 18–19

4.3 Identifiability results

  • Theorem 2 establishes identifiability of all causal predictors under do-interventions, noise interventions, or simultaneous noise interventions satisfying specified conditions.The result applies to linear Gaussian SEMs and remains valid under the weaker null hypothesis.Source: §4.2 · p. 17, §4.3 · p. 19
  • Theorem 2 requires at least one single intervention on every non-target variable for do- and ordinary noise interventions, while simultaneous noise interventions form a separate sufficient case.Noise interventions require nontrivial scaling or shifting conditions.Source: §4.3 · p. 19
  • Counterexamples show that omitted intervention-value conditions or unit noise-scaling factors can prevent identification, while descendants provide limited information and independent variables provide none.The assumptions are sufficient rather than necessary, and stronger results may hold under faithfulness or genericity.Source: §4.3 · pp. 19–20
  • Theorem 3 shows that, generically, one intervention on a youngest parent identifies all causal predictors with probability one under either a do- or noise intervention.The intervened parent need not be known, and the result extends when that setting is part of a larger collection.Source: §4.3 · pp. 20–21

5 Instrumental and hidden variables with confounding

The invariance approach extends causal inference to hidden-variable settings by testing cross-environment residual-distribution invariance rather than residual independence. It guarantees coverage under weaker conditions than classical instrumental-variable identification and can identify causal variables in specific variance-shift models.

5 Instrumental and hidden variables with confounding

  • With a univariate instrument I, environments can be defined by I=0 and I=1, provided the split is not based on Y, its descendants, or hidden variables.This reframes the instrument’s two values as experimental environments for invariant prediction.Source: §5 · p. 22
  • Because hidden variables prevent residual independence from causal predictors, testing replaces the usual null with invariance of Y^e−X^eγ’s distribution across environments.Testing this hidden-variable null is computationally harder because γ cannot be obtained by regressing Y^e on X^e.Source: §5 · pp. 22–23
  • The procedure guarantees coverage for hidden-confounding model (23), even when classical two-stage instrumental-variable identification assumptions are violated.The causal residual equals g(H^e, ε^e), whose distribution remains invariant because I is independent of H and ε.Source: §5 · p. 23
  • Under a full-rank variance shift in X between I=0 and I=1, γ∗ and the causal predictor set S∗ are identifiable in the population limit.This remains true when X has arbitrary dimension and E(X|I) does not shift, violating classical instrumental-variable conditions (ii) and (iii).Source: §5 · pp. 23–24
  • Unlike conventional instrumental-variable regression, the invariance approach needs no weak-instrument test; weak influence produces wider confidence intervals and smaller selected causal-variable sets.Conventional weak instruments can amplify bias, whereas this procedure reflects weak influence through uncertainty.Source: §5 · p. 24
  • The general hidden-variable procedure is computationally demanding because it tests the null for every predictor set and may require exponentially costly grid searches over γ.More efficient tests are needed, although stricter intervention structures can provide alternative implementations.Source: §5 · p. 24

6 Further extensions and model misspecification

The approach extends to nonlinear structural equation models and sparse interventions on the target, while retaining coverage guarantees under specified conditions. Under model misspecification, coverage can persist, but the method may lose power or select alternative invariant sets involving only ancestors.

6.1 Nonlinear models

  • In nonlinear structural equation models, Propositions 1 and 4 remain valid when an invariant causal set S∗ exists.The nonlinear model has the form Y_j = f_j(X_PA(j), ε_j), and the assumption is that some S∗ satisfies the nonlinear invariance condition.Source: §6.1 · p. 25
  • Testing all candidate sets with the nonlinear invariance null hypothesis yields a confidence set satisfying P(Ŝ(E) ⊆ S∗) ≥ 1 − α.This guarantee holds for any S∗ satisfying the nonlinear invariance condition when individual tests are conducted at level α.Source: §6.1 · p. 25
  • Nonlinear invariance tests can use environment-specific regression comparisons, pooled residual-distribution tests, or conditional independence tests Y ⟂ E | X_S.Regression-function and residual-distribution approaches are especially direct under additive noise; conditional independence tests avoid requiring noise additivity.Source: §6.1 · p. 25

6.2 Interventions on the target variable and its causal mechanism

  • When target interventions may occur in at most V ≪ |E| environments, invariance can be weakened to require consistency across some subset E′ with |E′| ≥ |E| − V.The weaker null is tested by examining all sufficiently large environment subsets and rejecting only when every such subset is rejected.Source: §6.2 · p. 26

