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Causal Effect Identification in Heterogeneous Environments from Higher-Order Moments

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Given two environments that differ in exactly one causal parameter, the treatment effect β can be recovered from higher-order moments: uniquely when the change is in a path coefficient, and up to two candidates when the change is in a…

desk verdict Useful four-way identifiability map for a two-environment linear confounding triangle, but the proofs rely on an unstated zero-mean noise assumption that the experiments also violate. read the letter →

arxiv 2506.11756 v1 pith:I25SB2O2 submitted 2025-06-13 cs.AI cs.LGmath.PR

Yaroslav Kivva , Sina Akbari , Saber Salehkaleybar , Negar Kiyavash This is my paper · ORCID
classification cs.AIcs.LGmath.PR MSC 62D2062H12
keywords causaleffectidentifiabilitylatentconfoundingheterogeneousenvironmentshigher-ordermomentstreatmentestimationstructuralmodelsnon-Gaussiannoisemulti-environmentinference
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 studies a common obstruction: a treatment T and an outcome Y are both influenced by an unobserved confounder U, so ordinary regression of Y on T does not give the causal effect β. It shows that if data come from two environments that are identical except for a single component of the data-generating mechanism, that causal effect can be recovered from higher-order moments of the observed variables. Recovery is unique when the varying component is the influence of U on T (α) or on Y (γ), and it yields at most two candidate values when the varying component is the exogenous noise of T or of U. It also proves that identifiability is lost when both noise terms vary, and gives a procedure to detect which component actually changed. This matters because multi-site observational data often exhibit exactly such single-mechanism shifts, and no experiments are required.

What carries the argument

The load-bearing object is a family of ratio identities built from cross-moments. For example, when only ϵt varies, the first moment k where the two noise distributions differ gives β = (E[Y⁽¹⁾(T⁽¹⁾)^{k−1}] − E[Y⁽²⁾(T⁽²⁾)^{k−1}]) / (E[(T⁽¹⁾)^k] − E[(T⁽²⁾)^k]), because all invariant terms cancel. The same difference-and-divide device is adapted to each of the four cases, with the cross-moment subroutine GetRatio recovering coefficient ratios such as γ/α from variables that share a common latent term. The non-identifiability proof uses an explicit reconstruction that swaps αϵu with ϵt and replaces γ with −γ/α, producing identical observed distributions with a different β.

What would settle it

Run Algorithm 1 on two environments generated with only ϵt changing, all noises drawn from uncentered exponential distributions with different rates; if the estimate does not approach the true β as sample size grows, the zero-mean premise is necessary and the stated identifiability result is incomplete.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3.1: for two linear structural causal models compatible with U→T, U→Y, and T→Y, if exactly one non-outcome parameter or noise distribution differs across environments, β is identifiable—uniquely when the difference is in α or γ, and up to two candidates when it is in ϵt or ϵu. Theorems 3.2–3.5 give constructive moment formulas and algorithms for each case, and Theorem 3.6 shows that if both ϵu and ϵt vary, two observationally equivalent SCMs with different β exist, so identifiability fails. The uniqueness cases require a mild non-Gaussianity condition, namely some n with E[ϵⁿ] ≠ (n−1)E[ϵⁿ⁻²]E[ϵ²]. The paper further claims a detection step, based on second moments, that locates the varying parameter and leaves only the ϵt-versus-ϵu distinction unresolved.

Load-bearing premise

The theorems assume both that exactly one mechanism varies and that every exogenous noise has zero mean, the latter being unstated in Assumptions 1–4; if either fails, the ratio estimators are no longer guaranteed to recover β.

Editorial extensions

If this is right

  • Given knowledge of which single parameter changed, Algorithms 1–4 estimate β consistently as sample size grows, while ordinary least-squares baselines remain systematically biased.
  • A change in α or γ identifies the causal effect exactly; a change in ϵt or ϵu leaves only two candidates, so the ambiguity is finite and searchable.
  • The Section 3.5 detection procedure distinguishes γ, α, and the noise cases from observed second moments alone; only the ϵt-versus-ϵu distinction is unresolvable.
  • If both ϵu and ϵt vary across environments, two environments cannot identify β at all, regardless of sample size.
  • Observed covariates can be absorbed by residualization, and more than two environments can be handled pairwise, making the two-environment analysis the core building block.

