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Falsification of Unconfoundedness by Testing Independence of Causal Mechanisms

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Pith's one-line read The paper proves that in multi-environment observational data, unmeasured confounding creates detectable dependence between treatment-assignment and outcome mechanism parameters, and it provides a two-stage test (MINT) that falsifies the…

desk verdict MINT is a new and practical falsification test, but Theorem 4.5's 'iff' is false as stated — a varying confounder parameter can drop out of the treatment mechanism, breaking the test's power. read the letter →

arxiv 2502.06231 v2 pith:YQDKWCIQ submitted 2025-02-10 stat.ME cs.LGstat.ML

Rickard K.A. Karlsson , Jesse H. Krijthe This is my paper · ORCID
classification stat.MEcs.LGstat.ML MSC 62D2062F03
keywords causalinferenceunmeasuredconfoundingfalsificationindependentmechanismsmulti-environmentdatatreatmenteffectestimationhypothesistestingMINT
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the no-unmeasured-confounding assumption, usually treated as untestable, can be falsified when data come from several heterogeneous environments. It argues that if changes between environments are driven by independent causal mechanisms, then unmeasured confounding is the one plausible source of dependence between the observed treatment-assignment and outcome mechanisms. The authors prove this for a linear model with a possibly unobserved confounder $U$: under their assumptions, the null hypothesis of mechanism independence is false exactly when $U$ confounds the treatment–outcome relation. They then give an algorithm, MINT, that estimates the mechanism parameters in each environment and tests their cross-covariance with a bootstrap-calibrated permutation procedure. The approach matters because it works without randomized data and remains valid even when treatment effects are not transportable across environments, a setting where earlier falsification strategies fail.

What carries the argument

The load-bearing object is the null hypothesis $H_0:P(\omega,\gamma)=P(\omega)P(\gamma)$ together with the MINT algorithm's test statistic $\hat{T}(\hat{\omega},\hat{\gamma})=\frac{1}{K}\sqrt{\sum_{i,j}\big(\sum_s(\hat{\omega}_{s,i}-\bar{\omega}_i)(\hat{\gamma}_{s,j}-\bar{\gamma}_j)\big)^2}$, the Frobenius norm of the cross-covariance between estimated treatment and outcome mechanism parameters. Under $H_0$ this statistic is zero in expectation; a bootstrap-then-permutation calibration converts it into a level-$\alpha$ test while accounting for first-stage estimation uncertainty. The theoretical engine is Lemma 4.4, which derives the exact induced dependence when $U$ is a confounder, and Theorem 4.5, which turns that shared-parameter dependence into an if-and-only-if statement under the independence-of-mechanisms assumption.

What would settle it

Simulate $K$ environments from model (3) with $U$ absent but with $\alpha_s$ and $\beta_s$ drawn from a joint distribution that violates Assumption 4.1, for example a shared latent factor driving both sets of parameters, and run MINT at level $\alpha=0.05$; if the test rejects in a large fraction of repetitions well above 0.05, then the claimed “rejection implies confounding” direction is unsupported in settings where mechanisms are dependent for reasons unrelated to confounding.

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Extended reading notes

Core claim

The central claim is that unmeasured confounding has testable implications at the level of mechanism parameters, not only at the level of observed variables. In the linear model $A=\alpha_s^\top\psi(X)+\alpha_s^{(U)}U+\varepsilon_A$, $Y^a=\beta_s^\top\phi(X,A=a)+(\beta_s^{(U)}+a\beta_s^{(AU)})U+\varepsilon_Y$, with $X\perp\!\!\perp U\mid S$, the regression parameters of $E[A\mid X,S=s]$ and $E[Y\mid X,A,S=s]$ both depend on the same underlying quantities $(\alpha_s^{(0)},\alpha_s^{(X)},\alpha_s^{(U)},\mu_s^{(U)})$. Under Assumption 4.1, which says the true mechanisms are drawn independently across environments, Theorem 4.5 establishes that $H_0:P(\omega,\gamma)=P(\omega)P(\gamma)$ is false if and only if $U$ is a confounder, provided at least one of those shared parameters varies non-degenerately across environments. A statistical test of $H_0$ is therefore a falsification test for the conjunction of unconfoundedness and independent causal mechanisms.

Load-bearing premise

The whole test rests on the premise that the causal mechanisms in different environments change independently of one another; if that premise is false, a rejection of the null does not point to unmeasured confounding.

