Causal Discovery for Linear Non-Gaussian Models with Disjoint Cycles

Elina Robeva; Marina Garrote-L\'opez; Mathias Drton; Niko Nikov; Y. Samuel Wang

REVIEW 4 major objections 5 minor 31 references

Causal Discovery for Linear Non-Gaussian Models with Disjoint Cycles

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two cyclic linear non-Gaussian graphs are equivalent exactly when one reverses disjoint cycles; a moment-based algorithm learns such graphs.

desk verdict A solid equivalence theorem, but the discovery algorithm's core lemma is false as stated; consistency claim unsound. read the letter →

arxiv 2507.10767 v1 pith:QQIGJR2X submitted 2025-07-14 math.ST stat.MEstat.MLstat.TH

Mathias Drton , Marina Garrote-López , Niko Nikov , Elina Robeva , Y. Samuel Wang This is my paper · ORCID

classification math.STstat.MEstat.MLstat.TH MSC 62H22

keywords linearnon-Gaussianstructuralequationmodelscausaldiscoverywithcyclesfeedbackloopscycle-disjointgraphsthird-ordermomentsdistributionequivalencetrekseparationstrongcomponents

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

This paper asks which directed graphs produce the same observational distribution in a linear structural equation model with independent non-Gaussian noise, and how to recover such a graph from data when feedback cycles are present. Its main theoretical result is that two graphs are distribution-equivalent exactly when one is obtained from the other by reversing a set of vertex-disjoint directed cycles and shifting the incoming edges of the reversed nodes accordingly. For graphs in which every node lies on at most one cycle, the paper shows that simple quadratic and cubic polynomial equations in the second and third moments identify the source cycles, their internal edges and weights, and the edges between cycles, yielding a consistent algorithm that avoids the permutation search required by earlier ICA-based methods. A sympathetic reader would care because causal discovery with feedback has resisted moment-based treatment, and this work gives the first complete equivalence-class characterization and a scalable discovery procedure for the disjoint-cycles case.

What carries the argument

The load-bearing mechanism is a pair of algebraic moment constraints: for each pair of vertices $u,v$, the $2\times 2$ determinant $d^{2\times 2}_{uv}$ formed from the row pair $(s_{uu},s_{uv})$ and $(t_{uuu},t_{uuv})$ vanishes identically exactly when $u$ and $v$ have no common ancestor and $v$ is not an ancestor of $u$; the $3\times 3$ determinant $d^{3\times 3}_{uv}$ of the matrix with rows $(s_{uu},s_{uv},s_{vv})$, $(t_{uuu},t_{uuv},t_{uvv})$, and $(t_{uuv},t_{uvv},t_{vvv})$ vanishes identically exactly when no simple 2-trek with non-empty sides connects $u$ to $v$. These constraints, derived from the trek rule for second and third moments, identify root nodes and root cycles; Lemma 4.8 then certifies a root cycle by checking that $E[X_c^2 (X_{D.C})_d]=0$ for all $c$ in the candidate cycle and $d$ in any other candidate. Because cycle-disjoint graphs have strong components that are simple cycles, the algorithm peels each root cycle away by regression, recovers internal edge weights from rank conditions on the matrices $A^{(2)}_{uv}$ and $A^{(3)}_{uvw}$, and finally recovers inter-cycle edge weights via multivariate regression using the block-topological order.

What would settle it

Simulate data from a linear non-Gaussian structural equation model on a graph with overlapping cycles, such as vertices $1\to 2\to 3\to 1$ and $2\to 3\to 4\to 2$, with skewed independent errors and generic coefficients: if Algorithm 1 returns a full graph rather than halting at the overlapping root component, the cycle-disjoint assumption would be violated, and if it recovers a graph not distribution-equivalent to the generating one under Theorem 3.5, the theorem would be false. For the theorem itself, searching for two directed graphs not connected by a disjoint-cycle-reversal permutation whose models coincide for some generic non-Gaussian independent noise would refute the completeness direction.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.5: for any directed graph $G$, the set of distributions generated by linear non-Gaussian structural equation models on $G$ equals the set generated on $G'$ if and only if there is a permutation $\pi$ of the vertices that factors into vertex-disjoint directed cycles of $G$, and $G'$ is obtained from $G$ by reversing those cycles and, for every edge $k\to j$ entering a node $j$ of a reversed cycle, redirecting it to the node $i$ that now points to $j$. The theorem also gives the parameter map: the new coefficients and the independent noise vector are determined by the old ones, so equality in distribution is realized by a permutation-dilation transformation. For the restricted class of cycle-disjoint graphs, the paper proves that the strong components are exactly the directed cycles, and that the determinants $d^{2\times 2}_{uv}$ and $d^{3\times 3}_{uv}$ computed from covariances and third moments vanish identically precisely under trek-separation conditions that mark root nodes and candidate root cycles; after regressing out a confirmed root cycle, the procedure recursively peels the graph and then recovers inter-cycle edges by regression. Algorithm 1 therefore consistently recovers a graph distribution-equivalent to the generating one under generic parameter values.

