Causal Effect Identification in lvLiNGAM from Higher-Order Cumulants
Authors: Daniele Tramontano
Yaroslav Kivva
Saber Salehkaleybar
Negar Kiyavash
Mathias Drton
Venue: ICML 2025 poster (2025)
Keywords: Causal Effect Identification
Structural Causal Models
Cumulants.
Primary area: general_machine_learning->causality

Abstract: This paper investigates causal effect identification in latent variable Linear Non-Gaussian Acyclic Models (lvLiNGAM) using higher-order cumulants, addressing two prominent setups that are challenging in the presence of latent confounding: (1) a single proxy variable that may causally influence the treatment and (2) underspecified instrumental variable cases where fewer instruments exist than treatments. We prove that causal effects are identifiable with a single proxy or instrument and provide corresponding estimation methods. Experimental results demonstrate the accuracy and robustness of our approaches compared to existing methods, advancing the theoretical and practical understanding of causal inference in linear systems with latent confounders.

Causal Effect Identification in IvLiNGAM from Higher-Order Cumulants
Daniele Tramontano'
Yaroslav Kivva? Saber Salehkaleybar' Negar Kiyavash Mathias Drton
Abstract
is associated with directed acyclic graph (DAG). Each vec- tor component is associated with a node in the graph and is function of the random variables corresponding to its parents in the graph and some exogenous noise.
This paper investigates causal effect identification in latent variable Linear Non-Gaussian Acyclic Models (lvLiGGAM) using higher-order cumu- lants, addressing two prominent setups that are challenging the presence of latent confound- ing: (1) a single proxy variable that may causally influence the treatment and (2) underspecified in- strumental variable cases where fewer instruments exist than treatments. We prove that causal effects are identifiable with a single proxy or instrument and provide corresponding estimation methods. Experimental results demonstrate the accuracy and robustness of our approaches compared to existing methods, advancing the theoretical and practical understanding of causal inference in lin- ear systems with latent confounders.
In general, latent confounders, i.c., unobserved variables affecting the treatment and the outcome of interest, often render the causal effect non-identifiable from the observa- tional distribution (Shpitser & Pearl, 2006). However, in some cases and under further assumptions the causal mechanisms, the causal effect may still be identifiable from observational data (Barber et al., 2022).
Linear models are among the most well-studied mechanisms and serve as a foundational abstraction in many scientific disciplines because they offer simple qualitative interpreta- tions and can be learned with moderate sample sizes (Pe' er & Hacohen, 201 1, Principle 1). When the exogenous noises in a linear SCM are Gaussian, the entire distributional in- formation is contained in the variables' covariance matrix. Consequently, the higher-order cumulants the distribution are uninformative (Marcinkiewiez, 1939, Thm. 2). As a re- sult, the causal structure and other causal quantities are often not identifiable from mere observational data. For instance, in the context of causal structure learning, this means the causal graph is identifiable only up to an equivalence class (e.g., Drton, 2018, $10). This motivated the widespread use of the linear non-Gaussian acyclic model (LiNGAM).
1. Introduction
Predicting the impact of an unseen intervention in sys- tem is crucial challenge in many fields, such as medicine (Sanchez et al., 2022; Michoel & Zhang, 2023), policy evalu- ation (Athey & Imbens, 2017), fair decision-making (Kilber- tus et al., 2017), and finance (de Prado, 2023). Randomized experimenns//nteevvntion studies are the gold standard for addressing this challenge but are often infeasible due to a variety of reasons, such as ethical concerns or prohibitively high costs. Thus, when merely observational data is avail- able, additional assumptions on the underlying causal sys- tem are needed to compensate for the lack of interventional data. The field of causal inference seeks to formalize such assumptions. One notable approach in causal inference is modeling causal relationships through structural causal mod- els (SCM) (Pearl, 2009). this framework, a random vector
The seminal work of Shimizu et al. (2006) showed that in the setting of LiNGAM, the true underlying causal graph is uniquely identifiable when all the variables are observed. Since then, rich literature on this topic has emerged, fo- cusing mainly on the identification and the estimation of the causal graph; see, e.g., Adams et al. (2021); Shimizu (2022); Yang et al. (2022); Wang et al. (2023); Wang & Drton (2023) for recent results that allow for the presence of hidden variables.
'Technical University Munich, Munich, Germany 'Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland 'Leiden Institute of Advanced Computer Science. Leiden Uni- versity, Netherlands "Munich Center for Machine Learning. Munich, Germany, Correspondence to: Daniele Tramontano wowpiicl.trmmontaooneeeeo
Within the LiNGAM literature, causal effect identification has received less attention; complete characterization of the identifiable causal effects was provided only recently by Tramontano et al. (2024b). The drawback of this char- acterization is that it is based on solving an overcomplete independent component analysis (OICA) problem, known to be non-separable (Eriksson & Koivunen, 2004). Hence, the approach of Tramontano et al. (2024b) does not translate
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
into a consistent estimatiin method for identifiable causal effects (Tramontano al., 2024b, $5.3).
Our second main contribution consists of practical methods to estimate identifiable causal effects in both considered setups. The methods build on the identifiability results and process finite-sample estimates of higher-order cumulants (Section 4). Our experiments show that the proposed ap- proach provides consistent estimators in causal graphs, for which previous methods in the literature fail (Section 6).
Recent works (Kivva et al., 2023; Shuai et al., 2023) have ex- ploited Con-Gausianitt by utilizing higher-order moments to derive estimation formulas for causal effects in specific causal graphs, avoiding reliance on the challenging OICA problem. A notable scenario involves the use of a proxy variable for the latent confounder (Tchetgen et al., 2024). In LiNGAM, causal effects are identifiable from higher-order moments if every latent confounder has corresponding proxy variable, and no proxy directly influences either the treatment or the outcome (Kivva et al., 2023). However, the method in Kivva et al. (2023) fails to produce consistent estimates when these assumptions are violated. Another important setup arises when an instrumental variable affects the outcome solely through the treatment (Angrist & Pis- chke, 2009, $4). For linear models, two-stage least squares (TSLS) regression can estimate causal effects when there is at least one valid instrument per treatment (Angrist & Pischke, 2009, $3.2). However, TSLS is based only on the covariance matrix, and in cases where the number of in- struments is fewer than the number of treatments-referreed to as underspecified instrumental variables-causee effects are not identifiable from the covariance matrix alone. This underspecification is often encountered in biological appli- cations (Ailer et al., 2023; 2024).
2. Problem Definition
2.1. Notation
A directed graph is pair G - (V, E) where V = [p] := {1, [1,...,p] .P] is the set of nodes and E S {(i.]) i,j E V. i # j} is the set of edges. We denote a pair (1, E E as i j.
A (directed) path from node node j in G is a sequence of nodes
(41 i,.... 1,...,,++.. fx+1 = j) such that i, + 1s41 E for s E {1. (1....,R, k). A cycle in G is path from node i to itself. A Directed Acyclic Graph (DAG) is directed graph without cycles. if + j E E, we say that is a parent of j. and j is a child of i. If there is a path from to j in g, we say that i is an ancestor of j and j is a descendant of i. The sets of parents, children, ancestors, and descendants of given node are denoted by pa(i), ch(i), an(i), and de(i). respectively. In our work, we distinguish between observed and latent variables by partitioning the nodes into two sets V = O C, of respective sizes Pa and pt- We write tensors in boldface. The entry (ii,..., ix) of a tensor T is denoted by t..........
This paper advances the field by providing identifiability results for causal effects using higher-order cumulants in two challenging setups: (1) a single proxy variable that may causally influence the treatment and (2) underspecified instrumental variables.
Cumulants are alternative representations of moments of a distribution that are particularly useful when dealing with lincar SCM (Robeva & Seby, 2021). Here, we formalize the definition and discuss their basic properties.
1.1. Contribution
Our first main contributions are identifiability results for the causal effects of interest in the aforementioned setups.
Definition 2.1. The k-th cumulant tensor of a random vec- tor N = [N NNJ is the k-way tensor in Rpxx-Xpp = (I*)** & whose entry in position (ii...., ig) is the cumulant
1. In the proxy variable setup (Section 3.1), unlike previ- ous work, our proposed method allows a causal edge from the proxy to the treatment. Additionally, it re- covers the causal effect for any 1 latent confounders using a single proxy variable, in contrast to Kivva et al. (2023, Alg. 1), which requires one proxy variable per latent confounder. Furthermore, we prove that for the proxy variable graph in Fig. 3, identification from the second and third-order cumulants alone is not possible.
Di....i :=
Σ (-1)4-1L - 1)IE (AL....AA)
II
N2 ... ...E II N₂
JEA1
JEAL
where the sum is taken over all partitions (AI...., AL) of the multiset fir....,is) (....
Cumulant tensors symmetric, i.e.,
(k) (N)...... = =c((()) (N)) := c (N) (N)(t)).. Yo E SE,
2. In the underspecified instrumental variable setup (Sec- tion 3.2), we demonstrate that the causal effects of mul- tiple treatments can be identified using only single instrumental variable. This relaxes the requirement in the existing literature on linear instrumental variables, which traditionally assumes the number of instruments to be greater than or equal to the number of treatments.
where Sy is the symmetric group on [k:). We write (y) for the subspace of symmetric tensors in (BP)).
Lemma 2.2 (Comon & Jutten 2010, $5). If the entries of N Ni,..., Ny) are jointly independent, then (kk) (N)
2