Editorial extensions

If this is right

  • A rejection of $H_0$ falsifies Assumption 3.1 and Assumption 4.1 jointly, so a practitioner who trusts independent causal mechanisms gains evidence against no-unmeasured-confounding.
  • The test needs no randomized arm and does not require treatment effects to be transportable, making it applicable to meta-analyses of observational studies and to clustered settings such as hospitals or schools.
  • Power to detect confounding grows with the number of environments $K$, and the non-degeneracy condition in Theorem 4.5 explains why a single environment cannot support this kind of falsification.
  • The test avoids conditional independence testing altogether, sidestepping known hardness results and the power loss that comes with larger adjustment sets.
  • Correct specification of the working models is essential: misspecified feature representations inflate false positives, while well-specified but more flexible models mainly reduce power rather than break error control.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because misspecified working models inflate the false-positive rate, MINT could plausibly double as a diagnostic for model fit, although the paper does not develop that use.
  • The kernelized sketch in Appendix C points toward a nonlinear version of the same logic; if completed, it could test confounding in settings where linear parameter estimates are unavailable.
  • In practice the test is best treated as a screening device: when independent causal mechanisms are plausible, a rejection justifies deeper sensitivity analysis, but when that assumption is doubtful the rejection is ambiguous by design.
  • The non-degeneracy requirement implies that study design should prioritize collecting data from genuinely different sites or policies, since variation in treatment assignment or confounder distribution across environments, not raw sample size, is what makes falsification possible.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper proposes MINT, a two-stage algorithm for falsifying the no-unmeasured-confounding assumption in multi-environment observational studies. The first stage estimates treatment and outcome mechanism parameters per environment; the second stage tests independence of these estimated parameter vectors across environments using a permutation test with bootstrap calibration. The authors prove (Theorem 4.5) that, under a linear model with an unmeasured confounder and independent causal mechanisms, a non-degenerate variation in certain parameters makes the null hypothesis of mechanism independence false if and only if unmeasured confounding is present. Experiments on synthetic and semi-synthetic data compare MINT with transportability-based falsification and a hierarchical-graph conditional independence test.

Significance. If the central theorem were correct, the paper would provide a practically useful falsification tool that avoids conditional independence testing and remains valid under transportability violations. The paper is clearly written, the experiments are extensive, and the code is public. However, the central theoretical claim is false as stated, and the reported experiments do not cover the regime in which the failure occurs. This substantially weakens the paper's contribution and requires a major revision before the manuscript can be considered sound.

major comments (2)
  1. [Theorem 4.5 / Appendix B.3] The 'if' direction of Theorem 4.5 is false as stated. The proof (Appendix B.3) asserts that if any of (α_s^(0), α_s^(X), α_s^(U), μ_s^(U)) varies non-degenerately, then both ω_s and γ_s depend on that parameter, hence are dependent. This implication is invalid because a shared parameter can affect one mechanism with zero coefficient. Concretely, set α_s^(0)=0, α_s^(X)=0, μ_s^(U)=0, β_s^(U)=1, β_s^(AU)=0, all other βs fixed at 0, and let α_s^(U) ~ N(1,1) vary across environments. Then U is a confounder for every s (α_s^(U) ≠ 0 and β_s^(U) ≠ 0), and Assumption 4.1 holds since β is degenerate and independent of α. But by Lemma 4.4, ω_s = [0,0]^T for all s, while γ_s varies with α_s^(U) (γ_s,3 = (σ^(U))^2/α_s^(U)). A constant vector is independent of any random vector, so H0 is true despite U being a confounder, and the test statistic T is identically zero. Thus MINT has zero power in this regime. The experiments in Section 6.2.2 and Appendix D.2 do not expose this because the default settings keep α^(0), α^(X), and μ^(U) nonzero while varying α^(U). The theorem needs an additional non-degeneracy condition on the coefficients of the shared parameter in both ω_s and γ_s, not merely on the parameter's own distribution.
  2. [Section 5] The Type I error guarantee for the proposed bootstrap-permutation threshold is not established theoretically. The text states that the threshold R is chosen to ensure Pr(T > R | H0) ≤ α, but the calibration uses bootstrap resamples of the estimated parameters followed by random permutations of ω within each resample. This procedure does not provably sample from the null distribution of T under the joint estimation error; the only evidence is the empirical ablation in Appendix D.5 (Figure 5). For a statistical methodology paper, a formal analysis of the bootstrap-based null calibration, or at least a statement of the conditions under which it is valid, is needed to support the abstract's claim of 'controlling false positives.'
minor comments (3)
  1. [Throughout] There are several typographical errors in the parameter lists: in Theorem 4.5, '(α(0) s α(X) s , α(U) s , µ(U) s )' is missing a comma; the same issue appears in Section 4.3 and the proof in Appendix B.3.
  2. [Appendix B.1] The proof of Theorem 4.3 appears to have the feature representations swapped: it uses 'eϕ(X,A)' for the outcome model and 'eψ(X)' for the treatment model, whereas the main text defines eϕ(X) for treatment and eψ(X,A) for outcome. Please clarify the notation.
  3. [Appendix D.2] The synthetic data generation in Appendix D.2 uses K=250 environments and 1000 samples per environment, which is a very favorable setting. The paper would benefit from a discussion of how the method behaves with smaller K (e.g., K=5 or K=10), which is the range shown in Figure 1 but is not reflected in the theoretical non-degeneracy discussion.