Load-bearing premise

The discovery algorithm assumes the true graph is cycle-disjoint, meaning every vertex lies on at most one directed cycle so that each strongly connected component is a simple cycle; if the graph contains overlapping cycles, the algorithm detects a root component that is not a simple cycle and halts, exactly as the paper acknowledges in its supplement.

Editorial extensions

If this is right

In any linear non-Gaussian causal model with cycles, observed data can identify the graph only up to reversing vertex-disjoint directed cycles and redirecting incoming edges; all graphs obtained this way form the full distribution-equivalence class.
For cycle-disjoint graphs, the block-topological order of cycles, the skeleton and orientation of each cycle, and all edge weights are identifiable from second and third moments alone, so no search over permutations is needed.
The algorithm is consistent: with generic coefficients and moments, as sample size grows it recovers a graph distribution-equivalent to the true one with probability tending to one, and all equivalent graphs can then be enumerated via Theorem 3.5.
In simulations the moment-based procedure handles graphs with 60 vertices in under 1.25 hours at 100,000 samples, while the ICA-based baseline method fails to finish within 24 hours at 24 vertices.

Reading between the lines

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

The equivalence characterization implies that any consistent cyclic causal discovery method, not just this one, can at best output an equivalence class of graphs related by disjoint-cycle reversal; comparing methods should therefore be done on equivalence classes rather than single graphs.
The root-peeling strategy could plausibly be extended to graphs whose strong components contain overlapping cycles by adding higher-order moment constraints that characterize larger strongly connected components, although the paper does not show this.
Because a cycle is recovered only up to orientation, feeding the estimated equivalence class into intervention-based or stability-based selection could pin down the actual feedback direction; the paper does not explore this downstream use.
The moment equations depend only on third moments, so an explicit sample-size versus skewness trade-off for near-Gaussian errors would be a useful testable extension of the consistency result.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here is the short version: the paper has a genuine contribution in Theorem 3.5, which characterizes distribution equivalence for linear non-Gaussian cyclic SEMs in terms of reversing disjoint cycles. The proof via Darmois-Skitovich is clean and I believe it. The rest of the paper, the discovery algorithm, has a load-bearing flaw: Theorem 4.3(a) is false.

For 1 → 2 with independent skewed noise, X1 = ε1, X2 = λX1 + ε2. Then s11 = σ1, s12 = λσ1, t111 = τ1, t112 = λτ1, so d_{1,2}^{2×2} = 0 identically. Under the paper's own definition of ancestor (paths of length ≥ 0), 1 is a common ancestor of 1 and 2, so the claimed 'only if' fails. The same example kills the use of Theorem 4.3(a) in the proof of Lemma 4.8, where it is asserted that d_{c,d}^{2×2} is not identically zero when d is an ancestor of c. That assertion is false. Since the algorithm's step of identifying root components and Corollary 4.13 rest on these statements, the consistency guarantee is unsound as presented. This is not a small gap; it is the algebraic engine of the algorithm.

What is genuinely good? Theorem 3.5 is a real advance for cyclic models. The moment-based strategy, using only second and third moments to find source cycles, is a nice idea and the authors are transparent about the cycle-disjoint restriction and the measure-zero genericity assumption. The supplement is thorough and the code appears to be provided.

The other soft spots are secondary: the simulations pick α and the multiple-testing procedure post hoc, which flatters the numbers; and the relationship between Theorem 3.5 and Ghassami et al. (2020) is not pinned down, so the novelty boundary is fuzzy.