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
diagonal, i.e., ( (N) &2... equal O unless i1 = i2 Ak for some [p].
i.e., b is the average total causal effect of j i. Hoyer et al. (2008) showed that for any IvLiNGAM model, an associated canonical model exists, in which, in the cor- responding graph, all the latent nodes have at least two children and have no parents. We refer to the graph corre- sponding to a canonical model as a canonical graph. The original and the associated canonical model are observa- tionally and causally equivalent (Hoyer et al., 2008, $3). Subsequently, without loss of generality, will assume our model is canonical in this sense.
We write Diag (p) for the space of order k diagonal tensors.
Lemma 2.3 (Comon & Jutten, 2010, $5). Let N
[N.....; Np) be any p-variate random vector, and A E
Rxxp for any N, then
(k) (A N)
(l (N (N) Ju Biih
In canonical models, AI,o = Ar, = 0, and in particular
In terms of the entire k-th cumulant tensor, this amounts to
B, (I Ao,0)
B (I -Ao,0)
(4)
CO (A (11) A
(1)
For every canonical G. let R be the set of p x p real matrices A such that a, Reexp be the set of all matrices B' [B. B:] that can be obtained from matrix A E R according to (4). Let NGP be the set of dimensional, non-degenerate, jointly independent non- non-Gaussian random vectors, and let M(G) be the set of all p. dimensional random vectors that can be expressed according to (3) with B' E R Morcover, we define (() (g) S Sym (Po) to be the set of symmetric k-th tensors that can be obtained k-cumulant tensor for distributions in .M(G). MM))..... i.c.,
j + i & g. Let IRC c
where is the Tucker product between (N) and A.
2.2. Model
Let G (V, F) be fixed DAG on p nodes. On fixed probability space, let V = .Vo.... Vp be a random vector taking values in I² and satisfying the following SCM:
V = AV +N BN,
(2)
where aj.; = 0 iffi - & E, matrix B = (I A) and the entries of the exogenous noise vector N are assumed to be jointly independent and non- Gaussian. V is partitioned into [V.. o. Vıl, where V. is observed dimension Po. while V is latent and of dimension pt-
M (=) (<()(() V. ((M)) = (K(K) a B' ((a) Diag*(1) B') E ERR)
We can rewrite (2)
where the set-equality is due to Lemma 2.3. Using the second equality, we can define the following polynomial parameterization for (G)
Aooo
Ao.
V. Vi
N₂ N
+
Vi
AI,o
Ar
S R 9 Diag (p) (
(k) g
(5)
which implies that the observed random vector satisfies
(B', (()
oR B'.
V. B'N BB
B.
No N
(3)
This map expresses the tensor of observed cumulants in terms of the tensor of exogenous cumulants and the mixing matrix. Finally, we define M = M x X
where B²: [I- A) is is known as the mixing ma- trix. This model for V. is known the latent variable LiNGAM (lvLiNGAM).
( and similarly Diag (Sk) Diag(SA) (p) and $ (Sk)
g
2.3. Identifiability
Salehkaleybar et al. (2020, $3) showed that the two parts of the matrix B' can be expressed as follows:
In this work, we are interested in identifying specific tries of the mixing matrix from finitely many cumulants of the observational distribution. We formalize the prob- lem as follows. We say that the causal effect from j to is generically identifiable from the first cumulants of the distribution there Lebesgue measure zero subset sg of RR x (iag(S*) such that for all (B', D(Šk)) (RC x D(SA)) \ SS we have by,5 = by.5 for every other mixing matrix B' E R that can define the same cumu- lants up to order k, that is, whenever / (SA) (B', (B,D(Ss)) =
B. (I - 11)-1 B- -I- A') Ao, -1,)))
with A' = A..o + (I -I,
The matrix B' = (bb,)) contains information on the interventional dis- tributions of V.. In !
(V; do
by
0v;
$ (Sh) B', (EE)) for some D(sk) E Diag (5k)
'See Pearl (2009, $3) for the definition do intervention.
3

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
For the remainder of the text, whenever we use the term generic, it is implied that the result holds outside the Lesbe- gue measure zero subset of the parameter space S.
expressed in terms first k(1 1) =4:
b²
+ b -itpiaoooViaaaa((iiiaaaa
Remark 2.4 (The scaling matrix). Equation (4) implies that as long as we are focused on identifying the causal effect between observed variables alone, the scaling of the latent columns does not make difference. Hence, without loss of generality, we assume subsequently that all mixing matrices are scaled so that the first non-zero entry in each column is equal to 1. In other words, a = 1 if is the first child of in a given causal order, where and / are observed and latent variables, respectively.
(1)222)
(c(V))2220( ( c(V) 12),2 0. (6)
Lemma 3.3 (Schkoda et al., 2024, Lemma 5). Con- sider the causal graph g' from Fig. 1. For ev- ery integer k 2 2, the exogenous cumulant vector [c_ (N........11 c²
(N) IN.............. (N) is solu- tion of the following linear system
(*(N) c c(NN
3. Main Results
bg.
b
This section presents our main identifiability results. Sec- tion 3.1 treats the case of a proxy variable. Section 3.2 de- tails our findings for underspecified instrumental variables case.
b
(N)
(7)
c (V(V.. 1,... (*(V.) 1,00112
Before presenting our results, we review two key results from Schkoda et al. (2024) pertaining to the causal graph g' depicted in Fig. 1, which includes two observed variables, V1 and V22 along with / latent variables Li...., Lt. They will be used to establish our identifiability results.
c² (V V..122....2
The solution is, generically, unique if 2 +
Let b be the vector [b2.1. be.L,, ,b,L,. We rewrite the system in (7) as
L
L
M(b, k) c c(, (Vo).
(8)
V2
The above lemma implies that after using Theorem 3.1 to recover [b21, bei, b2t,] up to permutation, it is pos- sible to estimate some cumulants corresponding to the ex- ogenous noises of Vi2 and the latent variables up to the same permutation.
Figure! 1. The causal graph G with latent confounders.
Theorem 3.1 (Schkoda et al., 2024, Thm. 4). Consider the causal graph G with two observed variables and I latent variables depicted in Fig. 1. There is a polynomial of degree 1+1 with coefficients expressed in terms of the first k(I)= := (1+2)+
3.1. Proxy Variable
12] cumulants of the distributions where the roots the polynomial are b b........ b2, We refer to this polynomial as (bb (see Remark B. I in the appendix for definition of the polynomial).
In this section, we first provide the identifiability result for a causal graph with a single proxy variable and latent variables where there is no edge from the proxy variable to the treatment. Then, we extend our result the case where there is edge from the proxy to the treatment.
The above theorem implies that in the causal graph g', one can identify the causal effect of interest, 12,1, up to a set of size + 1 using the first k(I) cumulants the distribu- tion. In Section 3.1 (and Section 3.2), we demonstrate how incorporating proxy (or instrumental) variable refine this result, enabling unique identification of the causal ef- fect. This approach involves deriving additional polynomial equations among the cumulants of the observed distribution, for which the true causal effect is a solution.
3.1.1. No EDGE FROM PROXY TO TREATMENT
L
Z
Y
Figure 2. The causal graph with a single proxy variable Z and I latent confounders Li,-.. La where there is edge from the proxy to the treatment.
Example 3.2 (Polynomial for the graph in Fig. 1). For the special case = 1, the polynomial equation described Theorem 3. I is defined as follows with the coefficients

