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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 effects are difficult to quantify with confidence after searching over causal structures. The paper exploits invariant conditional predictions across environments to construct confidence statements and study identifiability, with structural-equation results and empirical applications, while noting computational, distributional, and scope limitations.

  • Problem

    Existing causal discovery methods make confidence statements difficult after structure search, especially without relying on data splitting or prior causal ordering assumptions.

  • Method

    The method identifies causal predictors by collecting submodels whose target conditional distribution remains invariant across experimental settings, without requiring known intervention types.

  • Results

    The framework provides potentially conservative confidence sets and intervals, and establishes identifiability results for causal predictors in linear Gaussian structural equation models with interventions.

  • Takeaways & Limitations

    Invariant prediction provides a new framework for causal inference that links confidence statements with causal-predictor identification across changing environments.

  • Takeaways & Limitations

    The concrete procedure can be computationally slow, assumes Gaussian errors, and is not straightforward to extend to classification or nonlinear models.

Abstract

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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 proposes invariant prediction for causal discovery: across experimental settings, the conditional distribution given all direct causes remains unchanged, enabling confidence statements and causal identification. It develops the framework for multiple environments, structural equation models, and applications.

  • 1 Introduction: Causal predictors are characterized by an invariant conditional distribution of the target across interventions on variables other than the target.The invariance requires conditioning on the complete set of direct causes; omitting direct causes may break it.
  • 1 Introduction: The paper studies identifiability in linear Gaussian structural equation models, including settings with instrumental variables and hidden variables, while allowing only limited treatment of feedback.The main presentation assumes no hidden confounders influencing the target, although an instrumental-variable example illustrates broader applicability.
  • 1 Introduction: For structural equation models, the invariant set corresponds to the direct causes of the target under interventions that do not act on the target.The paper focuses mainly on linear models and relates the assumption to traditional interventions and structural equation models.
  • 1 Introduction: The method collects submodels whose predictive relationships remain invariant across experimental settings, using this property for causal discovery and inference.It is designed for data from multiple observational or interventional regimes, without requiring knowledge of how the regimes arose.
  • 1 Introduction: The framework constructs potentially conservative confidence sets for causal predictors and confidence intervals for causal coefficients without assuming a causal ordering or faithfulness.If causal effects are not identifiable from the data, the procedure is intended to detect this rather than make false causal discoveries.
  • 1 Introduction: The approach is evaluated through simulations and applications to gene perturbation and educational data, with implementations available in an R package.The paper also discusses violations, robustness, and possible extensions of the invariance assumption.

2 Assumed invariance of causal prediction

The paper defines causal predictors through conditional-distribution invariance across experimental settings and uses this property for inference. In structural equation models, parents of the target satisfy the assumption under interventions that exclude the target, while identifiability improves with more diverse environments.

  • Invariance assumption: The invariance assumption requires the conditional distribution of the target given a predictor subset to remain unchanged across experimental or intervention settings.The model uses a common regression function and error distribution across environments.
  • Structural equation models: In a linear structural equation model, the parents of the target satisfy the invariance assumption when interventions affect other variables but not the target.This includes do-interventions and soft-interventions, under the stated noise-independence condition.
  • Structural equation models: The approach can also accommodate more general interventions if the target’s conditional distribution given its direct causes remains unchanged across environments.This corresponds to a modularity or autonomy requirement, while general feedback remains an open extension.
  • Identifiability and inference: Inference is valid for any predictor set satisfying the invariance assumption, not only for parent sets in linear structural equation models.Under the assumption, true causal predictors are included among the plausible predictors used for confidence guarantees.
  • Identifiability and inference: The identifiable predictor set grows when environments are added, whereas the global set of plausible causal coefficients shrinks.For E1 ⊆ E2, S(E1) ⊆ S(E2) and Γ(E1) ⊇ Γ(E2).
  • Identifiability and inference: With only one environment, no causal variables can be identified, and observationally similar environments may leave multiple plausible coefficient vectors.Stronger interventions and more environments reduce the set of vectors satisfying the invariance null hypothesis.