Reading between the lines

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

  • A practitioner who runs the full pipeline and finds two candidate effects should treat the ϵt/ϵu ambiguity as a real modeling choice; a third environment with a different kind of shift is the natural way to break the tie.
  • Every proof cancels terms using zero-mean noises, so the effective theorem includes an unstated zero-mean condition; centering the data before applying the algorithms is a cheap, testable fix.
  • Because the algorithms need only the first moment index k where the changing distributions differ, they can be run without knowing the noise family, and the paper's histograms suggest k is usually small for common distributions.
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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 / 5 minor

Summary. The paper studies the problem of estimating the causal effect β of treatment T on outcome Y in a linear structural causal model with a latent confounder U, using observational data from two environments. Under the assumption that exactly one of the coefficients α, γ or one of the exogenous noises ε_t, ε_u varies across environments, the authors propose moment-based estimators (Algorithms 1–4) and prove that β is uniquely identifiable when the change is in α or γ, and identifiable up to two candidates when the change is in ε_t or ε_u. They also prove a non-identifiability result when both ε_u and ε_t vary, and provide a procedure to detect the source of the change. The paper reports synthetic experiments with exponential, Gamma, Gumbel, and Logistic noises.

Significance. The paper addresses a timely and relevant problem: estimating a causal effect under latent confounding when data come from multiple environments with a single unknown source of heterogeneity. The constructive moment-based estimators and the explicit non-identifiability theorem are useful contributions, and the paper is careful in distinguishing cases where β is uniquely identifiable from cases where only a finite set of candidates can be recovered. The reliance on the GetRatio subroutine from prior work is clearly flagged. If the zero-mean gap described in the major comments is fixed by amending the assumptions and adjusting the experiments, the results would be a valuable addition to the multi-environment causal inference literature.

major comments (2)
  1. [Assumptions 1–4 and Appendix A (Eqs. (10), (13), (24)–(25))] The proofs of Theorems 3.2–3.5 and the change-detection tests in Section 3.5 rely on the assumption that all exogenous noises have zero mean, but Assumptions 1–4 never state this. For instance, in the proof of Theorem 3.2, Eq. (10) drops terms involving E[ε_y] by stating that 'the exogenous noises are independent and mean zero.' If the means are nonzero, the ratio in Algorithm 1 is biased; analogous biases appear in Algorithms 2–4 and in the detection ratios of Section 3.5. For the γ case, letting μ_u=E[ε_u], μ_t=E[ε_t], μ_y=E[ε_y], the quantity computed in Algorithm 4, line 1, equals 2β+γ1+γ2 + [2μ_u μ_y − (γ1+γ2) μ_u μ_t]/[E[ε_u^2]+μ_u μ_t], not the claimed 2β+γ1+γ2. This is load-bearing: the estimators and classification procedure are not correct for the model class defined by Assumptions 1–4. The authors should either add a zero-mean assumption to the theorems, or modify the estimators/detection rules to account for nonzero means.
  2. [Section 5 (Experiments)] The experiments in Section 5 sample all noise variables from an exponential distribution with rate λ ∈ (0.9, 1.1), and the text does not mention centering. Exponential noises have mean 1/λ, so the data used in Figures 4–11 do not satisfy the (unstated) zero-mean condition under which the estimators are proven. As a result, the reported convergence of the algorithms does not validate the theorems as stated. The authors should either center the noises in the experimental protocol, add the zero-mean assumption to the theorems, or clearly report and justify the experimental regime, and then re-run or re-describe the simulations accordingly.
minor comments (5)
  1. [Theorem 3.1 vs. Theorem 3.5] Theorem 3.1 states that β is uniquely identified when F⊂{α,γ}, but the α case in Theorem 3.5 holds only 'almost surely' with respect to Lebesgue measure on coefficients; this qualifier should appear in Theorem 3.1 and the abstract.
  2. [Section 3.3, Step 5, Case 2] The expression 'ϕ(1)n∗ − ϕ(1)n∗ = 0' appears to be a typo; it should read 'ϕ(1)n∗ − ϕ(2)n∗ = 0', and likewise 'ϕ(1)n∗ + ϕ(1)n∗ ≠ 0' should read 'ϕ(1)n∗ + ϕ(2)n∗ ≠ 0'.
  3. [Algorithm 3 (Section 3.4)] Algorithm 3 has input 'F(M(1),M(2)) = {ϵt}', but the algorithm is for the α case and the input should be '{α}'.
  4. [Figure 4 caption] In Figure 4 (and similarly in Figures 5–11), the caption for panel (b) repeats 'only ϵt'; it should be 'only ϵu'.
  5. [Throughout] The paper should explicitly state that α≠0 and γ≠0, since several steps divide by these coefficients (e.g., the rescaling in Section 3.3 and the construction in Theorem 3.6).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the moment-ratio estimators are derived by direct algebra, and the cited subroutines (GetRatio, external counterexample) are not restatements of the target effect.