Who this is for: anyone interested in cyclic linear SEMs should know Theorem 3.5. But the algorithm section needs a corrected theorem (or a revised procedure) before its consistency claim can be trusted. I would send it to a serious referee, with a warning to check the algebraic lemmas carefully. My own verdict would be a risky revise-and-resubmit.

Referee Report

4 major / 5 minor

Summary. The paper studies linear non-Gaussian structural equation models with directed cycles. It first characterizes distribution equivalence of directed graphs (Theorem 3.5), showing that two graphs generate the same model exactly when one is obtained from the other by reversing a set of vertex-disjoint cycles and adjusting incoming edges. It then restricts to cycle-disjoint graphs and develops an algorithm that uses second- and third-order moment constraints (determinants d^{2×2} and d^{3×3}) to identify root nodes and root cycles, recover edge weights within cycles, and learn edges between cycles by regression. The algorithm is claimed consistent (Corollary 4.13) and is evaluated in simulations against an ICA-based alternative.

Significance. If correct, the equivalence characterization in Theorem 3.5 is a clean and nontrivial extension of LiNGAM identifiability to cyclic models, and the moment-based discovery algorithm offers a practical polynomial-time alternative to permutation-search ICA methods. The paper ships code, detailed supplementary proofs, and experiments demonstrating scalability to p=60, which are notable strengths. However, the current version contains a false statement in Theorem 4.3(a) and an index inconsistency in Algorithm 1, both in the load-bearing component-identification part, so the significance is conditional on repair.

major comments (4)

[Section 4.1, Theorem 4.3(a)] Theorem 4.3(a) is false as stated. For the graph 1→2 with X1=ε1 and X2=λX1+ε2, the determinant d^{2×2}_{1,2}=s11 t112 − t111 s12 vanishes identically, yet anc(1)∩anc(2) contains 1 because ancestors include length-0 paths. Thus the 'only if' direction fails for the case where the first argument is an ancestor of the second, which is exactly the case needed to certify that descendants of a root have zero d_{r,u}. The proof establishes the 'if' direction and treats v∈anc(u) and simple treks with non-empty sides, but not u∈anc(v). Since Corollary 4.4 and Lemma 4.8 rely on this theorem, the component-identification part of Algorithm 1 and hence Corollary 4.13 are not justified as written.
[Algorithm 1, line 2; Supplement A.8.1] Root detection in Algorithm 1 uses d^{2×2}_{ur}=0 for all u, but Corollary 4.4 states d^{2×2}_{ru}=0 for all u. With the printed index order, the root 1 in the graph 1→2 is not detected because d^{2×2}_{2,1}=λ(λτ1σ2−σ1τ2) is generically nonzero, while the non-root 2 would be detected since d^{2×2}_{1,2}=0 and d^{2×2}_{2,2}=0. The index order must be corrected to match Corollary 4.4, and the implementation in Supplement A.8.1 should be updated accordingly.
[Section 3, Definition 3.1 and Theorem 3.5] The model class P(G) is defined for arbitrary independent noise, but the proof of Theorem 3.5 invokes Darmois-Skitovich, which requires non-Gaussian components. If Gaussian noise is allowed, the 'only if' direction is false: for Gaussian errors, many different DAGs can represent the same covariance matrix. The definition and theorem should explicitly require non-Gaussian (or generically non-Gaussian) noise components, as the paper's title and setting otherwise suggest.
[Section 4.4, Corollary 4.13] The consistency claim for Algorithm 1 is not supported by a formal proof for the sample version of the procedure. Remark 4.2 asserts that a suitable threshold renders the procedure consistent, but no theorem specifies the threshold sequence or proves that the hypothesis-testing implementation in Supplement A.8.1 (delta-method p-values, empirical likelihood, multiple-testing corrections) recovers the exact-moment decisions with probability tending to 1. Since 'consistent' is a headline claim, this gap should be closed or the claim appropriately qualified.

minor comments (5)

[Algorithm 1, line 2] The line contains the duplicated phrase 'such that such that'; this should be corrected.
[Example 3.6] The notation 'M (2,3)(G)' appears where the text defines M_{2,3}(G); the notation should be unified.
[Supplement A.3] The phrase 'non/empty' should be 'non-empty' in the proof of Theorem 4.3.
[Section 5, Figure 4] The caption and text say the displayed performance uses the α and multiple-testing correction with the best average performance, but the selection procedure (e.g., whether α is chosen on a separate validation set) is not specified; this should be clarified for reproducibility.
[Section 2, Definition 2.2] The notation R_E and R_E^* is introduced but the paper later writes Λ∈R_E^* in Theorem 3.5 while Definition 2.2 requires λij≠0 for edges; this is consistent but should be stated explicitly at first use to avoid confusion with R_E.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the algebraic moment constraints are derived from the LSEM and the trek rule, not fitted; the paper's main weakness is a false theorem, which is a correctness issue rather than circular reasoning.