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
Theorem 3.4. In the IvLiNGAM for the causal graph in Fig. 2, with the proxy variable 7 and latent confounders Li...., Li, the causal effect from T to Y is generically identifiable from the first k(I) cumulants of the observational distribution.
the same in the original model (see Lemma B.2). Hence, once have one of these pairs, can use Theorem 3.4 to recover the causal effects between T' and Y.
To obtain the pairs, we apply Theorem 3.1 to [2, and Z Y1, finding
Proof. Considering the pairs [2, [Z, YJ and [7, Y] as pair [V1, V2] in Theorem 3.1, we obtain the vectors
b" [br,z, by, br.
(13)
by [by,2, by,,,..... by,
zr [0,
br.
up to some permutations of their entries. Moreover, using Lemma 3.3, we can align the pairs of solutions as we did in the proof of Theorem 3.4. this manner, we obtain
ZY = [0, by.t,y...,,,y....
(9)
bTY [by,r, "y,L//r,,,,... by,,,/br,uu
bz [bT,L.: by. Any b' allows us to identify the correct causal effect.
by [br,2, by,2l.
(14)
up to some permutations (notice that the ratios in the last equation are a consequence of the choice the scaling we discussed in Remark 2.4). Next, we recover the vector
The above result shows that estimating the first k(I) cumu- lants of the distribution sufficient to identify the causal effect. However, since estimating higher-order cumulants is statistically more challenging, it is important to understand whether the same result can be obtained with lower-order cumulants. The next result shows that this is not possible for the case =
|c IN),.... (N)((NNNN))))) (N) (is c (N,.,. (10)
using Lemma 3.3 twice (up to some permutations) with the vector b2X and then with b by solving the linear system in (7). Since the cumulants of different exogenous noises are generically distinct, we can match the entries in b to their corresponding entries in b ZY using the two recovered exogenous cumulant vectors. This allows to construct new vector
Theorem 3.6. Consider the causal graph depicted in Fig. 3 with 1 Then, the causal effect from T to Y is not identifiable from the first k(() = cumulants of the observational distribution.
b" = by, /tr. /br./r......... Dy,u//br,ul)
(11)
Finally, by.r is the only entry in bY that does not equal any entry of b".
Proof. Garcia et al. (2010, Prop. 3, 4) prove that, once a polynomial parametrization for a statistical model is known, the generic identifiability of any parameter can be verified through a Gröbner basis computation. We leveraged this fact as follows: we parameterize the model M (g) us- ing (5) and compute the vanishing ideal for the modified parametrization
3.1.2. WITH AN EDGE FROM PROXY TO TREATMENT
L
Z
$
RRC Diag Sk (p) R (SA)
T
(B',D (2) (3)) + [ 2 B'D 3 B").
Figure 3. The causal graph with single proxy variable Z and latent confounders Iı,".. La where there is an edge from the proxy the treatment.
Specifically, computing the reduced Gröbner basis for an climination term order (see Definition A.3), we find that by.r is determined merely as root of a degree two polyno- mial.² Since by unconstrained in R it is not generi- cally identifiable (Garcia et al., 2010, Prop. 3).
Theorem 3.5. In the !yLiGGAM for the causal graph in Fig. 3, the causal effect from T to Y is generically iden- tifiable from the first ke(l) cumulants of the observational distribution.
3.2. Underspecified Instrumental Variable
We now prove that in IvLiNGAM models, one valid in- strument suffices to estimate the causal effects of multiple treatments.
Proof. Let b be either equal [br,2, by.zl or to [br,L, by for some i E [[]. Then, the triple
"The computations were done using the computer algebra software Macaulay 2 (Grayson & Stillman, 2023). The code to replicate the computation can be found at http://gghhbb..coomaaeeettttannn Marmat/oooraaaeeeuul////uaaaaatt
vb_ = Z.T - b, 2, byZ]
(12)
follows a IvLiNGAM model compatible with the graph in Fig. 2 with the causal effect from T-b, Z to Y Ybb?? being
5

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[Z,, Tn. y,,, bound on the number of Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
T
L1
Remark 3.9 (Multiple instruments). For simplicity of nota- tion, stated the theorem in the most challenging context of a single instrumental variable. However, the result read- ily extends to cases with multiple valid instruments I, as long as each treatment is associated with at least one valid instrument. See Remark B.5 in the appendix for details on adapting the identification strategy to multiple instruments.
T2
L
Figure 4. An example of causal graph for the underspecified instrumental variable model.
4. Estimation
In causal graph 9, we say that I is valid instrument for the treatments T ..... T on Y if
In this section, we explain how to develop estimation tech- niques based on the identifiability results from the previous section. We assume access to an i.i.d sample V, E *** drawn from the distribution of a random vector Vo E M(g) for a fixed graph g. All algorithms will process unbiased estimates of the corresponding population cumulants, i.e. k-statistics (McCullagh, 1987, $4.2).
I pa(")) (an))) nC I Y
AiE E [m]. ViE E [n].
where I. denotes d-separation (Pearl, 2009, $1.2), and Gvr is the graph obtained by removing the edges T + Y from G for all E [ni] (Ailer et al., 2023, Eq. 1). Fig. illustrates an example with two treatments and one instrumental variable.
Algorithm Proxy Variable (Fig. 2)
INPUT: Data V,, n. latent variables 1.
Theorem 3.7. In the IvLiNGAM for the causal graph in Fig. 4, with instrumental variable I, treatments T ,...... T* and outcome Y, the causal effect I to Y is generically identifiable from the first k(() cumulants of the observational distribution, where 1 == max;g(n] ix] (T") O aY( \ T/.
1: b t- roots of P(z Pl = 0 (())
b t [0, b
3: bzY +- roots of PIz., = 0 {())
bžy t- [0, b0
5: bTY <- roots of PIr..Y.J, = 0 (())
The proof of the above result can be found in Appendix B. In the next example, we outline the identification strategy for the graph in Fig. 4.
6: ct solution to the linear system
M(b zT 1 1)- ()
c = 1 (1Z,, T,,) {(8)
7: c + solution to the linear system
Example 3.8 (Identification equations for the graph in Fig. 4). First, compute br., / and by.1
M(b +1+11) d' - (IZn Y,)) {(8)
8: on + arg miggmn.ee. E (llc a (ct )|))
, /c// Then, consider the vector
b, b on (bZY) [Under the convention 0/0 = 0.}
V [T' bb..b,,, IT, bT by
10: Th < arg min SES (I)b; -nb,, TY )
11: RETURN: b,y" [n,
The vector of causal effects [by:T by,ra) is the unique solution to the following polynomial system:
Algorithm outlines the estimation procedure for the causal effect for the graph in Fig. 2. This algorithm replaces the steps in the proof of Theorem 3.4 with their respective finite- sample versions. Specifically, lines I to 5 correspond to (9), where the in lines 1 and 3 results from the fact that, without an edge from Z to T, one of the roots of PTZ.. is known to be zero (Schkoda et al., 2024, Thm. 3). Lines 6 and correspond to (10), and lines and 8 correspond to (11). In particular, in line 8, we determine the permutation a E S that minimizes the by distance between cy2 and I+1
by VVV c(V L.a))
+by,r (CCV)sss.e(VVVnnn-ccVlyyyy.... (V333)))))) 1,3.3)
- 113333 (isassss((""sssss""llasss...""s" )13.3) = 0,
by.2,
(V <(V ).2 2.22)
+by,1, (.V3.3333. 12,2,2 c (V' (222 )2.33)
- (c((V') (((V'))2333(VV) 223 + cc(V') )2,3) 223,3)
acctt This step is necessary because, due to estimation error, we cannot perfectly align the entries of cT, Similarly, in line 9, we identify the permutation n/( ((c that minimizes the &a distance between b", and ()bb) ). Finally, we return the entry of b corresponding to the zero in b,,-
1+1 and cy
by. - bbr,, - br, =
where the first two equations are instances of (6), and the last equation can be derived by directly applying Lemma A.8 to the graph in Fig. 4.
The algorithms for the other graphs can be found Ap-
6