full rationale

The central derivation is not circular. Theorem 3.2's proof expands E[Y(T)^{k-1}] and E[T^k] in terms of noise moments and cancels invariant terms, giving beta directly as a ratio of observable moments (Eq. (11)); this is closed-form algebra, not a fitted parameter. Theorems 3.3-3.5 and Algorithms 2-4 similarly derive moment identities; the only imported subroutine is GetRatio from the authors' prior work (Kivva et al., 2024), used to recover gamma/alpha from a pair of variables sharing a hidden component. That is a cited external lemma with its own assumptions, not a restatement of the target beta, and the non-identifiability result (Theorem 3.6) uses an external counterexample from Salehkaleybar et al. (2020). The proofs do rely on an unstated zero-mean noise premise, for example Eq. (10) says 'exogenous noises are independent and mean zero' although Assumptions 1-4 do not state it, and the experiments use uncentered exponential noise; this is an omitted-assumption/correctness gap, not a circular reduction, because the moment formulas are not defined in terms of the quantity they estimate. The Appendix's admission that Algorithm 3 can use an incorrect n for logistic noise at small sample sizes is likewise an acknowledged limitation, not a circular step. No step was found in which an input is defined by the output or a prediction equals a fit by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper's theory depends on the standard linear-SCM machinery plus several unstated-but-used conditions: zero-mean noises, nonzero alpha and gamma, and the single-change promise. The non-Gaussianity conditions are explicit. No new entities are introduced.

assumptions (8)
  • domain assumption Linear SCM: each variable is a linear function of its parents plus exogenous noise (Assumption 1)
    The entire setup and all estimators assume linearity; without it the moment ratios do not have the stated meaning.
  • domain assumption Causal effect invariance: beta is identical across environments (Assumption 2)
    The target of estimation is assumed constant; all derivations use beta(1)=beta(2).
  • domain assumption Finite moments of all orders (Assumption 3)
    Higher-order moment formulas require existence of all moments.
  • domain assumption Distributions determined by moments (Assumption 4)
    Used to ensure distinct noise distributions differ in some moment, and to justify the minimal-k search.
  • ad hoc to paper Exogenous noises have zero mean
    Proofs (e.g., Eq. (10) of Theorem 3.2) rely on E[epsilon_y]=E[epsilon_u]=0 to drop cross terms, but this is never stated in Assumptions 1-4.
  • domain assumption Exactly one parameter or noise distribution varies between the two environments (|F|=1)
    The classification procedure in Theorem 3.1 requires this; if two things vary, the case may be covered by Theorem 3.6 (unidentifiable) or left open.
  • domain assumption Non-Gaussianity of epsilon_t (in Theorems 3.3 and 3.4) or of epsilon_u (in Theorem 3.5)
    The theorems require existence of n with E[epsilon^n] != (n-1)E[epsilon^2]E[epsilon^{n-2}], which fails for Gaussian noises; this is stated in the theorem statements.
  • ad hoc to paper alpha != 0 and gamma != 0 for rescaling and ratios
    The proofs divide by alpha (Theorem 3.6) and use gamma as a denominator; degenerate zero parameters are not discussed.