full rationale

The moment model M_{2,3}(G) is defined directly from the structural equations, and every constraint used by Algorithm 1 (d^{2x2}, d^{3x3}, residual independence, inverse-covariance zero pattern, cycle-edge equations) is derived via the trek rule and block matrix algebra rather than fitted to the evaluation data. The self-citations (Drton et al. 2019 Eq. (6.1) in Lemma 4.9; Robeva and Seby 2021 for the k-trek rule; Wang and Drton 2019 for the acyclic case) are published, externally checkable identities; none is an unverified uniqueness assumption that forces the paper's conclusions, so they do not raise the circularity score. Theorem 3.5 is proved from Darmois-Skitovich and matrix algebra, not assumed. The honest limitation in Supplement A.8.2 (the algorithm halts on overlapping cycles) is not circular. The notable flaw is correctness, not circularity: Theorem 4.3(a) is false as stated; e.g., for the two-node DAG 1->2, d^{2x2}_{1,2} is identically zero even though anc(1) and anc(2) share the common ancestor 1 and 2 is not an ancestor of 1. This invalidates the stated 'only if' direction and the proof of Corollary 4.4 as written. That is a mathematical error in the derivation chain, but it is not an equation reducing to its own input, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the standard mathematical toolkit of trek separation and Darmois-Skitovich, plus the modeling assumptions of non-Gaussian skewed noise, cycle-disjoint structure, and generic parameters. No extra fitted constants are introduced by the derivation itself; the only tuning parameter is the significance level alpha used in the implementation.

free parameters (1)

alpha (hypothesis test level) = varies, not specified (tuning parameter)
The algorithm's practical implementation uses a significance level alpha for the multiple hypothesis tests that certify zero/nonzero constraints. The experiments select the best alpha and multiple testing correction per setting (Section 5).

assumptions (6)

standard math Trek rule expansions for second and third moments (Eq. 7) hold.
Background result from the graphical-models literature (Sullivant et al. 2010; Robeva and Seby 2021; Améndola et al. 2023) used to express moments in terms of paths.
standard math Darmois-Skitovich theorem applies: independent non-Gaussian components force a permutation/dilation structure.
Used in the proof of Theorem 3.5 to deduce that an orthogonal (in distribution) transformation must be a permutation with scaling.
domain assumption Noise components are independent, centered, non-Gaussian and have nonzero third-order moments (skewness).
The paper assumes this throughout (Section 2 and the abstract). Needed for identifiability of the DAG case and for the moment constraints to carry information.
domain assumption The underlying graph is cycle-disjoint (each node in at most one directed cycle).
Definition 4.1. The discovery algorithm's main results (Theorem 4.3, Corollaries 4.4/4.5, Lemmas 4.8-4.10) are proven only for this class. The authors acknowledge in Supplement A.8.2 that overlapping cycles are not handled.
domain assumption Parameters Lambda, Omega(2), Omega(3) are generic, i.e., outside a measure-zero set that violates the rank/zero constraints.
Corollary 4.13 assumes genericity to ensure the determinants are nonzero exactly when they should be. This is a mild assumption but is load-bearing for consistency.
standard math Sample moments converge to true moments and the hypothesis tests can consistently decide zero vs nonzero constraints.
Remark 4.2 states that with a suitable threshold, the procedure is consistent. The practical implementation uses hypothesis tests with a tuning parameter alpha, whose consistency is not fully proven.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Causal Discovery for Linear Non-Gaussian Models with Disjoint Cycles." pith.science (2026). https://pith.science/paper/QQIGJR2X