3 Estimation of identifiable causal predictors

The method estimates identifiable causal predictors by testing whether regression relationships remain invariant across experimental settings, then constructs confidence sets for predictors and causal coefficients. Under valid testing and confidence-set coverage, it provides conservative coverage guarantees while exposing trade-offs between identifiability, power, and computation.

  • Generic method for invariant prediction: The procedure tests H0,S(E) for every candidate predictor set S and retains sets whose predictive relationships are not rejected as invariant.Confidence sets for regression coefficients are then constructed using pooled data.
  • Confidence guarantees: If the tests and pooled confidence intervals have the claimed error and coverage rates, the estimated causal predictors and coefficients retain valid coverage guarantees.Theorem 1 gives at least 1 − 2α coverage for the resulting coefficient set.
  • Confidence guarantees: With probability at least 1 − α, the estimator does not include non-causal predictors, although it may conservatively return an empty set.These guarantees hold for any collection of experimental or intervention settings.
  • Structural equation models: In linear structural equation models, the guarantees apply to causal coefficients and variables, with the parent set of the target providing the relevant causal predictor set under the stated assumptions.The framework also uses invariance of regression effects across settings for the causal predictor set.
  • Limitations: The linear confidence-region construction assumes Gaussian errors, may require costly covariance-matrix inversions, and is not straightforward to extend to classification or nonlinear models.Rank tests or other nonparametric procedures could address the Gaussian-error dependence.
  • Trade-offs and computation: Pooling improves statistical power but can reduce identifiability, while exhaustive testing over predictor subsets has super-exponential computational complexity.Screening variables before subset search can reduce complexity to O(q^s), provided causal predictors are retained with high probability.

4 Identifiability results for structural equation models

The paper studies when causal predictors in linear Gaussian structural equation models can be identified from data collected across observational and interventional settings. It gives sufficient identifiability conditions for several intervention types and illustrates conditions that are necessary or can be weakened.

  • Model setting: The analysis considers linear Gaussian structural equation models with data blocks representing different interventional setups, observed variables, and an acyclic directed graph.The target variable is denoted Y, with X containing the predictor variables; the parents of Y are the causal predictors of interest.
  • Intervention types: Three intervention classes are examined: do-interventions, noise interventions, and simultaneous noise interventions.Do-interventions fix selected variables, whereas noise interventions alter their error distributions; simultaneous interventions may affect multiple variables at once.
  • Identifiability results: Theorem 2 provides sufficient conditions under which all causal predictors are identifiable from the set of invariant predictors.The theorem applies to linear Gaussian SEMs with interventions and establishes identifiability under the specified intervention conditions.
  • Limits and extensions: The sufficient conditions are not generally necessary, and counterexamples show that removing particular intervention conditions can prevent recovery of the correct parent set.Examples include cases where an empty predictor set satisfies the null hypothesis, making the true parents non-identifiable.
  • Limits and extensions: Some assumptions can be weakened, while extensions may allow random coefficients or changing graph structures provided the target’s parents remain unchanged.The paper also notes that the proposed results include sufficient rather than exhaustive conditions for identifiability.
  • Identifiability results: Identifiability can also hold with one intervention setting on a single node, under suitable assumptions on the intervention and a generic data-generating model.The intervention target need not be known, and the result remains sufficient when that setting is part of a larger collection of interventional settings.

5 Instrumental and hidden variables with confounding

The paper extends invariant prediction to hidden-variable and instrumental-variable settings, providing coverage for causal coefficients and, under specific intervention structures, population identification. It also highlights computational and modeling limits relative to classical instrumental-variable methods.