Circularity Check

0 steps flagged · score 1.0 of 10

No derivational circularity: H0 is a derived consequence of the assumptions (Theorem 4.3) and the MINT statistic directly estimates the covariance; Theorem 4.5 is proved by self-contained algebra, and the self-citations to Karlsson and Krijthe (2023) are background, not load-bearing.

full rationale

No derivational circularity was found. The null H0:P(omega,gamma)=P(omega)P(gamma) is a derived consequence of the stated assumptions rather than an input: Theorem 4.3's proof (Appendix B.1) shows omega_s=(D^{-1})^T alpha_s and gamma_s=(C^{-1})^T beta_s and transfers Assumption 4.1's factorized prior to the observable parameters via the standard independence-preservation of invertible linear transforms. Lemma 4.4 then supplies closed-form expressions for omega_s and gamma_s under the confounded model (3), so the shared-parameter dependence invoked in Theorem 4.5 is obtained by explicit algebra, not assumed. 'Unmeasured confounder' is defined structurally (nonzero alpha^(U) with nonzero beta^(U) and/or beta^(AU)) and is logically separate from H0, so no self-definitional loop exists. The only-if direction uses 'no shared parameters' together with the explicitly stated Assumption 4.1, and the Discussion openly concedes the conditional nature of the test: 'our proposed algorithm is a joint falsification test that assesses both the conditions necessary for causal identification and the assumption of independent causal mechanisms.' That is an honest conditional premise, not a hidden circular step. Self-citations to Karlsson and Krijthe (2023) attribute the previously observed non-degeneracy condition and supply the HGIC baseline, but the proof of Theorem 4.5 (Appendix B.3) is written out in full without importing that citation as evidence, so the self-citation is not load-bearing; no uniqueness theorem is imported from the authors, and Assumption 4.1 is stated in-paper rather than adopted via citation. The skeptical concern about Theorem 4.5's if-direction (a shared parameter such as alpha^(U) can drop out of omega_s when mu^(U)=alpha^(0)=alpha^(X)=0, leaving omega_s constant while U remains a confounder) is a mathematical validity gap in the step from 'shared parameters' to 'omega_s not independent of gamma_s', not a circularity: the claim does not reduce to its own inputs by construction. Experiments are validated against external baselines (transportability test, HGIC) and the semi-synthetic Twins benchmark with released code, so the empirical claims are self-contained. Score 1 reflects only minor, non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No ad hoc fitted constants are introduced; the environment-specific mechanism parameters are random/model parameters, not free parameters tuned to make the central claim work. The bootstrap resample count M=1000 and significance level alpha=0.05 are standard calibration choices. No new particles, mediators, forces, or conserved quantities are postulated; U is a standard latent confounder used in the illustrative data-generating process.