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
pendix C. Furthermore, in Algorithm 4, we propose an optimization technique that improves the finite-sample per- formance for the graph in Fig. 3 with single latent variable (as shown in the right panel of Fig. 6).
the other variables being jointly Gaussian. Cui et al. (2024)
proposed alternative proximal identification procedure to that of Miao et al. (2018), again under the availability of two proxy variables. For IvLiNGAMs, Kivva et al. (2023) gave an explicit moment- based formula for the causal effect when there is no edge from the proxy to the treatment. For general introduction to proximal causal inference, see also Tehetgen et al. (2024).
5. Related Work
There is a substantial body of work on causal effect iden- tification in linear SCMs, with several graphical criteria developed for identification in fixed causal graph. For Gaussian models, Drton et al. (201 1); Kumor et al. (2020); Barber et al. (2022) provided conditions under which causal effects can be identified solely from the covariance matrix. In the non-Gaussian case, analogous results have been es- tablished by Tramontano et al. (2024a;b), with criteria that are both sound and complete but which require access to the full observational distribution.
Instrumental variables were first introduced in Wright (1928, App. B) and have since become fundamental identifica- tion strategy in both the social sciences (Cunningham, 2021, $7.1) and epidemiology (Didelez & Sheehan, 2007). In linear models, the standard TSLS equations (Angrist & Pis- chke, 2009, $3.2) have a unique solution only with at least one instrument per treatment. For cases with fewer instru- ments, Ailer et al. (2023) proposed estimating the causal effect using the minimum norm solution to the TSLS equa- tions, which is always unique but may introduce arbitrary bias. In contrast, Pfister & Peters (2022) showed that, un- der additional sparsity assumptions, causal effects can be identified by adding an lg penalty to the TSLS equations. For IvLiNGAMs, Silva & Shimizu (2017); Xie al. (2022) explored the testable implications of instrumental variables.
Results for the identification of the mixing matrix (i.e., with- out assuming knowledge of the causal graph) are provided in Salchkaleybar et al. (2020); Yang et al. (2022); Adams et al. (2021) and in Cai et al. (2023); Schkoda et al. (2024); Chen et al. (2024); Li et al. (2025). The former results are based on solving an OICA problem (hence, are not equipped with consistent estimation methods), and the latter results, similar to our approach, rely on explicit cumulant/moment equations. Notably, both Cai et al. (2023) and Chen et al. (2024) assume specific structural conditions-nande LeLaatnt-Cooponeee structure and a homologous surro- gate, respectively- which do not apply to the graphs con- sidered in Sections 3.1 and 3.2.
6. Experimental Results
a
This section presents experimental results on synthetic and experimental data for the graphs studied in Section 3.
As performance metric, we use the relative absolute error
In the context of proximal causal inference, Kuroki & Pearl (2014) explored two scenarios for determining causal ef- feets: (I) discrete finite variables 2 and L: It was shown that the causal effect can be identified if L)) is known (e.g., from external studies) or an additional proxy variable (W) is available and certain conditions on the conditional probabilities of PYY [T, L) and PZZ (T) are satisfied. (2) Linear SCMs: They proved that the causal effect of T on Y is identifiable using two proxy variables.
err(by:T, by.r) := = (6y:7 birr) /byrr
where by: is the true value of causal effect and by,r is its estimate. We report the median value of the relative estimation error over 100 random simulations; the filled area on our plots shows the interquartile range of the relative error distribution. Details on the experimental setup and experiments are provided in Appendix D.
Following their work, Miao et al. (2018) studied a scenario involving two proxy variables, Z and W. Unlike the pre- vious results, they allow Z and W to be parent nodes for T' and Y, respectively. They found that the causal effect can be identified for discrete finite variables if the matrix P(WIZ,T t) is invertible. They also provided analo- gous (nonparametric) conditions for continuous variables. Shi et al. (2020) extended these results, employing less stringent set of assumptions while still necessitating two proxy variables to identify the causal effect. Later Shuai et al. (2023) considered the setting with one proxy variable and proved that the causal effect is identifiable under the assumption that only the treatment is non-Gaussian, with
6.1. Proxy Variable
Z
Z T
T
91
G2
G
Figure 5. The causal graphs considered in the experiments.
We begin with experimental results for the proxy variable settings with the causal graphs illustrated in Fig. 5. We
"The code replicate the experiments can be found http:/gghw...comm/mmaeetttmntnno from---eetttsss....
7

---

Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
Distribution: Gamma
Graph: 0
Graph: 9
Graph: G3
100
Error (LOG-scale)
10-1
10
10-3
10
103
10
10 10
10a
10
107 101
103
10
107
Sample Size (LOG-scale)
Cumulant
Cumulant with Minimization
Cross Moment
GRICA
ReLVLINGAM
Figure 6. Relative error vs sample size for the graphs in Fig. 5.
compare our method (which we call Cumulant) with the Cross-Moment (Kivva et al., 2023, Alg. 1), GRICA (Tra- montano al., 2024b, $3.5), and ReLVLiNGAM (Schkoda et al., 2024) algorithms.
Distribution: Gamma, Graph: IV
10
Error (LOG-scale)
As can be seen in Fig. 6 (left). for the graph G1, the Cross- Moment algorithm outperforms all other methods. This is expected since provides a consistent estimate the causal
10-1
effect using third-order cumulants if there is no edge from the proxy variable the treatment. Although the Cumulant method also consistent, it uses fourth-order cumulants that are more challenging estimate.
Cumulant
Min Norm
10-2
GRICA
ReLVLiNGAM
For the graphs G2 and G3. which include either multiple latent variables or a causal edge from to T, our pro- posed method significantly outperforms other approaches (see Fig. middle and right). Additionally, an experiment involving both multiple latent variables and causal edge from Z T is presented in Fig. 10 in the appendix. For the graph Ga. we also provided the result for the Cumu- lant method with the minimization technique given in Ap- pendix C.1.1, which improves the performance of the Cu- mulant method since reduces the dependency on using the fourth-order cumulants. Notably, for these graphs, neither nor the GRICA algorithm provides consistent estimator of the true causal effect. This can also be seen from the experiments, as the relative error does not decay as the sample size increases. Furthermore, while the ReLVLiNGAM algorithm produces consistent estimators for the causal effect in graphs G1 and Gs. it performs poorly compared to our method. This results from ReLVLiNGAM performing causal discovery and causal effect estimation simultaneously, increasing its complexity.
10a
10?
103
10*
10a
10
107
Sample Size (LOG-scale)
Figure 7. Relative error sample size for the graph in Fig. 4.
6.2. Underspecified Instrumental Variable
In this part, we provide the experimental results for the un- derspecified instrumental variable graph depicted in Fig. 4. We compare our method (Cumulant) with the projection on instrument space proposed in Ailer et al. (2023, $3.1) (Min Norm), the GRICA, and the ReLViNGAM algorithm. Fig. 7 shows
the
(er(bby:r,, by:r: + err(by,
bū: brr(,,,,,.b /2
against sample size. As be seen, method is the only one that consistently estimates the causal effects for the two treatments having access to only one instrument.
Remark 6.1 (Small Sample Performance). From Figs. and 7, one can observe that for small sample sizes, the
8

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
GRICA method proposed in Tramontano al. (2024b) ex- hibits superior performance.
effects in the context of proxy variables and underspecified instrumental variable graphs, which cannot be handled with existing methods. Experimental results demonstrate the accuracy and practical utility of our proposed methods.
One possible explanation is that cumulant- based methods rely on unbiased estimators of high-order cumulants (typi- cally of order 4 or higher), also known as k-statistics. While these estimators are unbiased, they tend to exhibit high vari- ance when the sample size small.
A Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 483818) and supported in part by the SNF project 200021 1204355/1, Causal Reasoning Beyond Markov Equivalencies. DT's PhD scholarship is funded by the IGSSE/TUM-GS via Technical University of Munich- Imperial College London Joint Academy of Doctoral Stud- ies. MD acknowledges support by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Re- search.
In contrast, GRICA solves an optimization problem involv- ing the -norm of the observed data, which generally has lower sample variance. As a result, GRICA may achieve lower mean-squared error in small-sample regimes due to this variance reduction. However, because the GRICA so- lution is not asymptotically unbiased, it does not yield a consistent estimator, unlike our proposed method, which retains consistency in the asymptotic limit.
6.3. Experiments on Real Data
To assess the practical efficacy of our method, we conduct experiments on the dataset analyzed in Card & Krueger (1993), which contains information fast-food restaurants in New Jersey and Pennsylvania in 1992. The dataset in- cludes variables such as minimum wage, product prices, store hours, and other relevant features. The original study aimed to estimate the effect of an increase in New Jersey's 's minimum wage ffrm $4.25 to $5.05 per hour-on on employ- ment rates. Importantly, the data were collected both before and after the wage increase in New Jersey, while the mini- mum wage in Pennsylvania remained constant throughout this period.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
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Causal Effect Identification in IvLINGAM from Higher-Orde Cumulants
A. Notions of Non-Linear Algebra
In this section, we give the basic definitions of non-linear algebra we will need for the proofs; we refer the interested reader to Garcia al. (2010); Cox et al. (2015); Michatek & Sturmfels (2021) for more details.
Definition A.1. For every natural number nt, we denote the ring of polynomials in variables I1,.... , In by [[x,,... ,In] Let S be a, possibly infinite, subset of R(z1...., .T]] The affine variety associated to it is defined as V(S) = {i E R"
f(x) = 0, vf E S). The vanishing ideal associated to a variety V is I(V) = {s E [R(x,....,, Fnl [(x) = 0vr E V}.
The coordinate ring of V is defined as [ = [[ In]/I(V)).
Definition A.2. A term order- < the polynomial ring R[x] is a total ordering on the monomials in E[x] that is compatible with multiplication and such that is the smallest monomial; that is, 1 = x" I x" for all u E N" and if x" 3 x", then x" x" 3 x x". Since < is a total ordering, every polynomial g E R[x] has a well-defined largest monomial. Let in (g) be the largest monomial in g. For an ideal I C [x[x), let in (T) = (in~(g)
E T); this is called the initial ideal of I.
Among the most important term orders the lexicographic term order, which can be defined for any permutation of the variables. In the lexicographic term order, we declare x" x" if and only if the left-most nonzero entry of v u is positive.
Elimination orders partition A U B. In the elimination order, x" < x if x has a larger degree in the A variables than x If x" and x" have the same degree in the A variables, then some other term order is used to break ties.
a generalization of the lexicographic order. These are obtained by splitting the variables into
Definition A.3. A finite subset G C I is called a Gröbner basis for I with respect to the term order < if
in (in-(g) :g E G}.
The Gröbner basis is called reduced if the coefficient of (g) in is 1 for all g, each (g) is a minimal generator of (I), and no terms besides the initial terms of G belong =
Lemma A.4 (Okamoto, 1973, Lemma). Let f(x1...., Fn. be a polynomial in real variables Z 21....;2a Zn. which is not identically zero. The sel of zeros of the polynomial is a Lebesgue measure zero subset of RR..
Lemma A.5. Let IRA and R& defines as in Section 2.2. Then we have IRS R Rlel, where with the symbol ~e, we denote an isomorphism of affine varieties, see, e.g., Cox et al. (2015, Def. 6. for definition. Moreover R[G], R[GA]. and TR. [ j + i 9] are isomorphic as rings.
Proof. The isomorphism R ~ R comes directly from its definition. Indeed it is easy to see that IR is an lel-dimensional linear subspace of Rexp (0t.) (...)).eppppp defined by the linear equations ag,s = 1, and azj = 0. Vi,j E V such that j > i E g.
To prove the isomorphism R R& A. we need to prove that there a polynomial bijective map between the two spaces. From (4), and using [Bolk] [(A.,)) Jk.j = (-1)tts det((Ao.))v.ve)) where we used that det(Ao,o) = is clear that R2 is
the image of polynomial map of A- Let us call this polynomial map v and assume v6( (A) = r/ (A). Then from the definition
of db we have (T A,,o)- = (I ) that implies A Ao.. Moreover, (T Ao,o) 'o,, = (T- -A,o))) Ao,
that implies A., A and so A = A.
The isomorphisms between the rings come from Cox et al. (2015, $5, Thm. 9).
Corollary A.6. Let] f E [R( be a non-zero polynomial. Then the subset RR on which J vanishes is Lebesgue measure 0 subset RS..
Proof. Thanks to the isomorphism in Lemma A.5, we can apply Lemma A.4 to R$.
Definition A.7. Let π E P(j, i). The path monomial associated to it is defined as
at aig digdkii
E R[GA).
Lemma A.8. Let A defined as in (2), We have
B = B= (1- A)
Σ
A = T+
A
4 A2
AP,
1=0
b =
Σ ".
PEP(
12