@misc{pith2026250710767,
  author       = {Pith},
  title        = {Pith review of: Causal Discovery for Linear Non-Gaussian Models with Disjoint Cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQIGJR2X}},
  note         = {Machine review of arXiv:2507.10767}
}

read the original abstract

The paradigm of linear structural equation modeling readily allows one to incorporate causal feedback loops in the model specification. These appear as directed cycles in the common graphical representation of the models. However, the presence of cycles entails difficulties such as the fact that models need no longer be characterized by conditional independence relations. As a result, learning cyclic causal structures remains a challenging problem. In this paper, we offer new insights on this problem in the context of linear non-Gaussian models. First, we precisely characterize when two directed graphs determine the same linear non-Gaussian model. Next, we take up a setting of cycle-disjoint graphs, for which we are able to show that simple quadratic and cubic polynomial relations among low-order moments of a non-Gaussian distribution allow one to locate source cycles. Complementing this with a strategy of decorrelating cycles and multivariate regression allows one to infer a block-topological order among the directed cycles, which leads to a {consistent and computationally efficient algorithm} for learning causal structures with disjoint cycles.

Figures

Figures reproduced from arXiv: 2507.10767 by the authors.

**Figure 3.** The 3-cycles of Example 3.6 To prove the sufficiency of our condition, we can simply calculate that (I − Λ)−Tε =d (I − Λ ′ ) −Tε ′ , where Λ ′ is determined as above, and ε ′ =d (I − Λ ′ ) T(I − Λ)−Tε = DP Tε. In our Definition 3.1 of equivalence, we can rearrange the structural equations, and thus transform the graph only if the Λ coefficients we divide by in our procedure are nonzero, which is satisfied whenever Λ… view at source ↗

**Figure 4.** Horizontal axis shows sample size in thousands. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Average time (in seconds) required per replica [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Results when each graph consists of 5-cycles. In [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 29 canonical work pages

[1]

Identification of partially observed linear causal models: Graphical conditions for the non- G aussian and heterogeneous cases

Jeffrey Adams, Niels Hansen, and Kun Zhang. Identification of partially observed linear causal models: Graphical conditions for the non- G aussian and heterogeneous cases. In M Ranzato, A Beygelzimer, Y Dauphin, P S Liang, and J Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 22822--22833. Curran Associates, I...

work page 2021
[2]

Third-order moment varieties of linear Non- G aussian graphical models

Carlos Am\' e ndola, Mathias Drton, Alexandros Grosdos, Roser Homs, and Elina Robeva. Third-order moment varieties of linear Non- G aussian graphical models . Information and Inference: A Journal of the IMA, 12 0 (3): 0 1405--1436, 2023

work page 2023
[3]

Controlling the false discovery rate: a practical and powerful approach to multiple testing

Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society Series B: Statistical Methodology, 57 0 (1): 0 289--300, 1995

work page 1995
[4]

Causal discovery with latent confounders based on higher-order cumulants

Ruichu Cai, Zhiyi Huang, Wei Chen, Zhifeng Hao, and Kun Zhang. Causal discovery with latent confounders based on higher-order cumulants. In Proceedings of the 40th International Conference on Machine Learning, ICML'23. JMLR.org, 2023

work page 2023
[5]

Local causal discovery with linear non- G aussian cyclic models

Haoyue Dai, Ignavier Ng, Yujia Zheng, Zhengqing Gao, and Kun Zhang. Local causal discovery with linear non- G aussian cyclic models. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li, editors, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Learning Research, pages 154--162....

work page 2024
[6]

Analyse générale des liaisons stochastiques: etude particulière de l'analyse factorielle linéaire

Georges Darmois. Analyse générale des liaisons stochastiques: etude particulière de l'analyse factorielle linéaire. Review of the International Statistical Institute, 21 0 (1/2): 0 2--8, 1953

work page 1953
[7]

The maximum likelihood threshold of a path diagram

Mathias Drton, Christopher Fox, Andreas K\" a ufl, and Guillaume Pouliot. The maximum likelihood threshold of a path diagram. Ann. Statist., 47 0 (3): 0 1536--1553, 2019

work page 2019
[8]

Characterizing distribution equivalence and structure learning for cyclic and acyclic directed graphs

AmirEmad Ghassami, Alan Yang, Negar Kiyavash, and Kun Zhang. Characterizing distribution equivalence and structure learning for cyclic and acyclic directed graphs. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event , volume 119 of Proceedings of Machine Learning Research, pages 3494--3504. PM...

work page 2020

Show all 31 references

[9]

Integrative modelling reveals mechanisms linking productivity and plant species richness

James B Grace, T Michael Anderson, Eric W Seabloom, Elizabeth T Borer, Peter B Adler, W Stanley Harpole, Yann Hautier, Helmut Hillebrand, Eric M Lind, Meelis P \"a rtel, et al. Integrative modelling reveals mechanisms linking productivity and plant species richness. Nature, 52...

