Identification of Average Outcome under Interventions in Confounded Additive Noise Models
Authors: Muhammad Qasim Elahi
Mahsa Ghasemi
Murat Kocaoglu
Venue: Accepted by TMLR ()

Abstract: Additive noise models (ANMs) are an important setting studied in causal inference. Most existing works on ANMs assume causal sufficiency, i.e., there are no unobserved confounders. This paper focuses on confounded ANMs, where a set of treatment variables and a target variable are affected by an unobserved confounder that follows a multivariate Gaussian distribution. We introduce a novel approach for estimating the average outcome under interventions (AOIs) for interventions on any subset of treatment variables and demonstrate that a small set of interventional distributions is sufficient to estimate all of them. In addition, we propose a randomized algorithm that further reduces the number of required interventions to poly-logarithmic in the number of nodes. Finally, we demonstrate that these interventions are also sufficient to recover the causal structure between the observed variables. This establishes that a poly-logarithmic number of interventions is sufficient to infer the causal effects of any subset of treatments on the outcome in confounded ANMs with high probability, even when the causal structure between treatments is unknown. The simulation results indicate that our method can accurately estimate all AOIs in the finite-sample regime. We also demonstrate the practical significance of our algorithm by evaluating it on semi-synthetic data.

Published in Transactions on Machine Learning Research (10/2025)
Identification of Average Outcome under Interventions in Confounded Additive Noise Models
Muhammad Qasim Elahi
elahi08purdue..dl edu
School of Electrical and Computer Engineering
Purdue Universily
Mahsa Ghasemi
mahsa Dppardue. edu
School Electrical and Computer Engineering Purdue University
Murat Kocaoglu
mkocoogguujjhu. edu
Department of Computer Science Johns Hopkins University
Reviewed on OpenReview: //tpnrev/pperevveeee net/ forum? id- ybYnHzL
Abstract
Additive noise models (ANMs) are an important framework studied in causal inference. Most existing works assume causal sufficiency, i.c., no latent confounders, and focus on observational data for inference and discovery. In this paper, we study confounded ANMs, where treatment and outcome variables are influenced by an unobserved confounder, and we consider interventions on treatment variables to identify various causal effects. We introduce novel approach for estimating the average outcome under interventions (AOIs) for any subset of (reatment variables and show that small set of interventional distributions is sufficient to estimate all AOIs. Furthermore, we propose a randomized algorithm that reduces the number of required interventions to poly-logarithmic in the number of nodes. We also demonstrate that these interventions suffice to recover the causal structure among the observed variables. Consequently, a poly-logarithmic number of interventions is sufficient to infer the causal effects of any subset of treatments on the outcome in confounded ANMss with high probability, even when the causal relationships among treatments unknown. Simulation results indicate that our method can accurately estimate all AOIs in finite-sample settings, and we further demonstrate its practical significance using semi-synthetic data.
1 Introduction
Discovering cause and effect relationships is a paramount objective in data sciences, artificial intelligence, and machine learning. Through meticulous scientific study, identifying causal relations enhances the validity and generalizability of corresponding effects across various conditions (Le BBareinboim 2020). causal relations are deemed desirable and valuable for constructing explanations and for contemplating novel interventions
that were never experienced before (Pearl 2009 Lee Bareinboim 2020 significant area of interest in causal reasoning is the use of observations and series of interventions to identify various causal effects in a
structural causal model (SCM) (Pearl 2009 Peters et al. 2017) Kandasamy al. 2019 Stovitz & Shrier 2019). Based on the framework developed by Pearl, an SCM consists of endogenous and exogenous variables and functional mappings that determine the values of endogenous values (Pearl 2009). Every SCM can also represented as a directed acyclic graph (DAG), with variables represented as nodes, and relationships between variables represented as edges. Many fields, including medicine, advertising, oconomics, and others,
utilize multiple interventions [Worrall 2011 Rizzi Pedersen 1992). To account for unobservable variables,
interventional data and causal inference techniques may be required Tian Pearl 2002). While randomized

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controlled experiments can be used to identify causal relationships, they are often costly or unfeasible (Farmer et al. 2018). Estimating treatment effects from observational data has gained attention due to limitations and ethical concerns of randomized experiments (Yao et al. 2021]. Observational data enables researchers to investigate causal effects without actually running the experiments (Yao et al. 2021). If there are no latent variables, it possible to identify all causal effects from the observational distribution (Greenland & Robins, 1986; Spirtes et al., 2000). When there are unobserved variables, the obsorvational distribution may not be enough identify causal effects due to potential confounding.
In additive noise models (ANMs), it is assumed that the noise in the structural equations of variables is additive. If the noise terms in the ANM are statistically independent for different variables, it implics that there is no confounding present (Peters et al.) 2014). However, when the noise terms are not independent and have a joint distribution, the ANM is considered confounded (Jeunen et al.) 2022). Estimating causal effects from purely observational data in the presence of unobserved confounders, which may affect both the treatment and the outcome, is a challenging task. Independence of noise terms is sufficient condition for no confounding under the assumption of correctly specified additive noise model. Dependence among noise terms may arise either from hidden confounding or from model misspecification, such as assuming linearity when the true data-gen tenta-ggnrratigg process is nonlincar. In this work, we focus solcly on correctly specified ANMs and do not address model misspecification.
In this work, we aim to identify all AOIs in confounded ANMs with multivariate Gaussian noise. Identifying the causal effect of any subset of treatments crucial because real-world interventions often occur in com- binations, and their joint effects may differ from individual effects. This is particularly relevant in medicine, public policy, education, agriculture, and marketing, where multiple interventions are applied simultancously, and the goal is to identify the best optimal interventions through a series of experiments. The ability to determine effects for all possible intervention combinations, i.e., all possible subsets of treatments, using a much smaller number of interventional data sets, can be highly useful in such scenarios. Estimating these effects enables better decision-making resource allocation, and the design of effective intervention strategios. To achieve this, we make specific assumptions about the noise in our analysis. We assume that the noise follows a multivariate Gaussian distribution. Furthermore, we assumne that a treatment variable cannot be descendant of the outcome variable. A probability distribution is said to be faithful to a graph if and only if every conditional independence statement in the distribution can be inferred from the graph via d- separation, a graphical criterion that determines whether two sets of variables are conditionally independent given a third set. Faithfulness is a commonly used assumption in existing work on causal discovery Kocaoglu ct al. 2017): Hanser & Bühhmann (2014). We assume that the faithfulness assumption holds for both the observational and post-interventional distributions.
We propose an algorithm for estimating AOTs of any subset of treatments in confounded ANMs from a small number of interventional data sets. We identify conditions that the set of available interventional data sets needs to satisfy to enable this.
We propose a randomized algorithm that can be used to identify poly-lggarithmic number of inter- ventions (in the number of variables) that satisfy our identifiability condition for estimating all AOIs in confounded ANMs. We also show that the same set of interventions are sufficient for discovering the causal graph it is not available. Specifically, the algorithm requires (8dmaa log" n) interven- tions, where dmaa is the largest degree in the causal graph, and n is number of treatment variables in the graph.
Using randomly generated causal graphs, we demonstrate the effectiveness of our approach in accu- rately inferring AOTs, even when working with limited data samples from experiments. We showcase the practical significance of our inference scheme by evaluating it on semi-syuthetic data as well.
2
Related Work
In various disciplines, such as medicine, business, and advertising, the ability to provide causal explanations and answer causal queries is crucial for making informed decisions, understanding relationships ween
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Alvarez-Martinez & Pérez- variables, and evaluating the effectiveness of an intervention (Nabi et al. 2022
Campos 2004). This growing body of research highlights the increasing importance of causal inference across various fields. Causal inference is an important tool to infer the effect of interventions without actually carrying them out. An important area of research is to determine the conditions under which cerlain causal ellects can be estimated from the observational data and causal information transferred from other experiments. For example, Barcinboiim & Pearl (2012) provides an algorithm for fusing observational and experimental data to estimate causal query. In Hoyer et al. (2008), authors show that as long as the noise is additive, the linear non- non-Gaussian causal discovery framework can be extended to nonlinear functional dependencies among the variables.
Imposing certain limitations on the underlying causal structure can render causal quantities identifiable, even when they are generally unidentifiable. The model being an ANM is a common practical assumption. The ANM assumes that the latent variable additive the structural equation of the observed variables (Maclaren & Nicholson 2019 Kap ct al., 2021). ANMs do allow non-linear mappings between the observed variables. In Peters et. al.] (2014); the authors demonstrate that under mild conditions, the causal graph of model with additive noise structure can be identified from the joint observational distribution. They assume that the additive noise follows a joint normal distribution and is independent. Additionally, the structural equations in the model are nonlinear and atleast thrice differentiable (Peters et al. 2014). The paper Rolland et al. 2022 proposes a method for recovering causal graphs in a non-linear additive (Gaussian) noise model. The proposed approach utilizes score-matching algorithms as a building block to design a new generation of scalable causal discovery methods.