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
In particular ba. 0 R[9A] and only if 0.
B. Additional Proofs
Remark B.1. The polynomial PV.. (b) mentioned in Theorem 3.1, can be obtained as the determinant of an / + 2 x 1+2 minor the following matrix containing the first row
b
c (Vo).)
c (V (Vo)1....1.2
2 0 1.2.....
(4o) c4+3(Vo)2.) (V...
c 1........11
c ( 1,,12...... c ( 2,22,22....
c (V 1,2
cA (V.)(V) V... I.I.m.n
kll (V.) 1.n.d.ll ch(l) (V, I 1.1.
(1) (Vo)2............
el 2.
bll)) (Vo)2.....122.....
The proof of this fact can be found in Schkoda et al. (2024, Thm. 4).
Lemma B.2. Let V. Iz, T, Y] be vector generated from a IvLiNGAM model compatible with the graph in Fig, 3, and let be either equal to br by.z] or to [bT,L, l. for some E [1]. Then, the triple
yb_ = 7, b, - bb?
follows a IvLiNGAM model compatible with the graph in Fig. 2 with the causal effect from T b, Z to Y b2Z being the same as in the original model.
Proof. From (3), we know that
N
N
0
b by.z
br.
b
by,
by
by.
N
NT
Ny
From simple linear algebra manipulation, it follows that
N % NL
1
0
0
1
1
!0
bb_
b
V, =
-b, + b
-b; br.4
bi + br
b2
1
bb by.2
- -b,- by
ba + by
by.r 1
N
Nr
Ny
By setting b to either equal to [br.z, by.z] or to [bT,LE> by we set one of the first / + columns of the mixing matrix corresponding to vb [1, O, 0), hence removing the edge from Z T.
13

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
Lemma B.3. Let V - [Z, be vector generated from IvLiNGAM model compatible with the graph in Fig. 3 with one latent variable, and let (v)().)()) be the following univariate polynomial
(a(v.) = <VoYy - b b. (Vo)zy
(15)
(V. b b. (V.),.
Then, we have (v...) = by.r.
Proof. Direct computation, applying Lemma 2.2 and Lemma A.8.
Lemma B.4. Let V [I.T'..... T Y] be vector generated from IyLiNGAM model compatible with instrumental variable graph. Consider now the variables
T = T bTAII,
by,
obtained by regressing from T and Y, respectively.
Each one of the pairs [ Y can be represented by IvLiNGAM model with two observed variables and most I latent confounders, with the causal effect from T to Y being the same as in the original distribution.
Proof. From (3), we know that
N
N
1
0
0
0
T
=
br.
b
bT,
0
by.
by,
by
by,
1
N
Nr
Ny
From simple linear algebra manipulation, follows that
N
N
T
- bTi,
1
0
T
-bri. + DT,L,
-bri, + bT.
1 0
7
yl
by.
0
1
y
0 by. + by.
by. + by
byr 1
NL
Nr
Ny
N
bys, + br, - bby, + bT.ta
1
0
by + by. -by, + by
by:r 1
N
N
Ny
Which is indeed compatible with the graph in Fig. 1.
Theorem. Let Giv be an instrumental variable graph, with instrument I, treatments T'....," and outcome Y, and let l == maxiein) an(T)) ) n an(Y) 1\. Then, the causal effect from T to Y is generically identifiable from the first k(l) cumulants of the distribution.
Proof of Theorem 3.7. Since an(/) n an("") n C = an()) n an(Y) n L 0, we can identify by and by, from the covariance matrix through backdoor adjustment (Pearl al. (2016, $3.3), Henckel al. (2022, Prop. 1)). From Ailer et al. (2023, & 3.1), know that the causal effects of interest satisfy the following equation:
(b) := by.1 Σ bT. by, = 0 R(g)
(16)
14

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
where b by,ri, y.. by
Consider now the variables
T T br Y =/=Y- -y,,,,
(17)
obtained by regressing out I from T and Y, respectively.
Each one of the pairs [T' Y can be represented by a IvLiNGAM model with two observed variables and at most / latent confounders, with the causal effect from T to Y being the same as the original distribution (Lemma B.4).
Using Theorem 3.1, we know that the vector
b' = [b,,r by.t,..... by,4,
(18)
can obtained as roots degree I polynomial constructed using cumulants up to order k-()) of the observational distribution (up some permutations).
Consider the polynomial ry (y(b1.... ..... bn) R[b....... bn] defined in (16). For every choice of b b' x ... b", r>)
defines different polynomial in R(G]. We have already seen, that for b = by,r, this defines the zero polynomial. To conclude, it is only left to show that
ry (b) * E [g]
Abeb's x.. x b" fbb,Tt, by.r+
(19)
the result will follow by applying Lemma A.4.
Let us rewrite the entries of b' as by:rs + (by.L. by,rs) for some E {0, 13. This way, we can write ry (b)
bI.Y Σ (by cj (by,,, by,r=)) = Σ cj (by, by,r*),
using Lemmas A.5 and A.8 we can rewrite it as
)
( T
Σe Cj
Σ a
Σ
a Ty
E [R[GA
TY.EP(TYY
Notice that every summand in the above equation a monomial degree least two. If cf, for some then the degree term a,, y appears only once as a summand. This implies that the last equation defines a non-zero polynomial in R.C which concludes the proof.
Remark B.5 (Multiple Instruments). For simplicity, we stated and proved the theorem for the case single instrumental variable. However, the result naturally extends to scenarios with multiple valid instruments I, provided that each treatment has least one valid instrument.
To adapt the proof, (19) should be replaced with
T
T Σ brapp
y'=
Σ by jE(s]
(20)
JEI:
where Z; is the set of valid instruments for the treatment T
Additionally, the variety V1 == ((rr should be used place the single polynomial ry(b) in (19).
C. Estimation
C.1. Proxy Variable with an Edge from Proxy to Treatment
Algorithm 2 outlines the estimation procedure for causal effect estimation corresponding to the graph in Fig. 3. This algorithm replaces the steps in the proof of Theorem 3.5 with their respective finite-sample versions.
Specifically, lines and 3 correspond to (13). Lines 3 to align with (14), where the minimization step in line 5 is equivalent to that in line of Algorithm and further described in Section 4. The for loop spanning lines 7 to 17 corresponds to applying Algorithm for all possible choices of b in (12).
15