2016
[10]

A simple sequentially rejective multiple test procedure

Sture Holm. A simple sequentially rejective multiple test procedure. Scand. J. Statist., 6 0 (2): 0 65--70, 1979

1979
[11]

Grouplingam: Linear non- G aussian acyclic models for sets of variables

Yoshinobu Kawahara, Kenneth Bollen, Shohei Shimizu, and Takashi Washio. Grouplingam: Linear non- G aussian acyclic models for sets of variables. CoRR, abs/1006.5041, 2010

2010 arXiv
[12]

Ramsey, and Patrik O

Gustavo Lacerda, Peter Spirtes, Joseph D. Ramsey, and Patrik O. Hoyer. Discovering cyclic causal models by independent components analysis. In David A. McAllester and Petri Myllym \" a ki, editors, UAI 2008, Proceedings of the 24th Conference in Uncertainty in Artificial Intel...

2008
[13]

Learning linear non- G aussian graphical models with multidirected edges

Yiheng Liu, Elina Robeva, and Huanqing Wang. Learning linear non- G aussian graphical models with multidirected edges. J. Causal Inference, 9 0 (1): 0 250--263, 2021

2021
[14]

Handbook of graphical models

Marloes Maathuis, Mathias Drton, Steffen Lauritzen, and Martin Wainwright. Handbook of graphical models. CRC Press, 2018

2018
[15]

Linear models: a useful ``microscope'' for causal analysis

Judea Pearl. Linear models: a useful ``microscope'' for causal analysis. J. Causal Inference, 1 0 (1): 0 155--169, 2013

2013
[16]

Py-tetrad and rpy-tetrad: A new python interface with r support for tetrad causal search

Joseph Ramsey and Bryan Andrews. Py-tetrad and rpy-tetrad: A new python interface with r support for tetrad causal search. In Causal Analysis Workshop Series, pages 40--51. PMLR, 2023

2023
[17]

A discovery algorithm for directed cyclic graphs

Thomas Richardson. A discovery algorithm for directed cyclic graphs. In Eric Horvitz and Finn Verner Jensen, editors, UAI '96: Proceedings of the Twelfth Annual Conference on Uncertainty in Artificial Intelligence, Reed College, Portland, Oregon, USA, August 1-4, 1996 , pages ...

1996
[18]

A polynomial-time algorithm for deciding equivalence of directed cyclic graphical models

Thomas Richardson. A polynomial-time algorithm for deciding equivalence of directed cyclic graphical models. In Eric Horvitz and Finn Verner Jensen, editors, UAI '96: Proceedings of the Twelfth Annual Conference on Uncertainty in Artificial Intelligence, Reed College, Portland...

1996
[19]

Multi-trek separation in linear structural equation models

Elina Robeva and Jean-Baptiste Seby. Multi-trek separation in linear structural equation models. SIAM Journal on Applied Algebra and Geometry, 5 0 (2): 0 278--303, 2021

2021
[20]

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

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

2020
[21]

Causal discovery of linear non- G aussian causal models with unobserved confounding

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

2024 arXiv
[22]

Causal structure learning in directed, possibly cyclic, graphical models

Pardis Semnani and Elina Robeva. Causal structure learning in directed, possibly cyclic, graphical models. Journal of Causal Inference, 13 0 (1), 2025

2025
[23]

Statistical causal discovery: L i NGAM approach

Shohei Shimizu. Statistical causal discovery: L i NGAM approach . SpringerBriefs in Statistics. Springer Japan, Tokyo, 2022. JSS Research Series in Statistics

2022
[24]

arinen, and Antti

Shohei Shimizu, Patric. O. Hoyer, Aapo Hyv\"arinen, and Antti" Kerminen. A linear non- G aussian acyclic model for causal discovery. Journal of Machine Learninig Research, 7: 0 2003--2030, 2006

2003
[25]

Hoyer, and Kenneth Bollen

Shohei Shimizu, Takanori Inazumi, Yasuhiro Sogawa, Aapo Hyv \"a rinen, Yoshinobu Kawahara, Takashi Washio, Patrik O. Hoyer, and Kenneth Bollen. DirectLiNGAM: a direct method for learning a linear non- G aussian structural equation model. Journal of Machine Learning Research, 1...