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Cite this review

Pith. "Pith review of Causal Effect Identification in Heterogeneous Environments from Higher-Order Moments." pith.science (2026). https://pith.science/paper/I25SB2O2

@misc{pith2026250611756,
  author       = {Pith},
  title        = {Pith review of: Causal Effect Identification in Heterogeneous Environments from Higher-Order Moments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I25SB2O2}},
  note         = {Machine review of arXiv:2506.11756}
}
read the original abstract

We investigate the estimation of the causal effect of a treatment variable on an outcome in the presence of a latent confounder. We first show that the causal effect is identifiable under certain conditions when data is available from multiple environments, provided that the target causal effect remains invariant across these environments. Secondly, we propose a moment-based algorithm for estimating the causal effect as long as only a single parameter of the data-generating mechanism varies across environments -- whether it be the exogenous noise distribution or the causal relationship between two variables. Conversely, we prove that identifiability is lost if both exogenous noise distributions of both the latent and treatment variables vary across environments. Finally, we propose a procedure to identify which parameter of the data-generating mechanism has varied across the environments and evaluate the performance of our proposed methods through experiments on synthetic data.

Figures

Figures reproduced from arXiv: 2506.11756 by the authors.

Figure 1
Figure 1. Causal graph of a linear SCM in the i-th domain. More specifically, in each domain i ∈ {1, 2}, we consider the following linear SCM (with the corresponding causal graph in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Causal graph with observed covariates X. 3.1 THE CASE F M(1) ,M(2) = {ϵt} Here we consider the case where the distribution of ϵt is changing across the two environments. The corresponding SCMs for environments 1 and 2 can be simplified to1    U (i) := ϵu, T (i) := αU(i) + ϵ (i) t , Y (i) := βT(i) + γU(i) + ϵy. (2) Theorem 3.2. Suppose M(1) ,M(2) are linear SCMs compatible with the DAG of [PITH_FULL_IMAGE:figu… view at source ↗
Figure 3
Figure 3. The causal structure corresponding to SCM [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: Relative estimation bias given data from two domains, when only [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: Histogram of k in Algorithm 1 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Histogram of k in Algorithm 2 [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]
Figure 14
Figure 14. Figure 14: Histogram of n1 in Algorithm 3 [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Histogram of n2 in Algorithm 3 [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]
Figure 16
Figure 16. Figure 16: Histogram of n in Algorithm 4 [PITH_FULL_IMAGE:figures/full_fig_p032_16.png]

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Works this paper leans on

36 extracted references · 26 canonical work pages

  1. [1]

    Does compulsory school attendance affect schooling and earnings? The Quarterly Journal of Economics, 106 0 (4): 0 979--1014, 1991

    Joshua D Angrist and Alan B Krueger. Does compulsory school attendance affect schooling and earnings? The Quarterly Journal of Economics, 106 0 (4): 0 979--1014, 1991

    work page 1991

  2. [2]

    Invariant risk minimization

    Martin Arjovsky, L \'e on Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019

    arXiv 1907

  3. [3]

    Meta-Learned Invariant Risk Minimization

    Jun-Hyun Bae, Inchul Choi, and Minho Lee. Meta-learned invariant risk minimization. arXiv preprint arXiv:2103.12947, 2021

    work page Pith review arXiv 2021

  4. [4]

    Transportability from multiple environments with limited experiments: Completeness results

    Elias Bareinboim and Judea Pearl. Transportability from multiple environments with limited experiments: Completeness results. Advances in neural information processing systems, 27, 2014

    work page 2014

  5. [5]

    Differentiable causal discovery from interventional data

    Philippe Brouillard, S \'e bastien Lachapelle, Alexandre Lacoste, Simon Lacoste-Julien, and Alexandre Drouin. Differentiable causal discovery from interventional data. Advances in Neural Information Processing Systems, 33: 0 21865--21877, 2020

    work page 2020

  6. [6]

