Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables

AmirEmad Ghassami; Kun Zhang; Negar Kiyavash; Saber Salehkaleybar

REVIEW 3 major objections 4 minor 24 references

Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Causal order among observed variables is identifiable even when latent confounders are present, via support patterns of an overcomplete ICA mixing matrix.

desk verdict Causal-order identifiability under latent variables is a real result, but the 'faithfulness' assumption is actually a no-cancellation condition that the paper should flag clearly. read the letter →

arxiv 1908.03932 v1 pith:63GWR5V2 submitted 2019-08-11 cs.LG stat.ML

Saber Salehkaleybar , AmirEmad Ghassami , Negar Kiyavash , Kun Zhang This is my paper · ORCID

classification cs.LGstat.ML

keywords causaldiscoverylatentvariableslinearnon-GaussianacyclicmodelsovercompleteICAordertotaleffectsfaithfulnessidentifiability

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 claims that even when some causes are unobserved, the causal order among the observed variables in a linear non-Gaussian acyclic model can still be recovered, provided no total causal effect cancels to zero along multiple paths. The route is to run overcomplete independent component analysis on the observed variables and read directed-path information from which entries of the recovered mixing matrix are zero and which are nonzero. Once the order is known, the total causal effects among observed variables are shown to be non-unique in general; the paper gives an efficient way to enumerate exactly the set of all causal-effect matrices compatible with the observed distribution. It also gives a structural condition, that no latent variable may share its set of observed descendants with an observed variable, under which the effects are uniquely identified, and graphical conditions under which the number of variables in the system is identifiable.

What carries the argument

The load-bearing object is the set of support columns of the overcomplete-ICA mixing matrix $B''$, equivalently the observed-descendant sets $\mathrm{des}_o(V_i)$. Lemma 5 shows that for any two observed variables, the pattern of zeros in their two rows across all recovered columns, counted as $n_{0*}$ and $n_{*0}$, tells whether one is an ancestor of the other. Absorbing latent variables, defined by merging a latent noise into another variable when all paths from the latent variable to observed variables pass through one node, is the device that decides when the number of variables is identifiable: the graph is minimal when no absorption is possible, and minimality is equivalent to non-reducibility of $B'$ almost surely. The descendant-set matching in Theorem 15 then enumerates all data-compatible total-effect matrices.

What would settle it

Build the linear model $V_1 \to V_2$ with direct coefficient $1$ and $V_1 \to V_3 \to V_2$ with coefficients $2$ and $-0.5$, so the total effect of $V_1$ on $V_2$ is $1 + 2(-0.5) = 0$, with non-Gaussian noises and only $V_1,V_2$ observed. Run the proposed support-detection procedure: if it fails to declare a causal path from $V_1$ to $V_2$, as Lemma 5 would predict when the entry is zero, the method breaks under cancellation, showing that the guarantee depends exactly on Assumption 1.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the support pattern of the mixing matrix obtained from overcomplete ICA is a faithful mirror of ancestor–descendant relations among observed variables. Under Assumption 1, if $V_i\leadsto V_j$, then exactly one directional pattern occurs: the pair of rows $(i,j)$ in the recovered matrix has some column with a zero in row $i$ and a nonzero in row $j$, and no column with the reverse pattern. This converts causal order discovery into support-pattern inspection. Total effects are not uniquely recoverable in general, but Theorem 15 counts the candidates: there are $\prod_{i=1}^{p_o} r_i$ matrices $D$ consistent with the data, where $r_i$ is the number of variables, observed or latent, whose observed-descendant set equals that of observed variable $V_i$. Theorem 16 shows that when no latent variable's descendant set coincides with an observed variable's, the total causal effects are unique and are read directly from the normalized column of $\tilde{B}''$.

Load-bearing premise

The load-bearing premise is that a cause never acts with exactly zero total effect along all its paths to an effect: if two paths from $V_i$ to $V_j$ have coefficients that cancel, the support pattern the method reads will be wrong.

Editorial extensions

If this is right

A practitioner with non-Gaussian observations and hidden common causes can obtain a correct causal order among observed variables without modeling the latents explicitly, using only support recovery from overcomplete ICA.
When the structural condition of Theorem 16 holds, total causal effects are read off from a single normalized ICA column, giving point estimates rather than a set.
The number of latent variables is identifiable almost surely exactly for minimal graphs; latent variables that can be absorbed into another node's noise are genuinely undetectable.
The enumeration in Theorem 15 runs in $O(p_o^2 p_r)$ time, avoiding the combinatorial search over $\binom{p_r}{p_o}$ latent structures in earlier overcomplete-ICA causal discovery.
Because causal order is obtained before effect estimation, the method can be used as a preprocessing step for downstream effect-size analysis.