6.3 Model misspecification

  • The original method retains P(Ŝ(E) ⊆ S∗) ≥ 1 − α for any set satisfying the invariance assumption, even under model misspecification.If the considered causal set still satisfies the null despite violated assumptions, the desired confidence statement remains valid.Source: §6.3 · p. 26
  • If misspecification makes every candidate null false, the method returns Ŝ(E) = ∅ and preserves coverage but has no power to detect causal variables.Alternatively, if another invariant set exists, under additional assumptions the selected set can be replaced by one containing only target ancestors and no descendants.Source: §6.3 · p. 26

7 Numerical results

The method is evaluated on simulations, yeast gene perturbation data, and educational-attainment data. Across these applications, invariant prediction provides controlled error rates, identifies causal effects, and yields interpretable confidence intervals under stated modeling conditions.

7.1 Simulation experiments

  • Across 100 simulated scenarios, exact recovery was best for invariant prediction in 23, compared with 54 for LiNGAM, 14 for GIES with known interventions, and 3 for GES.Regression and marginal regression were dominated across all scenarios by GIES, LiNGAM, and invariant prediction; performance differences showed no clear dependence on any single parameter.Source: §7.1 · pp. 28, 31, Figure 4 · p. 29
  • The proposed invariant prediction controls false selections at 0.05 or below, whereas all other methods reach FWE rates of 0.4 or higher in simulations.No other compared method offers FWER control, making this the key simulation advantage.Source: Figure 5 · p. 30, §7.1 · p. 31

7.2 Gene perturbation experiments

  • In yeast perturbation experiments, invariant prediction identified six correct causal pairs among eight significant predictions, versus 9.2% expected success from random guessing.The eight predictions were significant at level 0.01 after Bonferroni correction, while alternative methods did not exceed random guessing among their eight highest-ranked pairs.Source: §7.2 · p. 34, Table 1 · p. 34
  • Sensitivity analyses found true-positive rates above random guessing across implementation choices, including 0.63 for Method II with Lasso preselection and 0.75–1.00 for selected Method I variants.Changing preselection, goodness-of-fit thresholds, or subsampling altered performance; using 500 rather than 1000 Method I subsamples increased speed but worsened performance.Source: §7.2 · p. 34

7.3 Educational attainment

  • In educational attainment data, 90% confidence intervals excluded zero only for test score, with a positive effect, and father lacking a college degree, with a negative effect.The analysis split pupils by median distance to a four-year college and required the conditional distribution of attainment given covariates to be invariant across groups.Source: Figure 7 · p. 35, §7.3 · p. 36
  • The educational example illustrates that instrumental-variable-type variables can create experimental groups for invariant causal prediction, although ground-truth validation and comparisons were not attempted.The distance variable was used only to split observations and was assumed independent of outcome noise and a non-descendant of the target.Source: §7.3 · p. 36

8 Discussion and Future Work · A An Example

The paper uses invariance across environmental changes to identify causal predictors and derive confidence sets, while establishing identifiability guarantees in Gaussian structural equation models with interventions. It frames invariant prediction as a new perspective on causal inference and identifies extensions involving nonlinear, feedback, hidden-variable, and computational challenges.

8 Discussion and Future Work

  • Causal predictors retain invariant influence on targets across environmental changes, unlike non-causal predictors, enabling their identification.The framework naturally yields confidence sets for causal predictors and confidence intervals for relevant parameters.Source: §8 · p. 36
  • Gaussian structural equation models with interventions admit identifiability guarantees for the set of causal predictors.Source: §8 · p. 37
  • Future work should develop tests for equality of conditional distributions in nonlinear models.Source: §8 · p. 37
  • Future work should address feedback models and improve computational efficiency with and without hidden variables.Source: §8 · p. 37
  • Invariant prediction introduces new causal-inference concepts and methods by viewing known concepts from a different angle.The approach offers a new understanding of causality and opens a novel class of theory and methodology.Source: §8 · p. 37

A An Example

  • Figure 8 illustrates the developed concepts and methodology using two environments generated from observational and interventional structural equation models.Source: A An Example · p. 42

B Hidden variables without confounding

This section extends invariant prediction to linear structural equation models with hidden variables, while initially excluding confounding between observable causal variables and the target. It gives conditions under which the observable direct causal variables satisfy the invariance assumption, including settings with hidden variables that directly affect the target.