    Invariance, causality and robustness

    Peter B \"u hlmann. Invariance, causality and robustness. Statistical Science, 35 0 (3): 0 404--426, 2020

    work page 2020

  7. [7]

    Minimum wages and employment: A case study of the fast food industry in new jersey and pennsylvania

    David Card. Minimum wages and employment: A case study of the fast food industry in new jersey and pennsylvania. American Economic Review, 1993

    work page 1993

  8. [8]

    Learning causal structures using regression invariance

    AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash, and Kun Zhang. Learning causal structures using regression invariance. Advances in Neural Information Processing Systems, 30, 2017

    work page 2017

Show all 36 references
  1. [9]

    Identifying invariant factors across multiple environments with kl regression

    Jaime Roquero Gimenez and James Zou. Identifying invariant factors across multiple environments with kl regression. arXiv preprint arXiv:2002.08341, 2020

    2002 arXiv

  2. [10]

    a gele, Jonas Rothfuss, Lars Lorch, Vignesh Ram Somnath, Bernhard Sch \

    Alexander H \"a gele, Jonas Rothfuss, Lars Lorch, Vignesh Ram Somnath, Bernhard Sch \"o lkopf, and Andreas Krause. Bacadi: Bayesian causal discovery with unknown interventions. In International Conference on Artificial Intelligence and Statistics, pages 1411--1436. PMLR, 2023

    2023

  3. [11]

    Invariant causal prediction for nonlinear models

    Christina Heinze-Deml, Jonas Peters, and Nicolai Meinshausen. Invariant causal prediction for nonlinear models. Journal of Causal Inference, 6 0 (2): 0 20170016, 2018

    2018

  4. [12]

    Causal discovery from heterogeneous/nonstationary data

    Biwei Huang, Kun Zhang, Jiji Zhang, Joseph Ramsey, Ruben Sanchez-Romero, Clark Glymour, and Bernhard Sch \"o lkopf. Causal discovery from heterogeneous/nonstationary data. Journal of Machine Learning Research, 21 0 (89): 0 1--53, 2020

    2020

  5. [13]

    Causal inference in the social sciences

    Guido W Imbens. Causal inference in the social sciences. Annual Review of Statistics and Its Application, 11, 2024

    2024

  6. [14]

    Causal discovery from soft interventions with unknown targets: Characterization and learning

    Amin Jaber, Murat Kocaoglu, Karthikeyan Shanmugam, and Elias Bareinboim. Causal discovery from soft interventions with unknown targets: Characterization and learning. Advances in neural information processing systems, 33: 0 9551--9561, 2020

    2020

  7. [15]

    A cross-moment approach for causal effect estimation

    Yaroslav Kivva, Saber Salehkaleybar, and Negar Kiyavash. A cross-moment approach for causal effect estimation. Advances in Neural Information Processing Systems, 36, 2024

    2024

  8. [16]

    General transportability--synthesizing observations and experiments from heterogeneous domains

    Sanghack Lee, Juan Correa, and Elias Bareinboim. General transportability--synthesizing observations and experiments from heterogeneous domains. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 10210--10217, 2020

    2020

  9. [17]

    Nonlinear invariant risk minimization: A causal approach

    Chaochao Lu, Yuhuai Wu, Jo \'s e Miguel Hern \'a ndez-Lobato, and Bernhard Sch \"o lkopf. Nonlinear invariant risk minimization: A causal approach. arXiv preprint arXiv:2102.12353, 2021

    2021 arXiv

  10. [18]

    Learning causal models under independent changes

    Sarah Mameche, David Kaltenpoth, and Jilles Vreeken. Learning causal models under independent changes. Advances in Neural Information Processing Systems, 36, 2024

    2024

  11. [19]

    Joint causal inference from multiple contexts

    Joris M Mooij, Sara Magliacane, and Tom Claassen. Joint causal inference from multiple contexts. Journal of machine learning research, 21 0 (99): 0 1--108, 2020

    2020

  12. [20]

    Causality

    Judea Pearl. Causality. Cambridge university press, 2009

    2009

  13. [21]