The work by Zhang et al. 22220] proposes a simultancous approach to discovery and identification for linear models. The proposed symbiotic approach leverages information gained from causal discovery to aid inference, and vice versa. The paper Saengkyongam & Silva (2020) investigates the identifiability of joint intervention from the conjunction of observational and single-variable interventional data. The authors show that without any restriction on the structure of the SCM, it is not identifiable. The complementary question is addressed by Jeunen el. al. (2022), which is to estimate the causal effect of a single treatment. from a conjunction of observational and joint intervention data regimes under the assumption that treatments do not have causal effects on one another, and Gaussian noise has a zero mcan. They show that the observational distribution and joint interventional distribution enable the identification of all AOIs in this scenario. Our work is focused on more general ANMs without any additional restrictions on the structural equations, where treatments can have causal effects on one another, and the mean of Gaussian noise is non-zero.
The main contribution of the paper is to identify the set of interventional distributions sufficient to determine all possible AOTs for all possible subsets of treatment variables in confounded ANMs. Our main identification scheme requires knowledge of the underlying causal graph for the ANM. In many practical scenarios, the causal graph is not available. To address this situation, we modify one of the algorithms proposed in [ooCooll et al. (017)) to not only learn the observed graph structure in the ANM between the treatments and the outcome but also simultancously return the interventional data sets for the inference task. Although there are many other proposed approaches to learning the causal graph with latent variables, including but not limited to Addanki et al. (2020]: Akkari et al. (022).. our current choice is due to the fact that we show we can use the same algorithm to simultaneously learn the causal graph and return the interventional data sets that satisfy a critcria enabling identification for all possible AOIs. This allows us to propose a randomizod algorithm that can be used to identify poly-lggarithnic number of interventions satisfying our identifiability condition for estimating all AOIs in confounded ANMs, even when the underlying causal graph is unknown.
3 Background
A Structural Causal Model (SCM) can be defined as the tuple M ((X,Y) Pa), where Y) are the observedd/ndoogeee /endogenouu variables separated into treatment set X and outcome Y, U are the exogenous variables (possibly confounders) with Pa defining their joint probability distribution and F are the structural equations (Paarl 2009). The SCM can be modeled as a DAG G = (V, E), where vertices V correspond to observed variables and edges E correspond to the causal relationships. In the DAG a directed edge (X + Y) denotes that X is direct cause of Y. On the other hand, bi-directed edge (X <> Y) indicates the presence
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of an unobserved confounder the common parent of X and Y. The set of nodes having outgoing edges to particular node X; is termed as parents of that node, denoted by Pa(X,). All the endogenous variables including the X; and outcome Y can be written as a function of the parent variables and the corresponding latent variable. Without loss of generality we assume that all treatments X; are parents of the outcome Y and the noise variables are correlated.
X = (((P(())).U)) Y= fx(,U)
(1)
We focus on an important class of SCMs called confounded ANMs where the latent variables appear additive noise in the structural equations. Similar to [Junenn et al. (2022), we assume that the latent variables, jointly represented as U, follow a multivariate Gaussian distribution.
:= = ((Pa(())) +U Y fr (X) +Uy
(2)
For an SCM M, an intervention do(W) on set of variables W C X refers to fixing their values irrespective of the values that the parent variables Pa (a) take. This can be encoded by replacing the original structural equations of W with some constant, which induces a submodel Mdo(W)- The intervention on set of variables W is equivalent to breaking all the incoming edges into W in the corresponding causal graph G. We denote the modified causal graph with incoming edges to set of nodes W removed as Gw. For a given intervention dW), the distribution observed over the rest of the variables P[X \ W]do (W)] is called the interventional distribution. We denote an interventional distribution in an SCM M with a set of endogenous variables X as PM W]do(W)). In our work, define the average outcome under intervention (AOT) EE[ | do( w)], where W is any subset of the treatment variables, i.e., W S X. In previous literature, the average treatment effect (ATE), also sometimes referred to as the average causal effect (ACE). is defined as the difference: Y do W1 = w1)] - E[Y do( W2 w2)], where W1, W2 X (Varian 2016 Naimi et al.
2017 Frauen et al., 2023). Since we propose a machinery to identify the AOIs the form E do No[W=> W w)]
our proposed machinery can also be used to identify the classic ATE or ACE, which is defined as the difference between two AOIs.
4 Identifiability of Causal Effects for Additive Noise Models
An estimand is identifiable if its value can be uniquely determined from unlimited data, samples. This means that any two models that agrec on data must also agree on the estimand when the query is identifiable. Our goal to construct a set of intervention largets denoted by I, such that we can identify any AOT of the form [[[ [W) where W X.
Definition 1 Given a causal graph G, and collection of interventional distributions [P(X W,))), an interventional distribution ((y) is not identtfiable iff among all models that entail the causal graph G and the interventional distributions (p() W.))}, there exists two models MI,M5 where pMi [y|do (z)] # PMa by) [))).)
The paper Jeunen et al. (2022) shows that for the confounded ANMs where there is no direct causal re- lationship among the treatments, all AOIs can be identified from the observational distrbution and joint interventional distribution on all treatments. However, in a general cases where treatments can have direct causal effects on other variables, the previously mentioned result may not be applicable. We can show this with help of simple example. Consider a intervention target set I that includes two interventions: the emply intervention for observational data and the joint intervention X for simultaneous interventions on all variables. We show that this set I = {0, X} is not sufficient to identify all the AOIs in ANMs.
The two SCMs M1,M2 in Table I agree on the interventional distributions for targets in the set I = {0, (2)}, but they can disagree on other interventional distributions. The interventional distribution do(X, = X2 = x2) is the same for both M1 and M2, Lee. 1))1 Similarly, the observational distribution for X, is the same, i.c. X, ~ NN(0, 1). For both models, the variable X2 a zero mean Gaussian with a variance of 3. i.c X2 3). Also, the covariance between X, and X2 is the same across both models, i.e. Coo(X,, X²) = 1.5. Consequently, the observational distribution is the same for
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Table 1: Two SCMs where U2, ~ (N and (U1,U2, Ur)Ma ~ ~N M.m.).
M1
M2
0
X1 U1
X1 U1
X₂ X1 + U2
X2 1.2X 1.2U2
ΣM,
1 1 0
ΣM²
4
1
7
0
Y = X1X2 + Uy
Y =1X2 Uy
0
0
1
0
0
U
X
X X2
X4
Xi
X2
X3
X4
(a) Causal Graph G1
(b) Cansal Graph G2
(c) Causal Graph Ga
Figure 1: Causal graphs for different confounded additive noise models.
both Mi and M2. Therefore, both SCMs agree on the interventional distributions for targets in the set I defined earlier. However, we have pMi [X₂ do(,)] & pMa [x]]ox)(o(())) which implies that interventional distribution do(X,) is unidentifiable from interventional distributions in I. This implies that we need a bigger intervention target set to make all average causal identifiable for confounded ANMs. This example demonstrates that it is possible to create two distinct SCMs that agree given observational distribution and intervention on all treatments do(X), but have differences in either their structural equations noise parameters. Consequently, these models will disagree on some interventional distributions. We show that with additional interventional datasets, including interventions on the parent sets of the treatment variables, we can make all possible AOIs in confounded ANMs identifiable.
4.1 Core Interventions for Identification
We first prove a sufficient condition to make all AOIs identifiable for an ANM with multivariate Gaussian noise in Theorem [ Later, we extend this result to provide the necessary and sufficient set of conditions that interventional data sets need to satisfy to ensure the identifiability of all AOIs in confounded ANMs with Gaussian noise in Theorem 2
Theorem (Identifiability of AOIs in confounded ANMs) Let (X,Y,, U,F,Pu) be an SCM where X, = fi(Pa(X.)) + U,, Y= = + Uy and Pa ~ (p, E). All estimands of the form E[] W)] where W C X are identifiable from the conjunction of the two data regimes; the observational distribution and interventional distrilulions given that the collection of interventional distribulions includes P[Y]do( X)) and (P[V) (a(XX))do((P((X)) Vi =i=1... L...I.