  • Instrumental-variable settings are reframed as environments by splitting samples according to the instrument, provided the split is not based on Y, its descendants, or hidden variables.
  • With hidden variables, the method replaces residual independence with invariance of the residual distribution across environments for some coefficient vector supported on S.
  • Testing hidden-variable invariance is computationally difficult because γ cannot be obtained by simple regression; brute-force testing over a coefficient grid is exponential in predictor dimension.
  • The invariant-prediction procedure retains coverage for the causal predictor set under the hidden-variable model when the corresponding null-hypothesis test is conducted at level α.
  • Under a variance-shift model for X, the causal coefficients γ∗ and predictor set S∗ are identifiable in the population with a perfectly powerful invariance test.
  • Unlike classical two-stage instrumental-variable analysis, identification can hold with a univariate instrument, arbitrary-dimensional X, and no shift in E(X | I).
  • Weak instrument influence produces naturally wide confidence intervals and consequently small selected causal-variable sets, avoiding a required weak-instrument test.

6 Further extensions and model misspecification

The paper extends invariant prediction to nonlinear models, hidden-variable-compatible tests, and sparse interventions on the target, then analyzes what coverage and power mean under model misspecification. These extensions preserve supported coverage but can reduce power or select noncausal yet ancestral variables.

  • Invariant-prediction propositions extend from linear to nonlinear structural equation models when a causal set satisfies the nonlinear invariance condition.
  • The nonlinear confidence set retains P(Ŝ(E) ⊆ S∗) ≥ 1 − α when the causal set satisfies the relevant null hypothesis and individual tests use level α.
  • Nonlinear invariance can be tested through environment-specific regression equality, pooled residual-distribution equality, or conditional independence of Y and E given X_S.
  • Sparse interventions on Y can be accommodated by requiring invariance across a subset of at least |E| − V environments rather than all environments.
  • Model misspecification: Under model misspecification, coverage remains valid when the considered causal set still satisfies the invariance assumption, but the procedure may lose power and return an empty set.
  • Model misspecification: Under additional assumptions, misspecification can instead yield a selected set containing only ancestors of the target and no descendants.

7 Numerical results

The method is evaluated on simulations, gene perturbation experiments, and educational-attainment data. It controls false selections while recovering causal predictors and producing confidence statements across applications.

  • 7.1 Simulation experiments: Simulations compare invariant prediction with alternative methods for recovering causal predictors from randomly generated linear Gaussian structural equation models.The study uses 100 settings and 1000 generated data sets per setting.
  • 7.1 Simulation experiments: The proposed method does not require knowledge of which nodes were intervened on, unlike methods that use intervention targets explicitly.For invariant prediction, the exact intervention information is discarded or presumed unknown.
  • 7.1 Simulation experiments: Across 100 scenarios, exact recovery was optimal for GES in 3, GIES with known interventions in 14, LiNGAM in 54, and invariant prediction in 23.No method performed uniformly best across all scenarios.
  • 7.1 Simulation experiments: The invariant prediction method controlled FWER at the desired 0.05 level, whereas all other methods reached FWE rates of 0.4 or higher.The authors identify FWER control and associated confidence intervals as the method’s key advantage.
  • 7.2 Gene perturbation experiments: In the yeast data, eight significant causal effects were tested after Bonferroni correction, with six of eight pairs judged correct; alternatives did not exceed random guessing.The data contain 6170 genes, 160 observational samples, and 1479 interventional samples.
  • 7.3 Educational attainment: In educational attainment, invariant logistic regression found significant effects for test score and having a father without a college degree.The analysis split pupils by distance to the nearest four-year college and assessed invariance between the resulting groups.

8 Discussion and Future Work

Invariant prediction reframes causal inference around stability across environmental changes and supports confidence statements for causal predictors. The discussion establishes identifiable special cases while highlighting unresolved nonlinear, feedback, hidden-variable, and computational challenges.