    Probabilistic reasoning in intelligent systems: networks of plausible inference

    Judea Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Elsevier, 2014

    2014

  14. [22]

    u gelgen, and Bernhard Sch \

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

    2022

  15. [23]

    Causal inference by using invariant prediction: identification and confidence intervals

    Jonas Peters, Peter B \"u hlmann, and Nicolai Meinshausen. 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

  16. [24]

    Invariant causal prediction for sequential data

    Niklas Pfister, Peter B \"u hlmann, and Jonas Peters. Invariant causal prediction for sequential data. Journal of the American Statistical Association, 114 0 (527): 0 1264--1276, 2019

    2019

  17. [25]

    Marginal structural models and causal inference in epidemiology, 2000

    James M Robins, Miguel Angel Hernan, and Babette Brumback. Marginal structural models and causal inference in epidemiology, 2000

    2000

  18. [26]

    The central role of the propensity score in observational studies for causal effects

    Paul R Rosenbaum and Donald B Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70 0 (1): 0 41--55, 1983

    1983

  19. [27]

    a usler, Peter B \

    Dominik Rothenh \"a usler, Peter B \"u hlmann, and Nicolai Meinshausen. Causal dantzig. The Annals of Statistics, 47 0 (3): 0 1688--1722, 2019

    2019

  20. [28]

    a usler, Nicolai Meinshausen, Peter B \

    Dominik Rothenh \"a usler, Nicolai Meinshausen, Peter B \"u hlmann, and Jonas Peters. Anchor regression: Heterogeneous data meet causality. Journal of the Royal Statistical Society Series B: Statistical Methodology, 83 0 (2): 0 215--246, 2021

    2021

  21. [29]

    Learning linear non-gaussian causal models in the presence of latent variables

    Saber Salehkaleybar, AmirEmad Ghassami, Negar Kiyavash, and Kun Zhang. Learning linear non-gaussian causal models in the presence of latent variables. Journal of Machine Learning Research, 21 0 (39): 0 1--24, 2020

    2020

  22. [30]

    Causal discovery of linear non-gaussian causal models with unobserved confounding

    Daniela Schkoda, Elina Robeva, and Mathias Drton. Causal discovery of linear non-gaussian causal models with unobserved confounding. arXiv preprint arXiv:2408.04907, 2024

    2024 arXiv

  23. [31]

    Invariant representation learning for treatment effect estimation

    Claudia Shi, Victor Veitch, and David M Blei. Invariant representation learning for treatment effect estimation. In Uncertainty in artificial intelligence, pages 1546--1555. PMLR, 2021

    2021

  24. [32]

    Permutation-based causal structure learning with unknown intervention targets

    Chandler Squires, Yuhao Wang, and Caroline Uhler. Permutation-based causal structure learning with unknown intervention targets. In Conference on Uncertainty in Artificial Intelligence, pages 1039--1048. PMLR, 2020

    2020

  25. [33]

    Causal invariance learning via efficient optimization of a nonconvex objective

    Zhenyu Wang, Yifan Hu, Peter B \"u hlmann, and Zijian Guo. Causal invariance learning via efficient optimization of a nonconvex objective. arXiv preprint arXiv:2412.11850, 2024

    2024

  26. [34]

    Optimization-based causal estimation from heterogeneous environments

    Mingzhang Yin, Yixin Wang, and David M Blei. Optimization-based causal estimation from heterogeneous environments. J. Mach. Learn. Res, 25: 0 1--44, 2024

    2024

  27. [35]

    Invariant causal prediction for block mdps

    Amy Zhang, Clare Lyle, Shagun Sodhani, Angelos Filos, Marta Kwiatkowska, Joelle Pineau, Yarin Gal, and Doina Precup. Invariant causal prediction for block mdps. In International Conference on Machine Learning, pages 11214--11224. PMLR, 2020

    2020

  28. [36]

    Causal discovery with heterogeneous observational data

    Fangting Zhou, Kejun He, and Yang Ni. Causal discovery with heterogeneous observational data. In Uncertainty in Artificial Intelligence, pages 2383--2393. PMLR, 2022

    2022

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