A sketch of the proof of Theorem is provided here, with a detailed proof available in the supplementary material. Consider two sets treatment variables such that: Xints Xo C X and Xo = xX) x X,nt- The struc- tural equation (X) is identified from the interventional distribution (X).. Consequently, the conditional AOI of the form E (do(Xint); Xo] can be identifiod follows:
[[ fr E(U,]
E|y|do Xol fy (Xint X E(U, Xo. do
EUU Xo, do X mn) = EUU, = Xo - (o((((X))))
We can estimate the conditional expectation E(U,, Xo. do( (iint)] by identifying the structural equations for Xo and the noise covariance matrix. To do so, utilize interventional distributions of the form
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do(Pa(Xo)) and rearrange equation 2 to estimate the noise covariance matrix. The detailed proof for this inference mechanism is provided in the supplementary material. The intervention target set I for identifying all possible AOIs in ANMs includes both the observational and interventional distributions: do(X) and do(Pa(Xi)). Our thcoretical results assume that the latent confounder follows a multivariate Gaussian distribution. The proposed methodology and proof techniques can be extended to distributions in which conditional expectations are fully determined by the mean and variance, such as certain exponential family distributions. For heavy-tailed or non-Gaussian distributions, it may be possible to bound causal effects; however, to achieve full identification, we impose these stricter assumptions to ensure that the theoretical guarantees hold.
Our identification result in Theorem I is applicable to a wide class of confounded ANMs where a multivariate additive Gaussian noise affects all variables. We explain with a simple example that while non- parametric identification will fail, our inference scheme will still be able to identify the average causal effect under our parametric assumption. Consider a causal graph on variables X2, Y of the form X1 + Y + X₂ with latent variable U Ber((..5) pointing to all of the variables. Consider two SCMs (Structural Causal Models) consistent with the above causal graph: M1, with structural equations X1 X2 U and Y = (X1 - (1 - U): and a sccond one, M2, with X1 = X₂ U and Y = (X1 - (U). It can be easily verified that both SCMs entail the same observational distribution and interventional distribution P|Y do(X, = 21, X2 = 2-)]_- However, they differ on the do(X, = 0) intervention. For M1, we have E| do( = 0)] = (l-( U'] = =0, but for M2, we have E| do(X, = 0) = E[-U²] = -0.5. Thus, E[Y do(X11] is not identifiable from the observational distribution and P|Y do(X,, X2)]. However, note that operating under our parametric assumptions on the underlying SCM, we can identify all the Average Causal Effects from just the observational distribution and the interventional distribution P|Y| | Xx) 8s implied by Theorem 1 Note that Pa(Xi) = Pa(X2)= =0 0 for this case.
In a previous study by Jeunen et al. (022), it was demonstrated that the combination of the observational distribution and interventional distribution (P[Y)d((X))) enables the identification of all AOIs within the framework of coufounded ANMs when the treatments have no causal ellects on each other. Furthermore, they assume the Gaussian noise to have a zero mean. However, in a more general scenario where treatments can have causal effects, interventions on all treatments and the observational distributions are insufficient to identify all AOIs. In this case, only the AOI for the sink treatment can be identified Jeunen et al. (2022). A sink ment is one that does not cause any treatments; for example, X4 is the sink treatment. in causal graphs G₁ and G2 in Figure]) [ Thus, to identify all possible AOIs for G1 and G2, we require access to more interventional data, as specified by identification theorem [ Also, the identification result in Jeunen et al.) (2022) for the case where treatments do not cause each other can be derived from Theorem I since here Pa(X.) = 0 for all i, which implics that the observational distribution and interventions on all treatments are sufficient to identify all AOIs. Thus, Theorem [ generalizes the identification results presented in Jeunen et al. 2022).
The Theorem 1 can be useful for ANMs with a large number of treatments because the size of the intervention set will be much smaller as compared to the total number of possible subsets of the treatment set X. For instance, consider an ANM with ten treatments and one outcome. There will be 200 = 1024 possible AOIs of the form B[Y [W)] where W C X. The number of sufficient interventional data sets for inference is at most linear in the number of variables of the graph. This implics that for ANMs with large number of ments, we can identify an exponentially large number of AOTs using much smaller number of interventional data sets. There are 16 possible AOIs for all causal graphs shown in Figure D Consider the graph structure G1 given in the Figure [a). Following the Theorem 1 the intervention target set I = (0, X2, Xs, X1). [X2]. [X1, XatF. Thus we can identify all possible 16 AOIs using the four interventional data sets. Similarly, for the cansal graphs in the Figure 1b) and Figure Ic), the intervention target sets are: {0, {X1, X2, X3, X4), [X.]. {X2), [Xa]] for G2, and (0, (X1, X3,X,) X4}} for G3. There is a possible extension of the result in Theorem D to the case when additive noise for a particular treatment variable say X, is independent of the additive noise for other treatments and the outcome variable. In this
case, can drop some intervention targets from the set I. The formal statement of this result is presented in the corollary [
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Corollary 1 Let ((X,Y), Pc. be SCM Where X; f.(Pa(X))) + UU fx( + Uy and Pa ~ N(n, Σ) and suppose. we have Un II Us vb + for some a E [m]- In this setling, any possible AOI is identifiabe from the conjunction of the two data regimes; the observational distribution and interventional distributions that includes [do(X) and {P(V)
Vi v a}.
The scope of Theorem I not limited to a specific graph structure, where there are treatments and one outcome variable. The established identifiability result can be generalized to other causal graph structures, provided that the noise or latent. variables satisly the conditions of being additive, and jointly Gaussian. For example, consider an SCM M = (x, F.F. where X; = fx, (Pa(X,)) + Ux. Let X; E X and W C X such that X; & W. Also assume T = Pa(X,) \ W where Pa(X;) is the parent set of the variable X; in the SCM M. The conditional AOI of the form [[X;]do((W) T] can be identified from the conjunction of the interventional distributions do())(()) and do(Pa())). We have B[X;]do(W)) T) = fx. (Pa()))) E(xx, (W), T). The structural equation fx, and the conditional expectation of the noise can be identified using the aforementioned interventional distributions. Theorem [ provides the sufficient conditions to identify all the AOIs in ANMs with Gaussian noise. We extend Theorem to give necessary and sufficient conditions to identify all the AOIs in confounded ANMs in Theorem 2
Theorem (Necessary & Sufficient Conditions for AOIs Identifiability in Confounded ANMs) Let (V= (), U.F.Pal be an SCM where X; = ((Pa(X))) + U, Y [X) + Uy and Pa ~ N E). All estimands of the form [d] W)] where W S X are identifiable if and only if we have access to interventional distribution P[Y[do(X) and the interventional distributions P(V S. such that X; & S; and Pa(X.) S S, for every treatment X, E X.
Although it appears as if Theorem states that, in the case of general confounded ANMs with n treatments, (n) interventional datasets are required to identify all possible AOTs, we show that is possible to reduce this poly-logarithmic order by using a randomized algorithm to select intervention targets similar to the one proposed in Kocaoglu et al. (2017). We also demonstrate that this set of interventions is adequate for learning the observable causal graph, i.., the induced graph between observed variables.
4.2 A Randomized Algorithm for Identifying AOIS
A deterministic approach to learning the observable graph will require n interventions (Addanki & Ka- siviswanathan, 2021 The observable graph is the induced subgraph of the original causal graph on the observed variables only, containing directed edges between observed variables while omitting edges involving unobserved (latent) variables. For every observed variable X,, the post-interventional graph Gxxx, is con- structed. A pair of observed variable X, and X, will be d-comnected in Gx,x, only when X₁ E Pa(X,). This approach enables thefn learning of the obscrvable graph using interventions, each involving n - variables. Compared to a deterministic approach, which would require n interventions, our randomized algorithm re- quires fewer interventions, i.e., O(8a dmax log"(n)). Additionally, it returns the necessary interventional datasets for inference with a probability of at least - - By utilizing ancestral graphs and incorporat- ing random interventions, the algorithm learns the observable graph and generates sufficient interventional datasets for inference purposes with high probability.
Definition 2 A collection of subsets S = (S,, S2, Sm) of the treatment variable set X is strongly separating set system if for any poir of variables X, and Xj, there crists St and S3 such that X, E S. S and X2 E S, S.
We can construct a strongly separating using the binary expansion of numbers from I to n. Specifically, we consider numbers of length [Iog(n]] in binary representation. For each bit i, we create a set S, that includes the numbers where the i-th bit is 1, and another set S, that includes the numbers where the i-th bit is O. The family of all such sets forms a strongly separating system (Kocaoglu et al. 2017). This implies that we can always find a strongly scparating set system S on the ground set X with |S S 2 [lgg())]. We can learn all the ancestral relations a causal graph with a maximum of 2 [log(n)] interventions using the separating system S.
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Algorithm 1: Learn the transitive closure of the graph given access to Cl-testing and query access to sample causal model under any intervention i.c. Mdo(S.).