  • 8 Discussion and Future Work: Causal predictors are distinguished from non-causal predictors by the invariance of their influence under environmental changes such as interventions.The paper exploits this invariance to identify causal predictors.
  • 8 Discussion and Future Work: The method provides confidence sets for causal predictors and confidence intervals for relevant parameters, with identifiability guarantees in Gaussian structural equation models with interventions.These guarantees concern the set of causal predictors in the stated special case.
  • 8 Discussion and Future Work: Open problems include conditional-distribution tests for nonlinear models, general feedback models, hidden variables, and improved computational efficiency.The paper explicitly identifies these areas as requiring more work.
  • 8 Discussion and Future Work: Invariant prediction introduces new causal-inference concepts and methods while relating existing ideas from a different angle.The authors present this as a new understanding of causality and a basis for further theory and methodology.
  • 8 Discussion and Future Work: Figure 8 illustrates the approach using observational and interventional structural equation models that generate two environments.The example demonstrates how invariant predictors can be estimated without requiring known structural equations.

B Hidden variables without confounding

The hidden-variable extension studies when invariant prediction remains valid without confounding between observable causal variables and the target. Under specified intervention and graph conditions, the method preserves an invariant target model.

  • B Hidden variables without confounding: The extension considers linear acyclic structural equation models containing observed variables and unobserved hidden variables.The hidden variables may have direct effects on the target, motivating separate conditions for validity.
  • B Hidden variables without confounding: The illustrative two-environment SEM has S∗ = {X2, X3}, while {X3} fails through differing coefficients and {X4} fails through differing residual variances.Intersecting the non-rejected invariant sets yields the conservative estimate Ŝ = {X2, X3} = S∗.
  • B Hidden variables without confounding: When hidden variables have no direct causal effects on Y, choosing S∗ as the observable direct causal variables satisfies the required invariance assumption.This is condition (i) in the stated proposition.
  • B Hidden variables without confounding: With hidden variables affecting Y, invariance can still hold when interventions are do- or soft-interventions, avoid Y and S0, and satisfy the stated d-connection restriction.The proposition gives these as conditions (a)–(c).
  • B Hidden variables without confounding: The target-noise distribution remains invariant across environments and independent of the predictors under the stated sufficient conditions.This invariant-noise property is the central condition used in the extension.

C Model Misspecification

Under model misspecification, invariant predictor sets may no longer contain only direct causes, but they generally remain within the ancestors of the target. Direct intervention on the target can eliminate all invariant predictor sets.

  • C Model Misspecification: Under misspecification, S(E) may include variables that are not direct causes of Y, while remaining a subset of Y's ancestors in most cases.The result is formulated using an environment variable E and a faithfulness assumption.
  • C Model Misspecification: Proposition 5 states that any set S satisfying H0,S,nonlin(E) can be replaced by a set within AN(Y) that preserves the required separation.The proof uses the equivalence between invariant prediction and conditional independence Y ⟂ E | X_S, together with faithfulness.
  • C Model Misspecification: If the direct-cause set S0 fails the invariance condition under some intervention, another set may still satisfy it, producing S(E) outside S0 but inside AN(Y).Figure 9 gives an example with S0 = {X2} and S(E) = {X1}.
  • C Model Misspecification: When Y is directly intervened on, no set S may satisfy the invariance condition, so S(E) can be empty.This follows under the assumptions of Proposition 5.

D Potential Outcomes and Invariant Prediction

The invariant-prediction principle extends to a potential-outcome framework: absent intervention on Y, conditioning on causal predictors yields an invariant conditional distribution across environments. Under FFR-CISTG assumptions, the parents of Y satisfy this principle.

  • D Potential Outcomes and Invariant Prediction: In a potential-outcome framework, conditional distributions of Y given causal predictors remain invariant across environments when Y is not directly intervened on.The paper develops the nonlinear version of invariant prediction in counterfactual language.
  • D Potential Outcomes and Invariant Prediction: The framework assumes consistency and recursive substitution, together with FFR-CISTG independence.These are labeled assumptions (C1) and (C2).
  • D Potential Outcomes and Invariant Prediction: For each environment, treatment variables receive assignments through counterfactual variables X_j(˜r_e), defining the environment-specific potential outcomes.The treatment sets exclude the target variable Y.
  • D Potential Outcomes and Invariant Prediction: Under the FFR-CISTG assumptions, the conditional distribution of Y(˜r_e) given its parents P(˜r_e) is identical across all environments.The equality holds whenever both conditional distributions are well-defined.
  • D Potential Outcomes and Invariant Prediction: The paper notes that some hidden variables can be accommodated by relaxing assumption (C2).This extension is discussed in Appendix B.