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
Algorithm 2 Proxy Variable with an Edge from Proxy Treatment (Fig. 3)
INPUT: Data Vn = I2, Ta, , bound on the number of latent variables (.
1: by roots of PlZ.. TT.....),.(
{(13)
2: by f- roots of PIz., Y.),
{(13)
3: et1 <- solution to the linear system Mb ((22T,) , + 1). = = c/1
(1.2) (1 Tx)) (())
R2
4: cy <- solution to the linear system M( 1 + +1) : c'+1=
= c
(Z Ya)) {(8)
5: on - arg ge (llc/t. o (1c2))111 )1))
6: mmin
7: for all in [1 1] do
8: br,z - [i]
9: by.2 + by [o(1]
10:
Vn [,,,T -br Zn, Y, by. Zm) {(12)
11:
by, Algorithm 1( 2) (Lemma B.2]
12:
r r 1by.2 - by,r * by,2l
13:
if < <min then
Jmin fr
15: by.r by,r
16: end if
17:
end for
18: RETURN: b
At the population level, any choice of b results in the correct causal effect. However, in practice, we observed that using the sample version of [br,z, by, yields better performance. Among the pairs in (14), [bT,z, by.z] is the only one satisfying the equation b[2] - -b,r b[1] = 0 (Lemma A.8). Therefore, we select the estimate derived from the pair that minimizes the sample version of this equation. This explains the steps outlined in lines to 18.
C.I.1. PROXY VARIABLE WITH AN EDGE FROM PROXY TO TREATMENT WITH ONE LATENT VARIABLE
In this section present two specialized estimation procedures for the causal effect in Fig. 2 with latent variable.
First, Algorithm 3 is simplified version of Algorithm 2, tailored for the case with single latent confounder. The key distinction between the two procedures lies in how the candidate value for the causal effect is computed: Algorithm 3 utilizes Lemma B.3 (lines 10-11 of Algorithm 3), whereas Algorithm relies on Lemma B.2 (line of Algorithm 2).
Next, we introduce an optimization technique that leverages cumulants up to degree three. While Theorem 3.6 establishes that the causal effect identifiable using second- and third-order cumulants alone, we observe that this procedure often achieves better finite-sample performance when initialized with a reliable starting point, compared to directly applying Algorithm 3.
Let Vo [2, be vector generated from IvLiNGAM model compatible with the graph in Fig. 3 with one latent variable. The following objective function is used:
lvo.byr (b) (b):= (b- cVVo) [[b)) - (Vo)z.)
(b. byT)²
(21)
where
(3) g(b)
22. (Vb)
1.3.3
vb -,,,
(3)
2.3.3 (Vb)
1.1.
Using Lemma 2.3, it can be shown that, ( (()) (L)),,.1 * 0, then (by,r) = by, As a result, Lemma B.3 guarantees that by.r minimizes the first term in (21). The second term in (21) serves as a regularization term to ensure the solution remains close to the initial estimate.
16

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
Algorithm 3 Proxy Variable with an Edge from Proxy Treatment with Latent (Fig. 3)
INPUT: Data Vn = I2,, , Ta, Yn..
1: by roots of TT.,,
0 ((13)
2: by f- roots of (,),a = 0 {(33)
3: C, &- solution to the linear system M( 2) c² = 11,2) 2 (I2n Tn]) {(8)
2
4: oi - solution to the linear system M(b 2) c² = (1,2) (1Zn, x,) {(8)
5: dn < arg min (11o7. - acc
6: by.2 - b
7: by,z - (A((a(1))
8: by,z + [TT
9: b - b [2)]
10: b + (e(v.)) (b).))))) ((15), Lemma B.3)
11: b + (v.) (b),2) 2) {(15), Lemma B.3)
12: rl by, bb. b p
13: 22 + , b b
14: rf <²² then
15: RETURN: b bkr
end if
RETURN: byT
Algorithm 4 Proxy Variable with an Edge from Proxy to Treatment with one Latent with Optimization (Fig. 3)
INPUT: Data Vn [Zn, Tni Yn].
1: by:r + Algorithm 3(V,))
2: by 4- arg min "vr,,br ((1))
3: RETURN: byT
In practice, solve the optimization problem using the Python implementation of the BEGS algorithm (Nocedal & Wright, 2006, $6.1) provided in Jones al. (2001-). The finite-sample version of this optimization process is detailed in Algorithm 4. Remark C.I. If c(3) (1,),,11 is zero, higher-order cumulants can be used to construct g(b). The existence such a polynomial is guaranteed as long as L1 is -an-Gausiinn; see, for example, Kivva al. (2023, Thm. 1).
C.2. Underspecified Instrumental Variable
Algorithm 5 outlines the estimation procedure for causal effect estimation corresponding to the graph in Fig. 8 with one instrumental variable. This algorithm replaces the steps in the proof of Theorem 3.7 with their respective finite-sample versions.
Specifically, lines 1 to 9 involve computing the covariance matrix and performing the regression adjustments required to derive the finite-sample versions of the vectors described in (17).
The for loop in lines 11 to 17 evaluates the finite-sample approximation of the polynomial (b) defined in (16). As the estimate of the causal effect, the algorithm selects the projection over the line defined by the equation ry (b) = of the tuple b, that minimizes ry (b,).
Algorithm 6 is an extension of Algorithm that accommodates the presence of multiple instruments together. It implements adaptations described in Remark B.5.
D. Details on the Experimental Setting and Additional Experiments
All the experiments in this subsection are done on the synthetic data generated according to the specific causal structure established for To generate synthetic data, we specify all exogenous noises from the same family distributions (with parameters sampled according to Table 1) and select non-zero entries within the matrix A through uniform sampling
17

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
X
L1
T
L
73
Y
1
T2
L
X²
Figure An example of a causal graph for the underspecified instrumental variable model.
Algorithm 5 Underspecified Instrumental Variables (Fig. 8)
INPUT: Data Vn [n, T2 ...,, Th Y,, X ....,,]J,, the causal graph G, bound on the number of latent variables /.
Σn t (V.)) (Sample covariance matrix} 2: a)( < an))) aY()) no {Valid adjustment set}
3: by. < (En)y, y) /En) {Regression adjustment (Henckel et al., 2022, Prop. 1))
4: Y < tY, by.,, In {(17)
5: for all i [k] do
6: ad(1, T") t an()) n (an(T)) no
7: br +
8: T 1 Th Jn In
9: b, + roots " (b) = 0 {(18))
10: end for
11: Tmin + 00
12: for all bn E b/ ... x by do
TO
(rr)),,) {(19)
rt < Mmin then
15:
Finin rb
16: by DYm,.... TM...TT
arg min ||b bnll3
(b)=
end if
18:
end for
RETURN: b% byr...... 19:
from [-0.9, 0.5] U 0.95
In the figures, we plot the median relative error over 100 independent experiments; the filled area on our plots shows the interquartile range.
18

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Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
Algorithm Underspecified Instrumental Variables with Multiple Instruments (Fig. 8)
INPUT: Data V, = [ I. T, ... Th Y X ..... X the causal graph G, bound the number of latent variables l.
1: Σn f ((2) ..) (Sample covariance matrix]
2: for all j [s] do
3: ad(I', Y) an(T)) (Y) no (Valid adjustment set}
4: by f (En)y, y)/ (Btn:/addti///n)nnnn) ( (Regression adjustment (Henckel et al., 2022, Prop. 1))
5: end for
6: Yn + +Y,, Yn -Eje(s] by. {(20)
7: for all i [b] do
8: TH -Th
9:
for all j do
10:
if 1 is a valid instrument for TV in 9 then ad(P), T 4-
no
bt Fin -- ( ( (a
T.t 4 Tū 1r, {(00)
end if end for
16: by roots of Pirt xg = 0 {(88)
end for
18: dmii
19:
for all bn by
b do dp f- min |b b, 1 (19))
20:
DEV;
if d6 < dmin then
22:
duin t do
23: byr
arg min |b b, 12 be
24:
end if
25:
end for
26: RETURN: bT.......
Table Summary of the experimental setups.
Figure
Causal Graph
Distribution
Parameters of Interest
Family
shape
scale
6 (left)
G in Fig.
Gamma
1
U(0.1,0.5)
T+Y
6 (middle)
G2 in Fig.
Gamma
U(0.1,
U(0.1,0.5)
T+Y
6 (right)
G3 in Fig.
Gamma
U(0.1, 1)
U(0.1,0.5)
T+Y
10 (left)
Fig. 9
Gamma
U(0.1,1)
U(0.1,0.5)
T
7
Fig. 4
Gamma
U(0.1,1)
U(0.1,0.5)
7 Y,T,2 +Y
Family
alpha
beta
11 (left)
9i in Fig.
Beta
U(1.5,2) 2)
U(2, 10)
T+Y
11 (middle)
G2 in Fig.
Beta
U(1.5,2)
U(2, 10)
T+Y
11 (right)
G3 in Fig.
Beta
2(1.5,2)
(22110)
T+Y
10 (right)
Fig. 9
Beta
U(1.5,2) 2)
102
T+Y
12
Fig.
Beta
U(1.5,2)
U(2, 10)
T. - Y,T2
19