Function LearmtransstiirellssreedCW
E =
Construct = { Sm. where < 2[log(n)] as strongly separating set system for 1 : 2[log(n)] do
for every pair X; E S. and X5 E V S; do Use samples from Maois for CI test.
if (X, M G)) then
L E BU EX,,
return The graph's transitive closure (V, E). End Function
Algorithm 2: Learn the observable graph: Accepts two paramcters a and maximum graph degree dmaz and outputs the observable sub-graph and sufficient interventional data sets for inference.
Function LearnDbservableeraph (M)): (MM):
E = 0
IData = Samples from observational distribution & intervention on all treatments i.e. do(X).
for i = 1 : 4a dmaz log(n) do
S 0
for X; E X do
S UUX, with probability 1
dmax
Gh Leaarrtranttiioooooureeeee
Compute the transitive reduction ion Tr(Gu) & add any missing edges from Tr(Gas to E Intervene and sample data from Mao(s) IData + IData Data from Mao(s)
return The observable graph structure (V, E) and interventional data samples in IDota.
End Function
Definition 3 For a causal graph G (V, E), we define a new DAG G (V, such that, for any pair of vertices V, and V, E V, the directed edge V, - V2 is included in G only when there is directed path from V, to V, in G. We refer to this new DAG, G represenling ancestral relations in G as the transitive closure of
Algorithm 1 can be employed to learn the transitive closure of the causal graph G corresponding to the SCM M with 2[log(n)] interventions (Kocaoglu et al. 2017
Definition 4 Consider DAG G = =V, with ils transilive closure Gts The transitive reduction of G, denoted by Tr(G)= = (V, E,), is a DAG with the minimum number of edges that has the some transitive closure as G
The transitive reduction of an acyelic graph is unique and can be computed in polynomial time (Aho et al. 1972). Furthermore the set of directed edges E, in Tr(G) is the subset of directed edges in G i.e., E, S E (Aho et al.) 1972). Algorithm randomly samples the intervention target sets and accumulates the edges learned from the transitive closure of the post-interventional graphs finally learn the observable graph structure. We demonstrate that Algorithm
2 not only learns the observable graph but also returns the required interventional datasets for inference with high probability in just polylogarithmic number of interventions (Theorem 3). In order to ensure the identifiability of all AOIs using Theorem 1 we need access to the observational distribution and interventional distributions, which include P|Y[do and P[V \ Pa(XX)))ooo (doXX))do(Pa(())) for all i 1 to n. This implies that O(n) interventional data sets are required infer all AOIs. However, to demonstrate that polylogarithmic number of interventional data sets from Algorithm 2 is sufficient infer all AOIs, we use more general criterion that the interventional data sets must satisfy ensure the identification of all AOIs, as stated in Theorem Once have access all the
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Olii 010
Shuchrrd Hamming Disterro (SHD)
100
Stratural hamminn Detarue
20
80
suffecieee sete for
16-
60
Properton
10
150
200
300
20
100
Number Treedmennt (n)
Sample
Number of Rar Intorvantinn
(a) SHD between the learned and actual graph structure using sample size of 300.
(b) SHD between the learned and actual graph structure vs.
(c) Proportion of DAGs with
sufficient data sets for inference interventions (n 20).
sample size (n 20).
Figure 2: Simulation results for the proposed randomized algorithm for ANMs with small number of treatments.
interventional datasets that meet the criteria mentioned in Theorem 2 we compute all possible AOls. Algorithm 2 takes two input parameters: dmax and a. The algorithm requires a polylogarithmic number of interventional data sels, specifically O(log"(n) (n)) when a = log(n). Note that the required nunber of interventional data sets scales linearly with the maximum graph degree (diaxx)
Theorem 3 Suppose that dmax is greater or equal to the marimum degree in the obsercable graph of the ANM. Algorithm 2 requires a marimum of 8a dmaz log" (m) interventions to learn the true observable graph with a probability of at least 1- - The algorithm also returns the required interventional data sets to compute the noise variance matrix and the AOIs with success probability 0
Corollary 2 If the parameter a is chosen as log(n) in Algorithm the mazimum namber of required interventions will be Sdmar log)(n) to relurn the observable graph with a probability of least mooute) The algorithm will also return required interventional data sets for inference with success probability of -1
3 is provided in the supplementary material. The randomized algorithm not only provides sufficient datasets for inference but also enables learning the underlying graph structure in ANMs when it is unknown. Note that we do not impose a bound on the maximum number of nodes in intervention set, allowing the total number of experiments to remain poly-logarithmic. If one wishes to restrict the size of each intervention set, an (n, k) separating system, where the size of each intervention set is bounded by k, from Theorem 1 SShammugam et al. (2015) can be used; however, in that case, the total mumber of experiments may no longer remain poly-logarithmic and may become linear in the number of nodes in the worst case,
The proof of Theorem
5 Empirical Validation
We empirically validate our method, which includes experiments on synthetic and semi-synthetic data based the HEALTHCARE Bayesian network from the bulearn library repository.
5.1 Synthetic Experiment
In this section, we evaluate our combined discovery and inference scheme using synthetic data. Our goal is to evaluate the algorithm's ability to accurately determine the graph structure, generate sufficient interventional data sets for inference, and provide precise estimates of AOls. We evaluate the combined causal discovery algorithm and inference by randomly generating DAGs with sets of treatments X, which all affect the outcome Y. A multivariate Gaussian noise is added to the structural equations of the ments and outcomes. We test different sizes of treatment sets, with n ranging from 3 to 20, and generate 50 random DAGs for each size. The curse of dimensionality presents challenge in testing conditional independence for continuous variables. We address testing conditional independence for continuous variables using an kernel matrix- method that outperforms discretization-bant approaches in terms of accuracy (Zhang et al. 2012

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E[ xxxxX.xx
[[ 2,x,XXX x
E ,
[ X.XxxxXXXXXXXxX x.xx
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[[Y]do(X, ) X x
[[ X x1 x
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Figure 3: Mean absolute error (MAE) in estimating AOIS under various interventions for randomly generated
ANMs with 4 treatment variables, across different sample sizes.
H D
C
O
Figure 4: Graph for HEALTHCARE Bayesian Network
Figure 2aa) plots the Structural Hamming Distance (SHD) for randomly generated DAGs against the number of treatments. The results show that the SHD increases along with the increase in the number of treatments, but at slower pace compared to the increase in the number of possible edges in the graph. This implies that the discovery algorithm being used performs well in determining the true structure of the DAGs, even when the number of treatments increases. Figure bb) plots sample size versus the performance of the discovery algorithm. The performance evaluated by the SHD between the learned and true DAGs. The results demonstrate that the SHD decreases as the number of samples increases. Figure presents the relationship between the number of random interventions and the proportion of DAGs that have sufficient data for inference. The results are based on 50 randomly generated DAGs, each with n 20 treatments. The plot confirms that as the number of interventions increases, the probability of having sufficient data for inference also increases. This indicates that a larger number of interventions can enhance the chances of obtaining adequate data for causal inference. Furthermore, the results suggest that the number of interventions required to obtain sufficient data for inference increases with the graph's 's maximum degree.
To demonstrate the performance of our inference scheme, we apply our combined discovery and inference algorithm to 50 randomly generated ANMs, each with fixed number of treatments (n = 4). We analyze and plot the error in approximating each of the 16 AOIs against the sample size in Figure
5.2 Semi-synthetic Experiment Based on HEALTHCARE Bayesian Network
In order to demonstrate the effectiveness of our inference scheme, we evaluate through a semi-synthetic experiment using the HEALTHCARE Bayesian network from bulearn repository (Scutari) 2009). The cor- responding causal graph is shown in Figure] 4 The variables and H are discrete, while the remaining
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E[T]do[C).D.co]]
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Figure 5: Mean absolute error (MAE) in estimating AOIs under various interventions for HEALTHCARE Bayesian Network.
variables are continuous. The conditional distribution of variables C, O, I and T given their parents follows a Gaussian distribution with a specific variance and mean determined by their parent variables' values. For instance, the conditional probability distribution of given its parents and D is NV00dd, of). The variance (a²) of the distribution depends on the realization (c) of the discrete variable (C). To in- troduce confounding into the network, we assume that we do not observe the variables and H, which now act as latent confounders, T, and the treatment variables include C, D, O, I. We apply our inference scheme to estimate all 16 possible AOIs using the three interventional datasets ( (I= fØ, fC, (), ), and we plot the cor- responding estimation errors in Figure 5 As our assumption of additive Gaussian noise remains unsatisfied, the estimation errors are higher when compared to the synthetic experiment. Nevertheless, the central finding remains that, even when the Gaussianity assumption is unmet, our inference scheme yields reason- ably precise estimates. The results of our experiment can be found at https://github. com www.ccoom/aaa.se /aaiamlammmmm/////amaaaaaaaaaaaa ortp::/ee-ppppettaaaaaaaaaaaaooo
indicated by the dashed edges in Figure The outcome variable is
6 Conclusion
We propose a scheme to estimate the average outcome under various interventions for confounded additive noise models. Our approach demonstrates that a relatively small number of interventional distributions is sufficient to determine all possible AOIs in ANMs. We also show that the interventional datasets charac- terized by our method are necessary to perform inference for all possible AOIs in confounded ANMs. By employing a randomized algorithm, we significantly reduce the number of interventions required, achieving a poly-logarithmic scale relative to the number of treatments. Through synthetic and semi-synthetic ex- periments, we validate the reliability and utility our approach for both causal discovery and inference. These results highlight the practical fcasibility of our method for learning observable causal structures and performing efficient. inference even in the presence of latent confounders.