Reading between the lines

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

The asymmetry between $n_{0*}$ and $n_{*0}$ suggests a robustness check: bootstrap the support matrix and require the asymmetry to be stable before declaring a directed path, which would give a direct test of the method's sensitivity to finite-sample ICA errors.
The enumeration of all compatible $D$ matrices gives a natural way to incorporate domain knowledge: any externally motivated constraint on effect signs or magnitudes can prune the product set, which could make the non-uniqueness example practically resolvable.
The method's use of i.i.d. noise and fixed causal structure might transfer to time-series causal discovery if each return series is pre-whitened; the authors' own stock-index experiment suggests this direction, but formal stationarity and lag treatment are not developed in the paper.

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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

Short version: the causal-order identifiability result is real, and the paper is worth serious referee time. The main thing to know is that Assumption 1 is doing more work than the name suggests.

What's actually new: under a linear non-Gaussian acyclic model with latent confounders, the paper shows the causal order among observed variables is identifiable, and it gives a polynomial-time enumeration of all observationally equivalent total causal effect matrices. The minimal-graph condition for identifying the number of variables (Theorem 11) is also new. These are genuine advances over Hoyer et al. 2008, which required combinatorial search, and over ParceLiNGAM, which fails on simple graphs like their Figure 1. The proofs in Sections 3 and 4 are mostly coherent; the non-identifiability example is credited correctly, and the ICA reduction is standard.

The soft spot, as your stress-test notes, is Assumption 1. It is not d-separation faithfulness; it is a no-cancellation condition on total effects. Lemma 5 and Theorem 15 read descendant sets directly off support patterns of B'', so an exact cancellation along multiple paths breaks the path test and the enumeration count. For generic coefficients this is a measure-zero event, so the theory still works in the usual generic sense, but the paper should say so explicitly. Calling it 'faithfulness' is misleading and will confuse readers who expect the standard graphical definition.

Two more issues, in proportion. Theorem 15's proof is terse: completeness of the enumeration is more asserted than shown, and the step assigning Aol and All needs more detail. The experiments are also weak: no code, no comparison with Hoyer et al. or ParceLiNGAM, and the real-data analysis is qualitative. None of this undermines the central theoretical claim, but it limits how far the practical claims can be trusted.

This paper is for researchers working on causal discovery with latent confounders, especially those who care about identifiability theory rather than applied benchmarking. It deserves a serious referee. My recommendation: send it out, and ask the authors to clarify the faithfulness assumption, tighten Theorem 15's proof, and add at least one comparison to existing methods on synthetic data.

Referee Report

3 major / 4 minor

Summary. The paper studies linear non-Gaussian acyclic structural equation models with latent variables. Under a support-type faithfulness condition (Assumption 1), it proposes to solve an overcomplete ICA problem on the observed variables, recover the reduced mixing matrix B'', and read the descendant set of each observed variable from the support of the corresponding column. From these descendant sets it obtains a causal order, gives graphical conditions for identifiability of the number of variables, derives a product formula for the number of observationally equivalent total-effect matrices, and provides structural conditions for unique total causal effects. Synthetic experiments and a stock-index application are presented as illustrations.

Significance. If the main claims hold, the paper is a useful contribution: it offers a polynomial-time alternative to the combinatorial search in Hoyer et al. (2008), gives a clean graphical characterization of absorbable latent variables, and carefully demonstrates that total causal effects are not uniquely identifiable in general. The paper is also honest about the non-identifiability example and credits it properly. However, the central identifiability results rest on an assumption that is stronger than the usual faithfulness notion, and the enumeration in Theorem 15 as stated omits observationally equivalent models with absorbable latent variables. These issues are load-bearing and need to be resolved before the results can be accepted at face value.

major comments (3)