Model setup

  • The generalization considers linear structural equation models with observed variables, target Y, and mean-zero hidden variables, using a directed acyclic graph.The initial setting excludes confounding between observable causal variables and Y; confounding is treated separately in Section 5.Source: B Hidden variables without confounding · p. 42

No hidden effects on Y

  • When hidden variables have no direct effect on Y, the observable direct causal variables satisfy the invariance assumption under do- or soft-interventions on observed predictors and hidden variables.The condition is that the hidden variables have no direct causal effects on Y.Source: B Hidden variables without confounding · p. 44

Hidden effects on Y

  • Hidden variables may directly affect Y if the experiments are do- or soft-interventions, avoid Y and the hidden variables or ancestors of the causal predictors, and have no d-connecting path between causal predictors and hidden variables.Under these conditions, the relevant residual distribution is invariant across environments and has mean zero.Source: B Hidden variables without confounding · p. 44

Proof

  • The invariance conditions follow from the structural equations and intervention assumptions, with the relevant distribution unchanged across environments and the residual independent of the predictors.The proof invokes the intervention definitions and conditions (a)–(c) to establish the required invariance relation.Source: B Hidden variables without confounding · p. 44

C Model Misspecification

Under model misspecification, the invariant predictor set S(E) need not contain only direct causes of Y, but under faithfulness it generally remains within Y’s ancestors. Some misspecifications yield no invariant set, especially when Y is directly intervened on.

C Model Misspecification

  • Under model misspecification, S(E) may include indirect causes but is generally a subset of the ancestors of Y under the proposition’s faithfulness assumptions.The proposition characterizes invariant sets through conditional independence between Y and the environment variable E given the selected predictors.Source: C Model Misspecification · pp. 44–45
  • Figure 9 shows S(E) = {X1} despite S0 = {X2}, demonstrating that misspecification can violate direct-cause containment while preserving ancestral containment.In the example, S(E) is not a subset of S0 but remains a subset of AN(Y).Source: Figure 9 · p. 45
  • For some misspecifications, including direct intervention on Y, no predictor set S satisfies the invariance condition, so S(E) may be empty.The proof relates the condition to Y ⊥⊥ E | X_S and shows that direct intervention on Y prevents such a set from existing.Source: C Model Misspecification · p. 45

D Potential Outcomes and Invariant Prediction

In a potential-outcome framework, invariant prediction holds when the target Y is not intervened upon: its conditional distribution given causal predictors remains invariant across environments. Under FFR-CISTG assumptions, the parents of Y therefore satisfy the nonlinear invariant-prediction condition, with some relaxation possible for hidden variables.

Potential-outcome formulation

  • Without intervention on Y, the conditional distribution of Y given its causal predictors remains invariant across environments in the potential-outcome framework.The section adopts counterfactual language based on FFR-CISTG graphs and considers the nonlinear invariant-prediction formulation.Source: D Potential Outcomes and Invariant… · p. 46

Invariant prediction result

  • Under consistency with recursive substitution and FFR-CISTG independence, the conditional distributions Y(˜r_e) | P(˜r_e) are identical across environments.Here, P denotes the parents of Y, and treatment variables may differ by environment.Source: D Potential Outcomes and Invariant… · p. 46
  • Consequently, the parent set P satisfies the nonlinear invariant-prediction condition (29).The equality is understood in distribution for all environments and quantiles where both sides are well-defined.Source: D Potential Outcomes and Invariant… · p. 46

Hidden variables

E Proof of Proposition 3 · F Proofs of Section 4.3

The proof shows that residual invariance across two environments identifies the true coefficient vector and yields the stated high-probability selection result.

E Proof of Proposition 3

  • E Proof of Proposition 3: Full-rank variation in Z makes residual distributions invariant across environments only when γ = γ∗, so every incorrect support is rejected.The environments correspond to conditioning on I = 0 and I = 1, with I, H, ε, η, and Z independent.Source: E Proof of Proposition 3 · p. 47
  • E Proof of Proposition 3: The true null is accepted with probability at least 1 − α, and the estimator therefore selects the true support by definition.Source: E Proof of Proposition 3 · p. 47

F.1 Proof of Theorem 2 (i) · F.2 Proof of Theorem 2 (ii) · F.3 Proof of Theorem 2 (iii)

The three proofs establish Theorem 2 by contradiction: any predictor set satisfying invariant residual conditions must recover the relevant parent set, under progressively different assumptions. Each part derives incompatible residual behavior from an omitted causal variable.