Acknowledgment
This research has been supported in part by NSF CAREER 2239375, LIS 2348717, Amazon Research Award, Adobe Rescarch and Intuit.
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A Supplementary Material
In the following subscctions, we provide comprehensive and formal mathematical proofs for the the theorems presented in the main paper.
A.1 Pearl's Rules of do-Calculan (Pearl (2009) )
Let G represent the causal DAG, and let P denote the probability distribution induced by the corresponding causal model. For any disjoint subsets of variables X, Y, Z, and W, the following rules apply:
Rule 1:
(Insertion/dellttion of observations): P(y]do(x), z, w) = Rule 2: (Action/observaation exchange): P(y]do(x), do(z), w) = P(y|do(x), 2, w) if (Y Z|X, W) W)GZZ'
w) if (Y Z W)
Rule 3: I of actions): P do(z), w) P(y(do(x), w) (Y TT ZX, W) G xzz(W) where Z(W) is the set of nodes in Z that not ancestors of any of the nodes in W.
A.2 Proof of Theorem I and Corollary
The section provides the proof of the Identifiablity of AOTs in symmetric ANMs. Consider two sets: Xint C X of the form Xint = {Xinti, Xint2; mny Xinta] and Xo = X Xine of the form Xo = {Xo1, Xos). Also note that a B n.
Xintl
X
Xint
Xoi
X02
X03
Figure 6: Causal graph (G) for a general confounded additive noise model (The odges between the treatments X/s can be arbitrary so long as no cycle formed)
[[[|do(Xint) Fintl; Xinta Tinta), Xo1 150,.... Xox 208.
Jy (x Tinta 401: ................................ TO8)) +
E(Uy|Xo1 To1 -Io......Xo. Xoo..... = Ioe, X Zintl .... Xinta = Finta)) (3)
From the intervention on all treatments, have access to the following:
Y [ddo( = Tintl Xinta Lintay Xo1 &01, Xoa = 203)l
B[U,] + (I ".,,,.,,,. (4)
The equation 5 gives us the conditional expectation of Uy given the observed treatment variables.
E ElUy 201 ..... Xos Tog, do(X intl Fintl ..... Xinto F(nta))
Uou #o1 for (oi(Pa(X)))), Vos zos fos(Pa(Xos)) (5)
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Where the assignment Pa(Xos) is consistent with the causal query specified in equation 3 In order to show how equation holds, we use the following approach:
Eou = zo1 - UUu = zor =oo Uos = zos fos (P(oo))))) do( (i = Fintl, ... Xinta Tinta )
Uos = ros - (os(ooXo))
(6)
E|Uy|Uo1 = xo1 - (oi((P((Xoo)) U/g too
UUUy|Uo1 Uo1 = z01 - foi (oo(PP(XXo)))) ........ U0:= zo3 (Pa(Pa() Xo1 = #01,.....
=0s,,do .....; Xos 20s, do (iit Fintl, Xinta Tinta )
-Elur|Xor 201 ... Xos 5os, do (Xintl Fintl, myy Xinta Tinta]] (7)
The equation holds because of Pearl's do-calculus rule 3 since Uy 1L Xint Uo in the post interventional graph where incoming edges to the set Xint are removed ie. Gx... Note that Uo (Vo Uo..... Uoa] and none of variables in the treatment set can be ancestors or parents of the noise variables. The equation 7
holds because Pearl's do-calculus rule since U, 11. Xo|Uo. Xint in the post. intervent ional graph where incoming edges to the set Xint are removed, i.e. Gx.,. Now, using the property of conditional expectation of correlated Gaussian random variables, the equation can be rewritten as:
BlUy Xo1 201, my Xos 8os, do Fintl, my. Xinta Tinta)) []nta]]=BBBp -B(U) E --1V
where Enz (1(0)) Or(0))))))...)((())
(01)(02)
((1))(0))
(O1)(88)
((2))1
002
((2))033)
Egz =
((8))(1))))
xo0 - foi(Pa(Xo))) -B(()))))) roz foz(Pa(Xoz)) E[Uoil
D03 - (Pa(Xo)))))) - E(Uos] If. (8)
In equation g the noise parameters (Eyz: Ege) and the structural equations for(.) must be identified from a combination of observational and interventional data. The term di denotes the covariance between the noise variables U; and Uy. Our results hold for any noise covariance matrix Exx- However, there may be cases where the covariance matrix is not invertible. For example, in the case of multivariate Gaussian noise, this occurs if and only if at least one component of the multivariate Gaussian noise is linear combination of other components. In equation g we invert the covariance matrix. This step might fail when the covariance matrix is non-invertible. If the noise covariance mairix is not invertible, we can apply Gaussian elimination lo identify the dependent noise component(s). These component(s) can then removed from the conditioning set. Consequently, the corresponding rows Σ,, and the corresponding entrics of E can also be removed from equations X This ensures that the resulting covariance matrix is invertible, and our identification scheme remains applicable. Thus, small clarification in our proof is required to address cases where there
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are dependent noise components, meaning the covariance matrix is not full rank. Specifically, need to identify the dependent noise components and remove the corresponding rows from the covariance matrix to ensure it is invertible.
Identifying the Structural Equations: We have access to the interventional distributions of the form Pr(V) \ Pa(X;) do(Pa(X))))- The structural equations foi(.), shifted by unknown Gaussian noise means (E(Uoi) can thus be identified by using Equation
(9)
The Equation 9 holds by Pearl's do-calculus rule 3 since Uoi Pa(Xoi) in the post interventional graph where incoming edges to the set Pa(Xos) are removed i.e. G "pa(Xo.)'
Identifying the Noise Co-variances: After identifying the structural equations, observational samples can be used once more to identify noise samples Us, which are again shifted by unknown noise means, by rearranging the equation X; (P(((X))) U, as follows:
Ú, = X1 [(xxx)) = X, (((a(()))) B[U,] U, E[U.]
(10)
A constant shift. noise values will not affect our noise co-variance estimates. This is because we have ()) =
- E[,,0,). We can finally identify the causal effect specified in equation
by combining information from equations and 9
follows:
(dY|do((ann) = Tintl: ..... Xinta = Dmnaa), Xo1 = z01 Xos = 20ol
[[y]do(Xine1 = Timel, Xi., Xinta = Iinta: Xo1 ro1. Ioi, Xo.. = xos)l + Eyz
where V= = E[Xoi do (o((o(()))) ....... 209 E[Xos do (o(Pa(Xo)))]]
(11)
This shows that E[Y (IY]oo( = Zintly Xinto = Vinta), Xo1 = _o.,...,,oo = zool is identifiable. Now, since our true objective is to identify the AOI of the form [|o|)d((Eau1 Fintl, ..... Xinta = Finta), we need to marginalize out the extra variables using the interventional distribution s(Xo 201, Xox zos (do( Tintl ..... Xinta zinta)), which can be identified follows:
f(Xo1 "o.....,.op = zomldo(Xin11 = Tintls Xinta = Iinta)
i=
II ((ro:lp((Xo)) (ot)))p uo2...., (oo))..,uu))) du (12)
Where the assignment Pa(Xoi) is consistent with the causal query specified in equation Notice that Xo: = f((Pa(X)))) + Uoi where fí(.) a deterministic function. This implies that the probability density function | f(xoilp((Xo))) (oo)) uos) will just be an impulse centered at uoi = Fox - ((Pa((o))). Thus we have:
f(Xo: zon. Xos 2oald Fintl, ..... Xinta Iinta)
fu(uoi = 301 (((aa(X)))) no2 = ro2 [(Pa(Xoo)):.., u03 = Jos [((P((Xss)) (13)
Where fu ~ N(uu, Enz). Let us define for ~ NO Egr). Notice that we cannot directly estimate the structural equation fi exactly but we identify shifted version i.c. say f((.) f. EUU (sce equation Thus, instead of the exact uo: ro: f((P((Xo)), we have access u/: ro: f((Pa(Xo)))
Fo: f(Pa(Xo:)) E[Uos} uoi E(Uoi.. We get around this issue by noticing that:
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(uou uO2,..., = (wo1 RR uo2 R|Uo2)..... MO33 IB(uos])
(14)
Thus, we can identify the any desired A01 with access to the interventional distributions of the form Pr[V] Pa(X.) do(Pa(X))) using the final expression below:
[[[]do((inn) = Fintl: Xinta = Tinta)]
18 Ido int1 Fintl Xinta = Iinta), Xo1 201. ... "o., Xoo 3 zool
f(Xo1 201 c. Xos 1ogldo((intt Lintl ... Xinta Dinta) dXo (15)
This completes the proof of Theorem The proof of the corollary T just follows from the proof of Theorem There are two possibilities, either X, E Xint or Xa Xo. In the first case, the interventional distribution of Pa(Xa) is not required to identify AOI given in equation 3 For the second case, we rely on the additional assumption in corollary | that Ua 11 Ub A b + a, we have dya = 0 and Tba = 0. Also, can use the observational data regime to identify the structural equation of variable X₂ which is required to evaluate The utructural equation fa(-) can be identified from observational data regime using equation
equation 5
E E[Xa|Pa(X,, = Sa (a(Pa(X.)) B(U./ Ja(Pa(X.)) B(U.)