assumptions (6)
  • domain assumption Shared causal DAG among (X, S, A, Y) across environments; observations i.i.d. within environment, not across environments.
    Section 3.1; needed so environment variation is mechanism variation rather than structural change.
  • domain assumption Assumption 3.1: consistency, positivity, and unconfoundedness for each environment, the target conditions to be falsified.
    Section 3.2; the test is conditional on these and rejects them jointly with ICM.
  • domain assumption Functional class (1): A = alpha_s^T psi(X) + epsilon_A and Y^a = beta_s^T phi(X,a) + epsilon_Y, with zero conditional mean noises and independent noises under unconfoundedness.
    Section 4.1; restricts treatment and outcome mechanisms to generalized linearity in known feature maps.
  • domain assumption Assumption 4.1: environment parameters (alpha_s, beta_s) are drawn i.i.d. from P(alpha, beta) = P(alpha)P(beta).
    Section 4.1; this is the independent-causal-mechanisms premise that makes dependence in estimated parameters attributable to confounding, and it is untestable and jointly falsified.
  • domain assumption Assumption 4.2: feature representations psi and phi are known up to invertible linear transformations, and z, z' < min_s n_s.
    Section 4.2; permits ordinary least squares estimation and identifies omega, gamma as invertible transforms of alpha, beta; misspecification inflates Type I error (Section 6.2.3).
  • domain assumption For the linear-model theorem: X is independent of U given S; U, epsilon_A, epsilon_Y are normal; at least one of (alpha_s^(0), alpha_s^(X), alpha_s^(U), mu_s^(U)) has a non-degenerate distribution across environments.
    Section 4.3 and Appendix D.2; normality allows a closed-form E[U|X,A,S], and the non-degeneracy condition turns shared parameters into statistical dependence.

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Cite this review

Pith. "Pith review of Falsification of Unconfoundedness by Testing Independence of Causal Mechanisms." pith.science (2026). https://pith.science/paper/YQDKWCIQ

@misc{pith2026250206231,
  author       = {Pith},
  title        = {Pith review of: Falsification of Unconfoundedness by Testing Independence of Causal Mechanisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQDKWCIQ}},
  note         = {Machine review of arXiv:2502.06231}
}
read the original abstract

A major challenge in estimating treatment effects in observational studies is the reliance on untestable conditions such as the assumption of no unmeasured confounding. In this work, we propose an algorithm that can falsify the assumption of no unmeasured confounding in a setting with observational data from multiple heterogeneous sources, which we refer to as environments. Our proposed falsification strategy leverages a key observation that unmeasured confounding can cause observed causal mechanisms to appear dependent. Building on this observation, we develop a novel two-stage procedure that detects these dependencies with high statistical power while controlling false positives. The algorithm does not require access to randomized data and, in contrast to other falsification approaches, functions even under transportability violations when the environment has a direct effect on the outcome of interest. To showcase the practical relevance of our approach, we show that our method is able to efficiently detect confounding on both simulated and semi-synthetic data.

Figures

Figures reproduced from arXiv: 2502.06231 by the authors.

Figure 1
Figure 1. Comparison of falsification rate when varying either the number of environment [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. (a): Comparison of falsification rate when different mechanisms vary across the environment. The parameters on the x-axis correspond to those of the data-generating process in (3). (b): Our algorithm’s performance is evaluated using polynomial basis functions as feature representation. The falsification rate is plotted against polynomial degree, with the true data-generating process including polynomials up to degre… view at source ↗
Figure 3
Figure 3. Comparison on the Twins semi-synthetic dataset. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Same experiment as in [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: An ablation study showing the falsification rate our proposed algorithm using permutation-based testing with [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Uncovering Bias Mechanisms in Observational Studies

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Reference graph

Works this paper leans on

56 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

  2. [2]

    Y., and Lee, D

    Almond, D., Chay, K. Y., and Lee, D. S. The costs of low birth weight. The Quarterly Journal of Economics, 120 0 (3): 0 1031--1083, 2005

    work page 2005

  3. [3]

    and Imbens, G

    Athey, S. and Imbens, G. W. The state of applied econometrics: Causality and policy evaluation. Journal of Economic perspectives, 31 0 (2): 0 3--32, 2017

    work page 2017

  4. [4]

    and Robins, J

    Bang, H. and Robins, J. M. Doubly robust estimation in missing data and causal inference models. Biometrics, 61 0 (4): 0 962--973, 2005

    work page 2005

  5. [5]

    Burauel, P. F. Evaluating instrument validity using the principle of independent mechanisms. Journal of Machine Learning Research, 24 0 (176): 0 1--56, 2023

    work page 2023

  6. [6]

    Double/debiased machine learning for treatment and structural parameters

    Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21 0 (1): 0 C1--C68, 2018

    work page 2018

  7. [7]