2011
[26]

V. P. Skitovich. On a property of the normal distribution. Doklady Akademii Nauk SSSR (in Russian), 89: 0 217--219, 1953

1953
[27]

Using path diagrams as a structural equation modeling tool

Peter Spirtes, Thomas Richardson, Christopher Meek, Richard Scheines, and Clark Glymour. Using path diagrams as a structural equation modeling tool. Sociological Methods & Research, 27 0 (2): 0 182--225, 1998

1998
[28]

Trek separation for G aussian graphical models

Seth Sullivant, Kelli Talaska, and Jan Draisma. Trek separation for G aussian graphical models. Annals of Statistics, 38 0 (3): 0 1665--1685, 2010

2010
[29]

Learning linear non- G aussian polytree models

Daniele Tramontano, Anthea Monod, and Mathias Drton. Learning linear non- G aussian polytree models. In James Cussens and Kun Zhang, editors, Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, volume 180 of Proceedings of Machine Learning Re...

1960
[30]

Samuel Wang and Mathias Drton

Y. Samuel Wang and Mathias Drton. High-dimensional causal discovery under Non- G aussianity . Biometrika, 107 0 (1): 0 41--59, 2019

2019
[31]

Samuel Wang and Mathias Drton

Y. Samuel Wang and Mathias Drton. Causal discovery with unobserved confounding and non- G aussian data. Journal of Machine Learning Research, 24 0 (271): 0 1--61, 2023

2023

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Identification of partially observed linear causal models: Graphical conditions for the non- G aussian and heterogeneous cases

Jeffrey Adams, Niels Hansen, and Kun Zhang. Identification of partially observed linear causal models: Graphical conditions for the non- G aussian and heterogeneous cases. In M Ranzato, A Beygelzimer, Y Dauphin, P S Liang, and J Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 22822--22833. Curran Associates, I...

work page 2021

[2] [2]

Third-order moment varieties of linear Non- G aussian graphical models

Carlos Am\' e ndola, Mathias Drton, Alexandros Grosdos, Roser Homs, and Elina Robeva. Third-order moment varieties of linear Non- G aussian graphical models . Information and Inference: A Journal of the IMA, 12 0 (3): 0 1405--1436, 2023

work page 2023

[3] [3]

Controlling the false discovery rate: a practical and powerful approach to multiple testing

Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society Series B: Statistical Methodology, 57 0 (1): 0 289--300, 1995

work page 1995

[4] [4]

Causal discovery with latent confounders based on higher-order cumulants

Ruichu Cai, Zhiyi Huang, Wei Chen, Zhifeng Hao, and Kun Zhang. Causal discovery with latent confounders based on higher-order cumulants. In Proceedings of the 40th International Conference on Machine Learning, ICML'23. JMLR.org, 2023

work page 2023

[5] [5]

Local causal discovery with linear non- G aussian cyclic models

Haoyue Dai, Ignavier Ng, Yujia Zheng, Zhengqing Gao, and Kun Zhang. Local causal discovery with linear non- G aussian cyclic models. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li, editors, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Learning Research, pages 154--162....

work page 2024

[6] [6]

Analyse générale des liaisons stochastiques: etude particulière de l'analyse factorielle linéaire

Georges Darmois. Analyse générale des liaisons stochastiques: etude particulière de l'analyse factorielle linéaire. Review of the International Statistical Institute, 21 0 (1/2): 0 2--8, 1953

work page 1953

[7] [7]

The maximum likelihood threshold of a path diagram

Mathias Drton, Christopher Fox, Andreas K\" a ufl, and Guillaume Pouliot. The maximum likelihood threshold of a path diagram. Ann. Statist., 47 0 (3): 0 1536--1553, 2019

work page 2019

[8] [8]

Characterizing distribution equivalence and structure learning for cyclic and acyclic directed graphs

AmirEmad Ghassami, Alan Yang, Negar Kiyavash, and Kun Zhang. Characterizing distribution equivalence and structure learning for cyclic and acyclic directed graphs. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event , volume 119 of Proceedings of Machine Learning Research, pages 3494--3504. PM...

work page 2020