(16)
Thus, the case where there is no edge pointing from latent variable U to a certain treatment Xa, the intervention do( Pa(X.)) is longer required to identify the structural equation of X,- This concludes the proof of corollary 1
A.3 Proof of Theorem 2
We prove the sufficiency and necessity of the conditions for identifiability of AOIs in confounded ANMs with Gaussian noise from Theorem 2 in separate subsections below:
A.3.1 Proof of Sufficiency:
Theorem states that any estimand of the form B [do W)] where W C can be identified from collection of interventional distributions including P[Y]do(X)) and PIV \ S,)do(S,)), such that X, & S, & Pa(X,) S S., Vi I, Theorem D states that access to the observational distribution, interventions treatments, and interventional datasets targeting the parents of each treatment variable is sufficient to identify all AOIs. The only diffcrence that interventions on parents of treatments Pa(X;) are replaced by interventions on S, such that X; & S, Pa(X,) C S, Vi I,..,..., n. The Proof of Sufficiency can be inferred from the proof of Theorem D except that we need to show we can still identify the structural equations of the treatment variables shifted by unknown Gaussian noise means using interventional distributions P[V \ S,)d)(S,)] instead of the interventional distribution of the form P(V (a(x)))do(Pa((x)).
all
E[X, do( f. E(U, =
E(Uo]]
(17)
Equation 17 holds due to the fact that X, & S, & Pa(X,) C S; and by Pearl's do-calculus rule 3, since U JL S, in the post-interventionall graph where incoming edges to S are removed, ie., Gg.: Thus, we can still identify the structural equations shifted by unknown Gaussian noise means using the interventional distributions P(V \ S.)do(S.)), such that X; S, & Pa(X.) C S,, Vi I...., ni and the rest of the identifiability proof follows from the proof of Theorem [
A.3.2 Proof of Necessity:
Consider the ANMY = fy (X) and X₁ = (Pa(()))) for all e [n], where U = [U1, U2.... Un. Uy is multivariate Gaussian noise. Suppose there exists a treatment X, E X such that do not have access
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to an interventional dataset S, for which Pa(X,) S S and X; & Sj. while we have access to all other possible interventional data. Using the interventional data of the form do(S; = 8;) and do(S; = s() for every treatment X, such that i # j, where Pa(Xi) S S; and X; $ Si, we have the following:
E[X; do(S; = s.)] = f (((a((X))) + B[U]]
(18)
B|X; do(S, = s' = (pa' (x:)) + E[U
(19)
Using the above equations, we can identify the noise mean as well as the structural equation fi- We can also rearrange the equation to recover the noise samples and identify the noise distributions. In fact, given access to interventional datasets S, such that Pa(X;) S S₁ and X; $ S. for every treatment X; E X {X,}, we can identify the joint distribution of the noise variables {U1, U...... U2-1, U.s1......... Un, Uy} and all the structural equations except for ʃز- The parameters that can differ across the two ANMs are the equation Sj and the noise distribution parameters B[Un]: مjj. and Øij for all i i j.
For the treatment Xj: we don't have access to an interventional dataset S, such that Pa(X,) C and Xj & Sy. Now, we want to construct two ANMs, (1) and (2), which have different interventional distributions for an intervention Sj such that Pa(X,) S S3 and X; $ Sj, but agree on all other interventional distributions in the collection.
x 1, (Pa(x,)) +
(20)
x/0 12 (Pa(X;)) +0
(21)
y@)= = Σ +U
(22)
x. EX
y(2) Σ
x(2) +
(23)
X EX
Suppose that the functions () and (() are equal everywhere except at Pa(X;) = B. One possible choice is f, (Pa(X,)) = a x ((Pa(X,) = 3} and f!(')( (Pa(x)) = x ({))(X,) = B), where a * b. Additionally, assume that the multivariate Gaussian noise distribution is the same across the ANMs. This implies that the distribution of X2 is different across the two ANMs under the intervention do(S, sj), B. Therefore, the two ANMs have different interventional distributions for do(S)). Note that Sy is any subset of the treatment set X such that Pa(X,) E S, and X, & Sy. Given that the distribution of X, is different across the two ANMs under some intervention do = 8j). and that both ANMs share the same noise distribution and structural equations fj for all j * i, the AOIs (ly do(S, = s,)] and (2 do(S = s, will be different, assuming that both ANMs have the same structural equation for the outcome [x( = Exiex X, as in equations 22 and 23
where Pa(X,)
We still need to show that the ANMs will have the same interventional distributions other than Sj, where Xj itself is not intervened on. Under any intervention other than S, where X, itself is not intervened on, the probability that Pa(X,) B is almost surely zero because of the additive Gaussian noise in the structural equations. Thus, the two ANMs constructed will almost surely have the same interventional distributions for all the larget sets except. S, where S, C X such that Pa(X,) C S₁ and X, & Sj. Thus, the given two ANMs can only be distinguished using the intervention on Sj. This proves that all AOIs can't be identified unless we have access to an interventional dataset S, for every treatment X, E X, such that Pa(X,) C S, and X; & S. This concludes the proof of the necessary conditions for the identifiability of AOIs in ANMs with Gaussian noise, stated in Theorem 2
18

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Published in Transactions on Machine Learning Research (10/2025)
A.4 Proof of Theorem 3
Recall that in mathematics, specifically in order theory, partial order on set is an arrangement where certain elements have a defined precedence over others. The term "partial" indicates that not every pair of clements needs to be comparable in terms of their ordering. In order to prove Theorem 3 we rely on the following lemma from Kocaoglu et al. 20177
Lemma 1 Kocaoglu et al. (2017) Consider a graph with observable variables X and an intervention set SCX. Consider the post-interventional observable graph G3 and variable X; X S. Let V Pa(X;) be such that all the parents of X; above V in partiol order are included in the intervention set S. This implies that {W : (WW)= (V), W E Pa(X;)) S Then, the direcled edge Xi) E E(Tr(Gg)). The properties of transitive reduction yields Tr(Gg) Tr Consequently, the transitive reduction of G may be used learn the directed edge Xi). (Note: F.G) denotes the edges of the DAG G)-
The proof of Theorem 3 is split into two parts. In the first part, show that the probability of learning the true graph is 1 7. In the second part, we show that the probability of having sufficient interventional data sets for inference is 1- no-T. Consider a directed edge (V, X,) in graph G. Assume that the number of the direct parents of X; above V is d, where d, I dmaz. Let (V) be the following event: X, $ S & {W (W) > (VV), W E Pa(X;)) S S. The probability of this event for one run of the inner loop in Algorithm with the assumption that tmaz >= 2 is given by:
PlE((V)) = dmax (1
(1
dmaa
(24)
dmna
dmaa
dmaz
dmnax 4
The last inequality holds for dmax >= because (1 - 1)== 2 0.25, Ve 2 2. Based on Lemma | the event (V) implics that the directed edge (V,X,) will be present Tr(G[S]). The outer loop runs for madmaz log(n) iterations and elements of the set are independently sampled. Hence, the probability of failure for all runs of the loop is given by:
tachnace log(n) I e ea 1 n
P((E((V)) <
(25)
4 dmax
For graph with total number of variables n, the total number of such bad events will be bounding the probability of bad events for every pair of variables, we have:
Union
n1 2
1
P[Failure] <
(26)
n
12-2
For inference, we need all interventional data sets S, S x such that Pa(X)) S & X; o S; for every treatment X, E X (Theorem ). Suppose p. is the number of parents of a treatment variable X,. We know that p. & dmar- Let us define the event & = Pa(X,) S S, & X, $ S.]. The probability of this event for one iteration of the inner loop given by:
P(&)] = dmaz
dmax
(27)
dmax
dmaa
dmax
dmax 4
The last inequality holds for dmaax >= of the set S are independently sampled. The probability of failure for all runs of the loop is given by:
2. The outer loop runs for 4admax log(n) iterations and the elements
P[g < (1 dmnax
Jaddmax log(n) S Log(n)
1
(28)
n1aa
lg is any total order that consistent with the partial order implied by the DAG, Lee, a(x) oY) ancestor of
Y.