    Causal inference methods for combining randomized trials and observational studies: a review

    Colnet, B., Mayer, I., Chen, G., Dieng, A., Li, R., Varoquaux, G., Vert, J.-P., Josse, J., and Yang, S. Causal inference methods for combining randomized trials and observational studies: a review. Statistical science, 39 0 (1): 0 165--191, 2024

    work page 2024

  8. [8]

    C., Lilienfeld, A

    Cornfield, J., Haenszel, W., Hammond, E. C., Lilienfeld, A. M., Shimkin, M. B., and Wynder, E. L. Smoking and lung cancer: recent evidence and a discussion of some questions. Journal of the National Cancer institute, 22 0 (1): 0 173--203, 1959

    work page 1959

Show all 56 references
  1. [9]

    J., Robertson, S

    Dahabreh, I. J., Robertson, S. E., Steingrimsson, J. A., Stuart, E. A., and Hernan, M. A. Extending inferences from a randomized trial to a new target population. Statistics in medicine, 39 0 (14): 0 1999--2014, 2020 a

    1999

  2. [10]

    J., Robins, J

    Dahabreh, I. J., Robins, J. M., and Hern \'a n, M. A. Benchmarking observational methods by comparing randomized trials and their emulations. Epidemiology, 31 0 (5): 0 614--619, 2020 b

    2020

  3. [11]

    Detecting critical treatment effect bias in small subgroups

    De Bartolomeis, P., Abad, J., Donhauser, K., and Yang, F. Detecting critical treatment effect bias in small subgroups. In Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence, pp.\ 943--965. PMLR, 2024 a

    2024

  4. [12]

    A., Donhauser, K., and Yang, F

    De Bartolomeis, P., Martinez, J. A., Donhauser, K., and Yang, F. Hidden yet quantifiable: A lower bound for confounding strength using randomized trials. In International Conference on Artificial Intelligence and Statistics, pp.\ 1045--1053. PMLR, 2024 b

    2024

  5. [13]

    M., Oberst, M., Philippakis, A

    Demirel, I., De Brouwer, E., Hussain, Z. M., Oberst, M., Philippakis, A. A., and Sontag, D. Benchmarking observational studies with experimental data under right-censoring. In International Conference on Artificial Intelligence and Statistics, pp.\ 4285--4293. PMLR, 2024

    2024

  6. [14]

    On multi-cause approaches to causal inference with unobserved counfounding: Two cautionary failure cases and a promising alternative

    D’Amour, A. On multi-cause approaches to causal inference with unobserved counfounding: Two cautionary failure cases and a promising alternative. In The 22nd international conference on artificial intelligence and statistics, pp.\ 3478--3486. PMLR, 2019

    2019

  7. [15]

    M., Vankadara, L

    Faller, P. M., Vankadara, L. C., Mastakouri, A. A., Locatello, F., and Janzing, D. Self-compatibility: Evaluating causal discovery without ground truth. In Dasgupta, S., Mandt, S., and Li, Y. (eds.), Proceedings of The 27th International Conference on Artificial Intelligence a...

    2024

  8. [16]

    Fisher, R. A. Statistical Methods for Research Workers. Oliver and Boyd, 1925

    1925

  9. [17]

    Data analysis using regression and multilevel/hierarchical models

    Gelman, A. Data analysis using regression and multilevel/hierarchical models. Cambridge university press, 2007

    2007

  10. [18]

    Multilevel modelling of medical data

    Goldstein, H., Browne, W., and Rasbash, J. Multilevel modelling of medical data. Statistics in medicine, 21 0 (21): 0 3291--3315, 2002

    2002

  11. [19]

    Greenland, S., Pearl, J., and Robins, J. M. Confounding and collapsibility in causal inference. Statistical science, 14 0 (1): 0 29--46, 1999

    1999

  12. [20]

    Kernel methods for measuring independence

    Gretton, A., Herbrich, R., Smola, A., Bousquet, O., and Sch \"o lkopf, B. Kernel methods for measuring independence. Journal of Machine Learning Research, 6: 0 2075--2129, 2005

    2005

  13. [21]

    and Stirzaker, D

    Grimmett, G. and Stirzaker, D. Probability and random processes. Oxford university press, 2020

    2020

  14. [22]

    Causal de finetti: On the identification of invariant causal structure in exchangeable data