F.1 Proof of Theorem 2 (i)

  • Theorem 2(i) begins with S(E) ⊆ PA(Y) and assumes S(E) ≠ PA(Y) to derive a contradiction.A violating set S omits part of PA(Y) while retaining the null hypothesis and has prediction coefficients different from the causal coefficient γ∗.Source: §F.1 · p. 47
  • Theorem 2(i) therefore rules out any invariant predictor set that does not contain PA(Y).The argument uses residuals satisfying constant beta and the same error distribution across environments.Source: §F.1 · p. 47
  • For Theorem 2(i), choosing a youngest omitted variable and intervening on it makes the residual distributions differ across experiments.The contradiction is that Re0(S) and R1(S) cannot have the same distribution.Source: §F.1 · p. 47

F.2 Proof of Theorem 2 (ii)

  • Theorem 2(ii) repeats the youngest-node intervention argument and obtains a contradiction when the relevant residual variance differs from one.The proof also applies analogously to shifted noise distributions.Source: §F.2 · p. 48
  • In Theorem 2(ii), the contradiction follows because R1(S) and Re0(S) cannot share a distribution despite the null hypothesis being correct.The proof iteratively uses the structural equations after selecting the youngest node with nonzero αk0.Source: §F.2 · p. 48

F.3 Proof of Theorem 2 (iii)

  • Because R1(S) and R2(S) are supposed to have the same distribution, the argument invokes Cramér’s theorem [Cramér, 1936] and considers Ak0ε1.The ˜Dk are continuous functions of the random variables As and βe=2j,s.Source: §F.3 · p. 48
  • Theorem 2(iii) then forces Ak0 to take only the values −c and c almost surely, contradicting the assumption that Ak0 has a density.Thus the assumed equality in distribution of R1(S) and R2(S) cannot hold.Source: §F.3 · p. 48

F.4 Proof of Theorem 3

The proof combines Lemma 1 with faithfulness, which holds with probability one, to show that accepting the null hypothesis while omitting true causal parents leads to a contradiction. Lemma 1 establishes that a youngest omitted parent can have zero residual coefficient only under conditions ultimately ruled out by the structural-equation argument.

Proof of Theorem 3

  • Theorem 3 follows because, if the null hypothesis is accepted while true parents are missing, Lemma 1 gives a nonzero coefficient that yields the same contradiction as in Theorem 2 through faithfulness [Spirtes et al., 2000, Theorem 3.2].Faithfulness holds with probability one, while the alternative construction would imply an unblocked path whose conditional relationship vanishes.Source: §F.4 · pp. 48–49

Lemma 1

  • Under the structural equation model with continuously distributed nonzero parameters and variances, Lemma 1 shows that an omitted youngest parent can have zero residual coefficient only if it is included in S or S contains one of its descendants.The result holds for almost all parameter values; exceptional cancellations require fine-tuning and therefore occur with probability zero.Source: §F.4 · pp. 49–50
  • The lemma rules out the descendant case by repeatedly removing the youngest descendant from S, ultimately showing that the omitted parent must belong to S and has the corresponding nonzero regression coefficient.The argument uses path coefficients, nonsingularity, and the fact that the remaining path-coefficient vector has only the parent’s nonzero entry.Source: §F.4 · pp. 49–50

G Experimental settings for numerical studies

The numerical studies sample paired observational and interventional data from randomly generated linear DAG models, varying graph size, density, coefficients, noise, and intervention scope across broad parameter ranges.

Data-generating process

  • The experiments use a shared random DAG for observational and interventional settings, with random topological order, average degree k, signed nonzero linear coefficients, and independently sampled parameters.They draw nobs and nint samples and model normally distributed noise; the graph has p nodes and edge probability k/(p −1).Source: G Experimental settings for numerical… · pp. 50–51

Intervention design

  • Interventions simultaneously modify noise and potentially linear coefficients on a random set A, which may contain one node or a fraction θ of nodes, including the target Y.Interventional coefficients can be identical or redrawn within intervention-specific bounds.Source: G Experimental settings for numerical… · p. 51

Parameter ranges

  • Across simulations, nobs and nint range from 100 to 500, p from 5 to 40, and average degree k from 1 to 4.Coefficient and noise bounds are also varied over discrete ranges, while intervention noise-multiplication differences may be fixed or variable.Source: G Experimental settings for numerical… · p. 51

Intervention scope

  • Intervention scope is one randomly chosen variable with probability 1/6; otherwise, the intervened fraction θ ranges from 1/3 to 1/1.1.The intervention coefficient bounds are generated from ordered uniform values between 0.1 and 2.Source: G Experimental settings for numerical… · p. 51

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