19

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Published in Transactions on Machine Learning Research (10/2025)
Table 2: Two SCMs where Uy)m, ~ ~N and Ur)Ma ~ N1 M.M.)
M1
M2
1 I 0
1 1
0
X1 U1
X1 U1
X2 2X, U2
X2 X1 U2
EM,
2 0
EM3 =
7 0
Y X1 + X2 +Uy
Y X1 X2 + UUY
0
0
0
0
Table 3: Two SCMs where (Un, U2 Ur)Ms ~ (0(0,,MM))) and (U1, U2, Uy U.M NO M.M.)
M3
M4
1 0
1 32
0
X1 U1
X1 U1
X; 3X, + U2
X2 2X1 + U2
MM
0
M.
5
0
Y = 2X1 X2 Uy
Y = =2X1 + X2 +Uy
For graph with total number of variables n, we will have of these bad events. The overall probability of missing any one of required data sets for the inference scheme can be determined by union bounding the probability of all such bad events:
S
nPEEg
(29)
na-1
This shows that the probability of having all interventional data sets for inference after the algorithm runs is al least
A.5
Additional Examples Showing Non-ldentifiability of AOIs from Intervention on all Treatments and Observational Distribution in ANMs
In this section, we present additional examples of additive noise models that exhibit agreement on both the joint obscrvational and joint interventional distributions. However, these modols differ in their interventional dist rissttitoonn of subsets of treatment variables. Consider Table 2 which presents two SCMs labeled as Mi and M2.
The joint interventional distribution do(X; ø1, X₂ x2) is the same for both Mi and M2, i.c. W(x1 + 22, 1). The joint observational distribution for both models is a multivariate Gaussian i.e. (2,, Y)M, N(0, Σ) and X2, Y)M, ~ (O E). This is formally stated in equation 30 It is worth noting that due to the different structural equations for X2 in the models, the interventional distribution do(Xi) is different for the two models.
13
(X,, (..X2,Y)M,,un ~(0.E), ==
7
37
(30)
37
27
4 2
Table 3 presents another example of SCMs that exhibit agreement on both the joint observational and interventional distributions but differ in the interventional distribution do(X)). The joint interventional distribution do(X; ø1, x1.X2= = .x2) is the same for both, i.c. N((x, #2, 1). Similar to the previous example, the joint observational distribution for both models multivariate Gaussian:
20

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Published in Transactions on Machine Learning Research (10/2025)
T
11
M..M ~ ()), ==
15
22
(31)
22
34
These examples highlights the possibility of constructing multiple such examples of additive noise models (ANMs) with Gaussian noise, where they align on the joint observational and joint interventional distribu- tions but diverge on certain interventional distributions.
21


Reviews:
[{"review_id": "J38q1Zu705", "reviewer": "Reviewer_ScDV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The AOI formalism and its identifiability are solid and extend the prior work of Jeunen et al. (2022) to settings where causal relationships between the treatment variables $X$ are allowed ($X \\rightarrow X$ edges).\n- By using strongly separating set systems, the paper provides a randomized algorithm that reduces the number of interventions to $O(d_{\\max} \\log^3 n)$ with high probability.\n- Synthetic and semi-synthetic experiments show that the Structural Hamming Distance (SHD) and AOI error decrease as the number of samples or interventions increases, trending in the right direction.\n\nWeakness:\n- The main contributions, the identifiability results in Theorems 1 and 2, heavily rely on the Gaussianity of the noise. The key step in the identification procedure (page 14) depends on the linear properties of Gaussian conditioning, i.e., $\\mathbb{E}[U_Y \\mid X] = \\mathbb{E}[U_Y] + \\Sigma_{yx} \\Sigma_{xx}^{-1} V$, and without the joint Gaussian assumption, the derivations break down.\n- The non-identifiability example illustrated in the middle of page 5 does not rule out the non-Gaussian noise settings; instead, it seems that the non-identifiability stems from a violation of the additivity of the noise variable $U$.\n- A major concern is the way “cost” is defined solely as the number of interventions, regardless of the size of the intervention set (number of knocked out variables in each intervention). Is this a valid criterion, and how can it be justified in practical applications? For example, if simultaneous interventions on all treatments are infeasible, can the results be adapted?\n\nI defer my other concerns to the following section.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper studies confounded additive noise models (ANMs) with jointly non-isotropic Gaussian exogenous noise. This setting corresponds to a causal structure where treatment variables are confounded by hidden variables. The paper shows that all average outcomes under interventions (AOIs) are identifiable if and only if we have access to:\ni) observational data,\nii) the joint intervention $P[Y|do(X)]$, and\niii) for each treatment $X_i$, an interventional distribution $P[V - S_i | do(S_i)]$ where $X_i \\not \\in S_i$ and $S_i$ contains all causal parents of $X_i$, i.e., $Pa(X_i) \\subseteq S_i$.\n\nFinally, the authors propose a randomized design which, given a bound on the maximum degree, requires only a polylogarithmic number of interventions to both learn the observable graph (via a direct application of Kocaoglu et al., 2017) and compute all AOIs. The paper assumes Gaussian noise and that no treatment is a descendant of the target variable $Y$.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- The identifiability condition in Theorem 2 is existential, yet in practice parents are unknown. The algorithm argues random $S$ will cover each parent set while excluding $X_i$ often enough, however the probability calculations with a transpernt mapping from these events to the returned $S$-collection needed by Theorem 2 is not discussed in the main text (only in appendix). Please make this link more concrete in the main body. \n- Theorem 2 requires access to the full interventional distribution $P[Y \\mid do(X)]$ in addition to the $S_i$ interventions. It is not clear whether the randomized routine that samples $S$’s can also recover the full intervention cases. Even though this would add only one additional deterministic intervention, it is worth mentioning.\n- Starting from Definition 3, the paper becomes somewhat difficult to follow, both in the main body and in the appendix. I suggest that the authors include the preliminaries from Kocaoglu et al. (2017) and provide more detailed explanations. \n- The paper alternates between the terms “transitive closure,” “transitive reduction,” and “observable graph.” With latent confounding, it is unclear what exactly is being learned? Moreover, the conditional independences used by Algorithm 1 implicitly require a weak faithfulness-like assumption after interventions, but this is not explicitly stated. The authors should clarify both the target graph and the assumptions in the post-interventional graph.\n- Please include the details of the experiments, for example, the choice of estimators, as well as the source code for reproducibility purposes.", "broader_impact_concerns": "No comment.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "JABT5sDydF", "reviewer": "Reviewer_AFDw", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n- More General Setting: Unlike prior work that often assumes no causal relations among treatments or no latent confounding, this paper accommodates treatment–treatment dependencies and more general Gaussian confounding.\n- Identifiability Results: Theorem 2 establishes rigorous sufficient and necessary conditions for identifying all average outcomes under interventions (AOIs).\n- Algorithmic Contribution: The randomized algorithm (Theorem 3 and Algorithm 2) reduces the required number of interventions from linear to poly-logarithmic in the number of treatments.\n\n**Weaknesses**\n- Gaussian Assumption: The framework depends heavily on latent confounders following a multivariate Gaussian distribution, leaving its performance under non-Gaussian or heavy-tailed noise unexplored.\n- Additive Noise Restriction: The assumption of additive noise throughout may limit applicability to real-world systems with more complex functional dependencies.\n- Scalability in Practice: Although the number of experiments is poly-logarithmic, the size of each intervention set within an experiment is not clearly bounded.\n- Limited Empirical Scope: The empirical evaluation focuses on small synthetic graphs and a semi-synthetic healthcare network.\n- Motivation and Writing: The paper could more clearly articulate why identifying the causal effect of any subset of treatments on the outcome is practically important and in which applications. Additionally, some parts are difficult to follow.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors studied a setting with multiple treatment variables, a single outcome, and potentially latent confounders. They assumed that the underlying causal model follows an additive noise model (ANM), where the latent confounders are jointly Gaussian, and it is allowed for the possibility of causal dependencies among the treatment variables. The main goal is to identify the causal effect of any subset of treatments on the outcome, referred to as the Average Outcome under Intervention (AOI), by carefully designing a suitable set of interventions. In Theorem 2, they established necessary and sufficient conditions on the interventional distributions required to identify all AOIs. Building on this, Theorem 3 demonstrates that a randomized procedure (Algorithm 2) can achieve this identification with high probability using at most $O(d_{\\max} \\log^2 n)$ interventions, where $d_{\\max}$  is the maximum degree in the causal graph and $n$ is the number of treatment variables. Finally, the authors validated their theoretical contributions through experiments on both synthetic and semi-synthetic data.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- How robust are the results if the latent confounders do not follow a multivariate Gaussian distribution—for example, in the presence of heavy-tailed or other non-Gaussian noise?\n\n- To what extent does the additive noise assumption limit the applicability of the proposed framework in real-world systems where functional relationships may be more complex or non-additive?