    Guo, S., T \'o th, V., Sch \"o lkopf, B., and Husz \'a r, F. Causal de finetti: On the identification of invariant causal structure in exchangeable data. Advances in Neural Information Processing Systems, 36, 2024 a

    2024

  15. [23]

    Do finetti: On causal effects for exchangeable data

    Guo, S., Zhang, C., Mohan, K., Husz \'a r, F., and Sch \"o lkopf, B. Do finetti: On causal effects for exchangeable data. arXiv preprint arXiv:2405.18836, 2024 b

    2024 arXiv

  16. [24]

    Heard, N. A. and Rubin-Delanchy, P. Choosing between methods of combining-values. Biometrika, 105 0 (1): 0 239--246, 2018

    2018

  17. [25]

    and Robins, J

    Hernan, M. and Robins, J. Causal Inference: What If. Chapman & Hall/CRC Monographs on Statistics & Applied Probab. CRC Press, 2020

    2020

  18. [26]

    Hern \'a n, M. A. and Robins, J. M. Estimating causal effects from epidemiological data. Journal of Epidemiology & Community Health, 60 0 (7): 0 578--586, 2006

    2006

  19. [27]

    Causal discovery from heterogeneous/nonstationary data

    Huang, B., Zhang, K., Zhang, J., Ramsey, J., Sanchez-Romero, R., Glymour, C., and Sch \"o lkopf, B. Causal discovery from heterogeneous/nonstationary data. The Journal of Machine Learning Research, 21 0 (1): 0 3482--3534, 2020

    2020

  20. [28]

    Falsification of internal and external validity in observational studies via conditional moment restrictions

    Hussain, Z., Shih, M.-C., Oberst, M., Demirel, I., and Sontag, D. Falsification of internal and external validity in observational studies via conditional moment restrictions. In International Conference on Artificial Intelligence and Statistics, pp.\ 5869--5898. PMLR, 2023

    2023

  21. [29]

    M., Oberst, M., Shih, M.-C., and Sontag, D

    Hussain, Z. M., Oberst, M., Shih, M.-C., and Sontag, D. Falsification before extrapolation in causal effect estimation. Advances in Neural Information Processing Systems, 35: 0 6161--6174, 2022

    2022

  22. [30]

    and Sch \"o lkopf, B

    Janzing, D. and Sch \"o lkopf, B. Detecting confounding in multivariate linear models via spectral analysis. Journal of Causal Inference, 6 0 (1): 0 20170013, 2018

    2018

  23. [31]

    Information-geometric approach to inferring causal directions

    Janzing, D., Mooij, J., Zhang, K., Lemeire, J., Zscheischler, J., Daniu s is, P., Steudel, B., and Sch \"o lkopf, B. Information-geometric approach to inferring causal directions. Artificial Intelligence, 182: 0 1--31, 2012

    2012

  24. [32]

    and Krijthe, J

    Karlsson, R. and Krijthe, J. Detecting hidden confounding in observational data using multiple environments. Advances in Neural Information Processing Systems, 36, 2023

    2023

  25. [33]

    Putting causal identification to the test: Falsification using multi-environment data

    Karlsson, R., Creast a , S., and Krijthe, J. Putting causal identification to the test: Falsification using multi-environment data. In Causal Representation Learning Workshop at NeurIPS 2023, 2023

    2023

  26. [34]

    L., Jimenez, F., Kaya, Y., Stapleton, L

    Leite, W. L., Jimenez, F., Kaya, Y., Stapleton, L. M., MacInnes, J. W., and Sandbach, R. An evaluation of weighting methods based on propensity scores to reduce selection bias in multilevel observational studies. Multivariate behavioral research, 50 0 (3): 0 265--284, 2015

    2015

  27. [35]

    T., and Cohen, T

    Lipsitch, M., Tchetgen, E. T., and Cohen, T. Negative controls: a tool for detecting confounding and bias in observational studies. Epidemiology, 21 0 (3): 0 383--388, 2010

    2010

  28. [36]

    Learning causal models under independent changes

    Mameche, S., Kaltenpoth, D., and Vreeken, J. Learning causal models under independent changes. Advances in Neural Information Processing Systems, 36, 2024 a

    2024

  29. [37]

    Identifying confounding from causal mechanism shifts

    Mameche, S., Vreeken, J., and Kaltenpoth, D. Identifying confounding from causal mechanism shifts. In International Conference on Artificial Intelligence and Statistics, pp.\ 4897--4905. PMLR, 2024 b