---

Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
L1
Z
L2
T
Figure 9, Proxy variable graph with an edge from proxy to treatment and two latent confounders.
Graph: C4
Disribution: Gamma
Disribution: Beta
100
Error (LOG-scale)
10-1
10
10a
10a
10
10"
101
10a
10
10
Sample Size (LOG-scale)
Cumulant
Cross Moment
GRICA
ReLVLiNGAM
Figure 10. Relative sample size for the graphs in Fig. 5.
Distribution: Beta
Graph: C1
Graph: C2
Graph: G3
100
Error (LOG-scale)
10
10
-3
10
10
10
10
10
10
10
10'101
10
10
10
10
10'
Sample Size (LOG-scale)
Cumulant
Cumulant with Minimization
Cross Moment
GRICA
ReLVLiNGAM
Figure 11. Relative error vs sample size for the graphs in Fig. 5.
20

---

Causal Effect Identification in IvLINGAM from Higher-Order Cumulants
Distribution: Beta, Graph: IV
10
Error (LOG-scale)
10-1
Cumulant
10-2
Norm
GRICA
ReLVLİNGAM
10
102
10a
10a
10a
10
107
Sample Size (LOG-scale)
Figure Relative error vs sample size for graph in Fig.
21


Reviews:
[{"review_id": "hBJhlMCkWr", "reviewer": "Reviewer_6fLa", "summary": "This paper explores causal effect identification in latent variable Linear Non-Gaussian Acyclic Models (lvLiNGAM) using higher-order cumulants, addressing two challenging scenarios involving latent confounding: (1) a single proxy variable that may influence the treatment and (2) underspecified instrumental variable (IV) cases with fewer instruments than treatments. The authors theoretically prove that causal effects are identifiable under these conditions and propose corresponding closed-form estimation methods. Experimental results demonstrate the accuracy and robustness of the proposed approaches.\n\n## update after rebuttal\n\nI thank the authors for their rebuttal. My score remains unchanged.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed comments and helpful suggestions.\n\n- *While the findings support the effectiveness of the proposed methods, applying them to a real-world dataset may further strengthen the validation and highlight their practical applicability.*\n\n\nFollowing the input of the reviewer, we have evaluated our methods on the data from Card and Krueger (1993), similar to the experiments conducted by Kivva et al. (2023). We provide a brief overview of the results here, while a more complete discussion will be included in the revised version of the manuscript.\n\nThe goal of the study is to estimate the effect of the minimum wage on the employment rate. In our experiments, we utilized the same preprocessing procedure in Kivva et al, (2023) and found that the results of our method, when assuming graphs $\\mathcal{G}_1$ or $\\mathcal{G}_2$, are consistent with findings in the existing literature. Specifically, the estimated causal effects under $\\mathcal{G}_1$ and $\\mathcal{G}_2$ are 2.68 and 2.71, respectively. Previous methods, such as the cross-moment approach (Kivva et al., 2023) and the Difference-in-Differences method, also provide an estimate of 2.68.\nIn contrast, when assuming $\\mathcal{G}_3$ as the true graph, we estimate a causal effect of 8.26. While this result still indicates a positive effect of the treatment on the outcome—consistent with prior work—the value of the estimated causal effect diverges from the ones reported in the literature. This suggests that, for this dataset, a causal graph excluding an edge from the proxy to the treatment (as in $\\mathcal{G}_1$ or $\\mathcal{G}_2$) may provide a better representation of causal relationships. \n\nThe code to reproduce the results can be found at https://anonymous.4open.science/r/CEId-from-Moments-20AC/estimation/real_data.ipynb.\n\n**Card and Krueger (1993)** - D. Card and A. B. Krueger. Minimum wages and employment: A case study of the fast food industry\nin new jersey and pennsylvania, 1993. \n\n**Kivva et al., (2023)** - Y. Kivva, S. Salehkaleybar, N. Kiyavash, A Cross-Moment Approach for Causal Effect Estimation, NeuriPS 2023.", "claims_and_evidence": "The claims in this paper are clear and convincingly supported.", "methods_and_evaluation_criteria": "This paper builds on the existing literature on LiNGAM-based structure discovery, shifting focus toward causal effect identification, which has received comparatively less attention. Following the lvLiNGAM framework, the study aims to identify specific entries of the mixing matrix using finitely many cumulants of the observational distribution under two challenging settings.  \n\nBy leveraging higher-order cumulants, the proposed method introduces the following advancements:  \na). Single Proxy Variable Setting: It allows a causal edge from the proxy to the treatment, thereby relaxing the previous assumption that each latent confounder must have exactly one proxy variable.  \nb). Underspecified Instrumental Variable Setting: It relaxes the assumption that the number of instrumental variables must be at least equal to the number of treatments.  \n\nThe proposed approach appears methodologically sound, with a well-justified use of cumulants for causal effect identification.", "theoretical_claims": "I have looked through the theorems and proofs and did not find evident issues. However, as I am not an expert in the causal effect identification domain, I will defer to other reviewers for further validation during the rebuttal phase.", "experimental_designs_or_analyses": "This paper evaluates its proposed methods through experiments on synthetic data across several representative causal graphs. For each of the two settings, the authors compare their approach against baseline methods, with results demonstrating its advantages.  \n\nWhile the findings support the effectiveness of the proposed methods, applying them to a real-world dataset may further strengthen the validation and highlight their practical applicability.", "relation_to_broader_scientific_literature": "This paper focuses on relaxing assumptions and enhancing identifiability in causal effect estimation, which contributes to the broader literature on causal inference. By improving identifiability, the proposed approach may increase the applicability of causal effect estimation methods to real-world problems, making them more practical in settings where strict assumptions are difficult to satisfy.", "essential_references_not_discussed": "I didn't find other essential references that are not discussed.", "other_strengths_and_weaknesses": "All are listed above.", "other_comments_or_suggestions": "All are listed above.", "questions_for_authors": "All are listed above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yJAuBmKWUv", "reviewer": "Reviewer_oHRC", "summary": "The paper studies the problem of estimating causal effects in lvLiNGAM via higher-order cumulants. Specifically, the authors consider two setups where a single proxy variable exists and a instrumental variable (IV) exists with multiple treatments. In both settings, the authors provide the effect identification results and corresponding estimation methods. The conducted experiments verify the effectiveness of proposed methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed comments and helpful suggestions.\n\n- *Lack of intuitive example or explanations for the main theorems. It could be better to\nmove the proofs in the main text into the appendix and add more discussion.*\n\nWe will use the additional page in the final version of the paper to provide further explanations and enhance the clarity of our theorems.  \n\nIn short, all our theorems are based on the following intuition. Theorem 3.1 establishes that there exists a polynomial of degree $l+1$ (where $ l $ is the number of latent variables in Fig. 1) whose coefficients can be computed using only the cumulants up to order $k(l)$. The roots of this polynomial correspond exactly to the causal effects of the variables $L_i$ for $1\\leq i \\leq l $ and $ V_1 $ on $ V_2 $. Solving this polynomial system reduces the space of possible solutions for the causal effect from $ V_1 $ to $ V_2 $ to a finite set of size $ l+1$.  \n\nTo further refine this solution and identify the correct causal effect uniquely, we need to derive additional equations that can only be satisfied by the true causal effect. This requires analyzing the nonlinear relationships among the cumulants of the observed distribution. Theorems 3.4--3.7 detail how this approach applies to the respective graphs.\n\n- *What is the key role of the additional variables (proxy and IV) compared with the results in [1]? How does it benefit the identification?*\n\nThe problem addressed in [1] pertains to *causal discovery*—learning the causal structure from observational data—which is distinct from the problem of causal effect identification. Specifically, [1] seeks to recover all causal graphs consistent with the observed data, but multiple such graphs may exist, leading to different possible causal effects. In contrast, causal effect identification assumes that the causal structure is known and focuses on determining the effect of a treatment on an outcome. This distinction is fundamental in the literature (see, for example, the seminal work by Shpitser \\& Pearl, (2006). Following this framework, we assume that the causal graph is given and aim to uniquely identify the target causal effect.\n\n**Shpitser \\& Pearl, (2006)** - Shpitser \\& Pearl,  Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models, AAAI, 2006.\n\n- *Why does the method of 'Cumulant with Minimization' only occur in the experiments regarding the graph?*\n\nAs mentioned in the final paragraph of Page 6, for the proxy setup shown in Figure 3 with a single latent variable (i.e., graph $\\mathcal{G}_3$), we proposed an optimization-based technique that relies on computing lower-order cumulants compared to those used in the standard \"Cumulant\" method. We refer to this approach as ``Cumulant with Minimization\", and it is tailored specifically to the structure of $\\mathcal{G}_3$. As such, its results are presented only for graph $\\mathcal{G}_3$ in Figure 6.", "claims_and_evidence": "The claims are generally well-supported by theoretical analysis.", "methods_and_evaluation_criteria": "The proposed method is reasonable and well-motivated.", "theoretical_claims": "The theoretical claims are clearly presented, and the proofs are well-structured.", "experimental_designs_or_analyses": "The experimental design and analyses are generally sound.", "relation_to_broader_scientific_literature": "NAN", "essential_references_not_discussed": "NAN", "other_strengths_and_weaknesses": "strengths\n\n1. The paper is well-written.