E Proof of Proposition 3

The proof establishes that, in the considered structural model, residual-distribution invariance across two environments forces the fitted coefficient vector to equal the true coefficient vector.

  • E Proof of Proposition 3: Residual distributions can be invariant across the two environments only when γ equals the true coefficient vector γ∗.The argument uses independence among I, H, ε, η, and Z, plus the full-rank covariance of Z.

F.1 Proof of Theorem 2 (i)

The proof of Theorem 2(i) shows that omitted causal variables make residual invariance fail under a suitable intervention, contradicting the invariant null hypothesis. The argument is extended across model variants.

  • F.1 Proof of Theorem 2 (i): The proof begins from S(E) ⊆ PA(Y) and assumes that S(E) differs from the full parent set, seeking a contradiction.The contradiction targets a set whose invariant null hypothesis is correct despite omitting parents.
  • F.1 Proof of Theorem 2 (i): The method represents prediction errors as residuals and compares their coefficients across environments under the constant-beta and same-error-distribution conditions.These conditions are imposed for all pairs of environments.
  • F.1 Proof of Theorem 2 (i): Selecting a youngest omitted node and intervening on it produces residual distributions that cannot match across environments, yielding the contradiction.The construction uses an experiment e0 with intervention target {k0}.
  • F.1 Proof of Theorem 2 (i): The same youngest-node intervention strategy is applied to shifted-noise distributions, where unequal scaling contradicts residual-distribution invariance.The proof invokes the condition (k0)^2 ≠ 1 in the displayed argument.
  • F.1 Proof of Theorem 2 (i): For the remaining case, equality of residual distributions and Cramér's theorem imply normality and then a discrete-support condition that contradicts the assumed density of A_k0.The contradiction arises because P[A_k0 ∈ {−c,c}] = 1 for some c ≥ 0.

F.4 Proof of Theorem 3

The proof of Theorem 3 relies on Lemma 1 and faithfulness to show that an accepted null hypothesis with missing causal parents leads to a contradiction except on a probability-zero set.

  • The remaining contradiction follows because the implied relationship violates faithfulness or requires fine-tuning of a variance parameter, an event with probability zero.
  • Lemma 1 assumes a structural equation model with jointly continuously distributed nonzero parameters and a youngest parent of the target variable.
  • Under these assumptions, α_k0 = 0 implies that the youngest parent is included in S or that S contains one of its descendants.
  • If S contains a descendant of the youngest parent, the proof removes such descendants iteratively and obtains a set containing no descendants while preserving the relevant regression relation.
  • The resulting relation shows that the youngest parent must belong to S and that its predictive regression coefficient equals the corresponding structural coefficient.

G Experimental settings for numerical studies

The numerical studies sample observational and interventional data from randomly generated linear structural models, varying graph, coefficient, noise, intervention, and sample-size parameters.

  • Each simulation uses n_obs observational and n_int interventional samples across two environments sharing a sampled directed acyclic graph.
  • Graphs are generated from a random topological order, with edges added independently at a probability chosen to yield average degree k.
  • The observational setting samples normally distributed noise and random linear coefficients, with sample sizes selected from specified discrete ranges.
  • Interventional settings apply simultaneous noise interventions to a random single node or a fraction θ of nodes, potentially including the target variable.
  • Interventional coefficients may be identical to observational coefficients or independently resampled within intervention-specific bounds, while noise multipliers and coefficient differences are varied.
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