    2024

  30. [38]

    Manski, C. F. Partial identification of probability distributions. Springer Science & Business Media, 2003

    2003

  31. [39]

    and Nelder, J

    Mccullagh, P. and Nelder, J. Generalized linear models. CRC press, 1989

    1989

  32. [40]

    On the testability of causal models with latent and instrumental variables

    Pearl, J. On the testability of causal models with latent and instrumental variables. In Proceedings of the Eleventh conference on Uncertainty in artificial intelligence, pp.\ 435--443, 1995

    1995

  33. [41]

    u gelgen, J., and Sch \

    Perry, R., Von K \"u gelgen, J., and Sch \"o lkopf, B. Causal discovery in heterogeneous environments under the sparse mechanism shift hypothesis. Advances in Neural Information Processing Systems, 35: 0 10904--10917, 2022

    2022

  34. [42]

    Causal inference by using invariant prediction: identification and confidence intervals

    Peters, J., B \"u hlmann, P., and Meinshausen, N. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society Series B: Statistical Methodology, 78 0 (5): 0 947--1012, 2016

    2016

  35. [43]

    Elements of Causal Inference: Foundations and Learning Algorithms

    Peters, J., Janzing, D., and Sch\" o lkopf, B. Elements of Causal Inference: Foundations and Learning Algorithms. MIT Press, 1st edition, 2017

    2017

  36. [44]

    Reddy, A. G. and Balasubramanian, V. N. Detecting and measuring confounding using causal mechanism shifts. Advances in Neural Information Processing Systems, 2024

    2024

  37. [45]

    D., Lambert, P

    Riley, R. D., Lambert, P. C., and Abo-Zaid, G. Meta-analysis of individual participant data: rationale, conduct, and reporting. Bmj, 340, 2010

    2010

  38. [46]

    Rubin, D. B. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology, 66 0 (5): 0 688, 1974

    1974

  39. [47]

    and Smola, A

    Sch \"o lkopf, B. and Smola, A. J. Learning with kernels: support vector machines, regularization, optimization, and beyond. 2002

    2002

  40. [48]

    On causal and anticausal learning

    Sch\" o lkopf, B., Janzing, D., Peters, J., Sgouritsa, E., Zhang, K., and Mooij, J. On causal and anticausal learning. In Proceedings of the 29th International Coference on International Conference on Machine Learning, ICML'12, pp.\ 459–466, Madison, WI, USA, 2012. Omnipress. ...

    2012

  41. [49]

    Shah, R. D. and Peters, J. The hardness of conditional independence testing and the generalised covariance measure . The Annals of Statistics, 48 0 (3): 0 1514 -- 1538, 2020

    2020

  42. [50]

    A distributional approach for causal inference using propensity scores

    Tan, Z. A distributional approach for causal inference using propensity scores. Journal of the American Statistical Association, 101 0 (476): 0 1619--1637, 2006

    2006

  43. [51]

    Tippett, L. H. C. The methods of statistics. 1931

    1931

  44. [52]

    VanderWeele, T. J. and Ding, P. Sensitivity analysis in observational research: introducing the e-value. Annals of internal medicine, 167 0 (4): 0 268--274, 2017

    2017

  45. [53]

    and Athey, S

    Wager, S. and Athey, S. Estimation and inference of heterogeneous treatment effects using random forests. Journal of the American Statistical Association, 113 0 (523): 0 1228--1242, 2018

    2018

  46. [54]

    and Blei, D

    Wang, Y. and Blei, D. M. The blessings of multiple causes. Journal of the American Statistical Association, 114 0 (528): 0 1574--1596, 2019

    2019

  47. [55]

    Kernel-based conditional independence test and application in causal discovery

    Zhang, K., Peters, J., Janzing, D., and Sch \"o lkopf, B. Kernel-based conditional independence test and application in causal discovery. In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence, pp.\ 804--813, 2011

    2011

  48. [56]

    Causal-learn: Causal discovery in python

    Zheng, Y., Huang, B., Chen, W., Ramsey, J., Gong, M., Cai, R., Shimizu, S., Spirtes, P., and Zhang, K. Causal-learn: Causal discovery in python. Journal of Machine Learning Research, 25 0 (60): 0 1--8, 2024

    2024

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.