[Section 3, Assumption 1 and Lemma 5] Assumption 1 is a no-cancellation support condition, not the standard d-separation faithfulness used in causal discovery. The proofs of Lemma 1, Lemma 5, and Theorem 15 all identify the support of a column of B' with the observed descendant set of the corresponding variable. This equivalence fails under exact cancellation of total effects. For example, in the DAG V1 -> V3 -> V2 and V1 -> V2 with structural equations V1 = N1, V3 = a V1 + N3, and V2 = b V1 + c V3 + N2, choosing b = -ac gives [B]_{2,1} = 0 even though V1 reaches V2. The n0*/n*0 test in Lemma 5 then reports no causal path from V1 to V2. If the paper intends to claim identifiability under ordinary faithfulness, this claim is unsupported; if the stronger condition is intended, it should be stated as a separate assumption (e.g., 'no-cancellation faithfulness') and its restrictiveness should be discussed explicitly.
[Section 4.2, Theorem 15] The enumeration formula in Theorem 15 is not correct as stated because the product is computed from B'', the reduced matrix obtained after deleting columns that are proportional to other columns. Models with absorbable latent variables generate exactly the same observed distribution but are not counted. A concrete case is the model of Example 4: V3 latent, V3 -> V1 -> V2. The column of B' corresponding to N3 is proportional to the column corresponding to N1, so after absorption B'' has only the two observed columns and the product in Theorem 15 equals 1. Yet both the original latent-variable model and the equivalent two-variable model are compatible with the same Vo. More generally, latent variables with no observed descendants can be added arbitrarily without changing the observed distribution, so the set of 'all possible D's' is infinite unless the statement is restricted to minimal representations or to a fixed number of variables. The theorem needs an explicit minimality or fixed-dimension restriction, or a revised statement of what is being counted.
[Section 4.2, proof of Theorem 15] The proof of Theorem 15 shows that each of the Pi r_i column selections can be realized by some assignment of the matrices A_oo, A_ol, A_lo, A_ll, but it does not prove completeness, i.e., it does not show that every D generating the same observed distribution corresponds to one of the enumerated selections. In particular, representations for which the associated B' is reducible are not covered by the identifiability argument based on Proposition 3. The completeness step needs to be made explicit, and in light of the previous comment it will require additional assumptions to be true.

minor comments (4)

[Lemma 5 proof] In the proof of Lemma 5, the phrase 'there is no causal path between Vi and Vi' should read 'there is no causal path between Vi and Vj'.
[Section 4.1] In the non-identifiability example, the sentence 'The direct causal effects from Vk to Vi, from Vk to Vj, and from Vi to Vi are α, γ, and β, respectively' should say 'from Vi to Vj' rather than 'from Vi to Vi'.
[Section 3, paragraph before Example 4] The statement that B' is not reducible if and only if the columns of [I | Aol(I-All)^{-1}] are 'not linearly independent' is confusing and, when there are more columns than rows, trivially true. The intended condition is that no two columns are linearly dependent (pairwise linear independence), which is the definition of reducibility used in the paper.
[Section 5] The experiments use RICA, a heuristic overcomplete ICA method, and the paper does not discuss the conditions under which RICA recovers the true mixing matrix. The empirical results should therefore be presented as illustrative rather than as a validation of the identifiability theorems.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main identifiability results are derived from explicit assumptions and an external ICA theorem, with no fitted input renamed as prediction.

full rationale

The paper's derivation chain is self-contained relative to its stated assumptions. Assumption 1, which says the total causal effect is nonzero whenever a directed path exists, is an explicit hypothesis rather than a conclusion smuggled in; Lemmas 1 and 5 use it to translate support patterns of the recovered mixing matrix into ancestor/descendant relations. This is a conditional identifiability argument, not a definitional equivalence. The column-recovery step rests on an external overcomplete ICA identifiability theorem from Eriksson and Koivunen (2004), which does not depend on the present paper's fitted values. Theorem 15 counts possible total-effect matrices by choosing columns with identical descendant sets; the count follows from the stated support identification and is not a fitted parameter later called a prediction. Theorem 16's unique-identification condition is an explicit structural assumption, Assumption 2, and the non-identifiability example is credited to Hoyer et al. (2008). Self-citations appear mainly in the introduction and as background, and they are not load-bearing for the central claims. The strong no-cancellation content of Assumption 1 is a substantive limitation: if two paths cancel, Lemma 5 and Theorem 15 would fail. But the paper states that assumption explicitly rather than deriving its conclusions from its own output, so this is a correctness or applicability concern, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theory relies on four explicit premises: the paper's strong faithfulness assumption, non-Gaussianity of noises, acyclicity of the linear SEM, and an external overcomplete ICA identifiability theorem. No free parameters are used in the theoretical derivations; experimental hyperparameters such as the RICA regularization weight, the number of columns chosen by model selection, and the bootstrap threshold affect empirical performance but not the central claims.

assumptions (4)

domain assumption Faithfulness (Assumption 1): total causal effect [B]_{j,i} is nonzero whenever Vi to Vj is a directed path, ruling out exact cancellation of effects along multiple paths.
Stated in Section 3 and used in Lemma 1, Lemma 5, and Theorem 15. Stronger than standard d-separation faithfulness.
domain assumption All exogenous noises are non-Gaussian.
Needed for ICA identifiability of the mixing matrix; used throughout Sections 3 and 4.
standard math Eriksson-Koivunen overcomplete ICA identifiability (Proposition 3): a non-reducible mixing matrix is identifiable up to column scaling and permutation.
External theorem cited and used as the basis for recovering B'' and for Theorems 11 and 15.
domain assumption The causal graph is acyclic and the structural equations are linear.
Standard model assumption from Section 2; needed for the expansion B = (I - A)^{-1} and for the causal order definitions.