\n\n2. This paper extends the original results of [1] that can only identify the effects up to some equivalence values. With the help of additional variables, proxy or IV, this paper shows the effect is identifiable.\n\n\n\nWeaknesses or Questions\n\n1. Lack of intuitive example or explanations for the main theorems. It could be better to remove the proofs in the main text into appendix and add more discussion. \n\n2. What is the key role of the addition variables (proxy and IV) compared with the results in [1]? How does it benefit the identification?\n\n3. Why does the method of 'Cumulant with Minimization' only occur in the experiments regarding the graph $\\mathcal{G}_3$. \n\nReference:\n\n[1] Schkoda, D., Robeva, E., and Drton, M. Causal discovery of linear non-gaussian causal models with unobserved confounding.", "other_comments_or_suggestions": "Typo:\n\nline 131: $\\mathcal{M}(G))$ should be $\\mathcal{M}(G)$\n\nline 153: contain should be contains\n\nline 665: fist should be first", "questions_for_authors": "See Weakness.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lrqtPKp5QN", "reviewer": "Reviewer_nMo5", "summary": "This paper propose causal effect identification method for proxy variable setup and underspecified instrumental variable setup based on high-order cumulants. In the proxy variable setup, both multiple latent confounders and a causal edge from the proxy variable to the treatment are allowed while only one proxy variable is required. In the underspecified instrumental variable setup, multiple treatments are allowed while only one instrumental variable is required.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed comments and helpful suggestions.\n\n- *There is no related work that is essential to understanding the (context for) key contributions of the paper but are not currently cited/discussed in the paper. However, there are some recent works [1,2] that also use high-order cumulants for identification of LiNGAM with latent variables. I recommend the authors discuss them.*\n\nWe will include a discussion of these works in the related work section. Both of these papers focus on the problem of *causal discovery*—that is, learning the causal structure from observational data—rather than on the problem of *causal effect identification*.  In the latter, the causal graph is assumed to be known, and the goal is to identify the causal effect of a treatment on an outcome (see, for example, the formal definition of this problem in the seminal work of Shpitser \\& Pearl,  (2006).\n\nIn particular, [1] proposed a procedure for using high-order moments to test for the presence of an edge between two observed variables where there is exactly one latent confounder. It is noteworthy that Schkoda et al. (2024) extended this result to settings with an arbitrary number of latent variables.\n\nIn [2], the authors considered specific causal graphs under the assumption that for each latent variable $L$ in the graph, there exists a ``homologous surrogate\" (see the definition in [2]). They then derived a set of structural properties using equations involving cumulants to recover the underlying causal structure. It is important to note that all the graphs considered in our work do not satisfy the assumption that every latent variable has a homologous surrogate except the graph $\\mathcal{G}_1$, with a single latent variable and no edge from the proxy variable to the treatment.\n\n**Shpitser \\& Pearl, (2006)** - Shpitser \\& Pearl,  Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models, AAAI, 2006.\n\n- *It seems that the proposed method assumes that the underlying causal graph is known. Please correct me if I’m wrong.*\n\nAs we mentioned above, in the problem of causal effect identification (the focus of the current work), the causal graph is known.\n\n- *Tramontano et al. (2024b) have already provided necessary and sufficient graphical conditions for generic identifiability, so this paper is not novel in terms of identifiability results; its novelty lies only in the identification method.*\n\nThe identifiability results in Tramontano et al. (2024b) rely on the assumption that the full observed distribution is known. In contrast, our identifiability results require only knowledge of finitely many moments of the distributions, which is a strictly weaker assumption. \n\nMoreover, the estimation method proposed by Tramontano et al. (2024b) is based on solving an overcomplete Independent Component Analysis (OICA) problem, which is inherently non-separable. This implies that the true mixing matrix cannot be recovered solely by optimizing for independence among the exogenous noise—precisely the approach taken by their algorithm. Consequently, their method fails to consistently estimate the correct solution. This limitation is evident in our experiments, where we observe that GRICA’s accuracy does not improve monotonically with sample size.\n\n- *(Minor) According to the experimental results shown in Figure 6, the proposed method is not superior to that proposed by Tramontano et al. (2024b) when the sample size is not very large.*\n\nOne possible explanation is that cumulant-based methods process unbiased estimates of high-order cumulants (order 4 or higher), also known as k-statistics. While these estimators are unbiased, they tend to have high variance for small sample sizes.\n    \nIn contrast, GRICA solves an optimization problem involving the $\\ell_1$-norm of the observed samples, which have lower sample variance. As a result, for small sample sizes, GRICA may yield a lower mean squared error due to reduced variance. However, since the solution obtained by GRICA is not asymptotically unbiased, it cannot provide an asymptotically consistent estimator—unlike our proposed method.\n\nWe will add a remark on this point in the final version of the manuscript.", "claims_and_evidence": "The claims made in the submission are supported by evidence.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria make sense for the problem.", "theoretical_claims": "I have tried my best to read the theoretical proofs. However, I'm not familiar with the concepts defined in Definition A.1~A.3. In fact, I have never encountered those concepts except for a recent work (Tramontano et al., 2024b). Besides, the proofs in this paper use theorems in (Schkoda et al., 2024), which are quite complicated. Therefore, I cannot provide a reliable assessment for the correctness of theoretical proofs in this paper.", "experimental_designs_or_analyses": "I have no concern about the soundness/validity of the experimental designs or analyses.", "relation_to_broader_scientific_literature": "The authors have discussed this in Impact Statement.", "essential_references_not_discussed": "There is no related work that is essential to understanding the (context for) key contributions of the paper, but are not currently cited/discussed in the paper. But there are some recent works [1,2] that also use high-order cumulants for identification of LiNGAM with latent variables. I recommend the authors to discuss them.\n\n[1] Identification of causal structure with latent variables based on higher order cumulants. AAAI 2024.\n\n[2] Recovery of Causal Graph Involving Latent Variables via Homologous Surrogates. ICLR 2025.", "other_strengths_and_weaknesses": "# Strengths\n\n1. This paper is well-motivated, Tramontano et al., (2024b) provides profound identifiability results, but their identification method is based on OICA, which has not yet been equipped with consistency guarantees. To overcome this limitation, this paper provides an identification method based on high-order cumulants.\n\n2. This paper is solid, it provides both theoretical results (although the proofs of which are difficult to check for me) and experimental results.\n\n# Weakness\n\n1. Tramontano et al., (2024b) have already provided necessary and sufficient graphical conditions for  generic identifiability, so this paper is not novel in terms of identifiability results, its novelty lies in only the identification method.\n\n2. (Minor) According to the experimental results shown as Figure 6, the proposed method is not superior to that proposed by Tramontano et al., (2024b) when the sample size is not very large.", "other_comments_or_suggestions": "Personally, I think this paper is hard to follow because the theoretical tools used in this paper are not common in the community of causal inference. I fully acknowledge that this is not a deficiency per se, but this may limit the impact of this paper.", "questions_for_authors": "It seems that the proposed method assumes that the underlying causal graph is known. Please correct me if I'm wrong.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper addresses the challenging problem of causal effect identification in latent variable linear non-Gaussian acyclic models (lvLiNGAM), focusing on two realistic and underexplored settings: identification using a single proxy variable, and underspecified instrumental variable scenarios. By leveraging higher-order cumulants, the authors establish identifiability results and propose corresponding estimation methods, supported by both theoretical analysis and empirical evaluation.

The reviewers agree that the paper is well-motivated, clearly written, and contributes meaningfully to the literature on causal inference with latent confounding. Reviewer 6fLa and oHRC appreciate the extension beyond existing LiNGAM frameworks and confirm the soundness of the theoretical and empirical results. Reviewer nMo5 notes that while some core identifiability conditions have been addressed in prior work (e.g., Tramontano et al., 2024b), the novelty of this paper lies in proposing a tractable, cumulant-based estimation method with clearer guarantees. Some concerns were raised about the complexity and accessibility of the theoretical tools, and the experiments are currently limited to synthetic data. A real-world application could further strengthen the practical relevance in future work. Overall, I recommend acceptance if there is room.
Decision: Accept (poster)