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Cite this review

Pith. "Pith review of Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables." pith.science (2026). https://pith.science/paper/63GWR5V2

@misc{pith2026190803932,
  author       = {Pith},
  title        = {Pith review of: Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/63GWR5V2}},
  note         = {Machine review of arXiv:1908.03932}
}

read the original abstract

We consider the problem of learning causal models from observational data generated by linear non-Gaussian acyclic causal models with latent variables. Without considering the effect of latent variables, one usually infers wrong causal relationships among the observed variables. Under faithfulness assumption, we propose a method to check whether there exists a causal path between any two observed variables. From this information, we can obtain the causal order among them. The next question is then whether or not the causal effects can be uniquely identified as well. It can be shown that causal effects among observed variables cannot be identified uniquely even under the assumptions of faithfulness and non-Gaussianity of exogenous noises. However, we will propose an efficient method to identify the set of all possible causal effects that are compatible with the observational data. Furthermore, we present some structural conditions on the causal graph under which we can learn causal effects among observed variables uniquely. We also provide necessary and sufficient graphical conditions for unique identification of the number of variables in the system. Experiments on synthetic data and real-world data show the effectiveness of our proposed algorithm on learning causal models.

Figures

Figures reproduced from arXiv: 1908.03932 by the authors.

**Figure 2.** Causal graph of Example 1. In the remainder of this section, we briefly describe LiNGAM algorithm, which is capable of recovering the matrix A uniquely if all variables in the model are observable and exogenous noises are non-Gaussian (Shimizu et al., 2006). The vector V in Equation (1) can be written as a linear combination of exogenous noises as follows: V = BN, (3) where B = (I − A) −1 . The above equation fits i… view at source ↗

**Figure 3.** Causal graph of Example 3. Recovering the Number of Variables in the System According to Proposition 3, the number of variables in the system can be recovered if and only if the matrix B0 is not reducible. Furthermore, Equation (6) implies that matrix B0 is not reducible if and only if the columns of the following matrix are not linearly independent: [Ipo×po |Aol(I − All) −1 ]. In the rest of this section, we will p… view at source ↗

**Figure 4.** Causal graph of Example 5. V1 and V2 are the only observed variables. Definition 9 We say a causal graph is minimal if none of its variables are absorbable. Based on above definition, a causal graph is minimal if none of the latent variables satisfy the conditions in Theorem 8. We borrowed the terminology of minimal causal graphs from Pearl (1988) for polytree causal structures. In (Pearl, 1988), a casual graph is c… view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: An example of non-identifiability of causal effects from observed variable [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: An example of casual graphs satisfying structural conditions. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Average normalized error versus number of samples. [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: The causal relationships among five world stock indicies obtained from the proposed [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Causality in linear nongaussian acyclic models in the presence of latent gaussian confounders

Zhitang Chen and Laiwan Chan. Causality in linear nongaussian acyclic models in the presence of latent gaussian confounders. Neural Computation, 25 0 (6): 0 1605--1641, 2013

work page 2013

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Bradley Efron and Robert J Tibshirani. An introduction to the bootstrap. CRC press, 1994

work page 1994

[3] [3]

Discovering unconfounded causal relationships using linear non-gaussian models

Doris Entner and Patrik O Hoyer. Discovering unconfounded causal relationships using linear non-gaussian models. In JSAI International Symposium on Artificial Intelligence, pages 181--195. Springer, 2010

work page 2010

[4] [4]

Identifiability, separability, and uniqueness of linear ica models

Jan Eriksson and Visa Koivunen. Identifiability, separability, and uniqueness of linear ica models. IEEE signal processing letters, 11 0 (7): 0 601--604, 2004

work page 2004

[5] [5]

Learning minimal latent directed information polytrees

Jalal Etesami, Negar Kiyavash, and Todd Coleman. Learning minimal latent directed information polytrees. Neural computation, 28 0 (9): 0 1723--1768, 2016

work page 2016

[6] [6]

Budgeted experiment design for causal structure learning

AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash, and Elias Bareinboim. Budgeted experiment design for causal structure learning. In International Conference on Machine Learning, pages 1724--1733, 2018

work page 2018

[7] [7]

Estimation of causal effects using linear non-gaussian causal models with hidden variables

Patrik O Hoyer, Shohei Shimizu, Antti J Kerminen, and Markus Palviainen. Estimation of causal effects using linear non-gaussian causal models with hidden variables. International Journal of Approximate Reasoning, 49 0 (2): 0 362--378, 2008

work page 2008

[8] [8]

Nonlinear causal discovery with additive noise models

Patrik O Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Sch \"o lkopf. Nonlinear causal discovery with additive noise models. In Advances in neural information processing systems, pages 689--696, 2009

work page 2009