\n\n- Although the number of experiments is poly-logarithmic, is the size of each intervention set within an experiment bounded, and what implications does this have for scalability in practice?\n\n- Why is it practically important to identify the causal effect of any subset of treatments on the outcome, and in which concrete application domains would this be most relevant?\n\n- In the last paragraph of page 3, the definition of the average treatment effect (ATE) is given over two possibly different sets $W_1$\n and $W_2$. Could the authors elaborate further on this definition and clarify its interpretation?\n\n- In Section 4, the phrase “joint interventional distributions” seems misleading, as it appears the authors are actually referring to a set of interventional distributions. Could this terminology be clarified?\n\n- In the last sentence of page 4, the authors write: “We show that with additional interventional datasets, including interventions on the parent sets of the treatment variables, we can make all possible AOIs in confounded ANMs identifiable.”\nIt seems that the intended meaning is interventions only on observed variables, since parent sets of treatment variables may include latent confounders, which are not directly intervenable. Could the authors confirm and rephrase this statement?\n\n- When the notation $X_{int}$ is first introduced, please provide a clear definition.\n\n- In the proof of Theorem 1 (Appendix A.2), please specify how the term $\\Sigma_{yx} \\Sigma_{xx}^{-1} V$ is derived.\n\n- On page 14 of the proof, it is noted: “However, since we estimate the noise covariance matrix from data, if it is not invertible, we can apply Gaussian elimination to identify the dependent noise component(s).” Given that the proof is framed as an infinite-sample argument with access to the full distribution, why is it necessary to discuss finite-sample issues such as non-invertibility of the estimated covariance matrix?", "broader_impact_concerns": "Nothing specific to be mentioned.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "cgMiQM0R3t", "reviewer": "Reviewer_Nve7", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper presents interesting results, rigorously establishing necessary and sufficient conditions for the identifiability of AOIs in confounded ANMs, supported by comprehensive mathematical proofs that leverage do-calculus and Gaussian noise properties. Furthermore, if the proposed randomized algorithm, which requires only a poly-logarithmic number of interventions, withstands tests, the paper significantly reduces the practical cost of experimentation. The tests on synthetic and semi-synthetic data based on the HEALTHCARE Bayesian network are non-trivial as well. \n\nThe main methodological limitation I find is the assumptions for the theoretical analysis. The identifiability proofs and methods rely on multivariate Gaussian noise assumptions, which limit their applicability to cases where noise deviates significantly from Gaussianity, a common occurrence in real-world data. A minor drawback is the size of the data used. The paper discusses scalability theoretically; empirical tests are limited to relatively small graphs (up to 20 treatment variables). The method’s performance on high-dimensional or densely connected graphs could strengthen the arguments in the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a method to estimate the Average Outcome under Interventions (AOIs) in confounded Additive Noise Models (ANMs) where unobserved confounders follow a multivariate Gaussian distribution, requiring only a poly-logarithmic number of interventions to identify all causal effects even when the causal graph is unknown. This study addresses causal inference in confounded ANMs by developing a new algorithm to estimate AOIs for any subset of treatment variables, assuming multivariate Gaussian noise. The paper establishes necessary and sufficient conditions for the identifiability of AOIs, showing that the combination of observational data and targeted interventional distributions is enough to recover all AOIs. Specifically, they prove that interventions on each treatment’s parent set, along with a joint intervention on all treatments, enable full identification of causal effects. To reduce the intervention burden, they propose a randomized algorithm that selects intervention targets using a strongly separating set system and transitive closure of the causal graph. This approach learns both the observable graph structure and identifies sufficient interventional datasets for inference, requiring only $O(d_{max} \\log^3n)$ interventions, where $d_{max}$ is the graph’s maximum degree and $n$ is the number of treatment variables. Empirical evaluations on synthetic data and a semi-synthetic healthcare Bayesian network validate the approach, showing accurate causal graph recovery and AOI estimation despite sample size limitations or deviations from Gaussian assumptions. Detailed proofs based on do-calculus establish the theoretical soundness of their identifiability conditions and algorithmic efficiency.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Can the method be evaluated on additional datasets from domains such as healthcare, economics, or genomics to assess its practical utility and limitations?\n\nCould you indicate how can the framework be extended or adapted to handle non-Gaussian noise distributions while retaining identifiability guarantees?\n\nWhat algorithmic optimizations or parallelization strategies can improve scalability to large, high-dimensional, or dense causal graphs?", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "JrQKoWzlrh", "reviewer": "Reviewer_PHE7", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "While the proposed problem is potentially interesting, I found the writing of the paper of low quality and the problem setting restrictive, which is not clearly articulated in the abstract or the introduction. The paper also contain inaccurate claims. The randomized algorithm is not clearly explained, hence i could not verify the validity of the approach under the presence of unmeasured confounders.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors study the identification of the expected outcome under a joint intervention in additive noise models with potential confounders. The authors provide the identification criteria under the assumption that the potential confounders are additive and follows a Gaussian distribution, which implies that the treatment or the outcome (or both) are linearly dependent on the unmeasured confounders. The authors additionally propose a randomized algorithm to identify the target parameter through a sequence of interventions where they aim to simultaneously learn the transitive closure of the causal graph.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The first three sentences of the abstract is very misleading. As the majority of the study on additive noise models in causal discovery focuses in the observational setting, the authors needs to make clear that they are using interventional dataset in the first three sentences. Saying that the unobserved confoudner follows a multivariate Gaussian is not an accurate description of the assumption. You need to highlight that it is additive. Your model in fact implies that the unmeasured confoudners follow linear Gaussian, which is a key distinction from observational settings, as it is known that this will not lead to a unique identificable causal graphs when using observational data alone.\n\n2. In the abstract, you mentioned that “these interventions are sufficient to recover the causal structure between the observed variables.” However, in your randomized algorithm section, it seems that you are discovery the transitive closure of the causal graph instead. Which is not an accurate description.\n\n3. The logic of the introduction is flawed: toward the second half of the first paragraph, the authors start by saying that RCTs are not feasible in real-world that’s why observational studies are useful. The authors then focus again on the interventional case, which contradicts in the logic earlier in the paper. The authors need to rewrite their introduction to focus on the interventional case, and clearly articulate the problem setting from the first paragraph. \n\n4. The second and third sentence of the second paragraph is inaccurate: for example, when assuming ANM is linear, while the true DGP follows a nonlinear distribution will also yield nonindependent noises in the observational settings. When making these claims, the authors need to clearly articulate the problem setting. \n\n5. The related work needs to be streamlined to categorize them into observational and interventional settings, and a direction contrast between the identifibility criteria is needed in this section to further clarify the setting of this paper. The current logic of this section again mixes the observational and internvetional cases without clear logic flow.\n\n6. The sentence “Our work is focused on more general ANMs without any additional restrictions on the structural equations” is not true. The authors make strong assumptions on the structure equation of the unmeasured confounders, which is the focus of this paper. \n\n7. It is useful for the authors to discuss in section 3 expicitly what their DGP on the unmeasured confounders assumes. For example, if the parents follow a nonlinear generating distribution, then this would entail that the parents of X_i must be all observed in this model. The authors currently present the structural equations as a less restrictive one. However, it makes strong assumptions of the allowable DGPs.\n\n8. I found Theorem 1 poorly worded. “Is identifiable” in the sentence could be describing both W\\in X and E[Y|do(X)]. Additionally, the conditions for identifiability is quite restrictive in the presense of the limited access to interventional data in the introduction, as it requires specific interventions on particular varaibles, which might not be feasible in practice.\n\n9. In section 4.2, the authors start the paragraph with a “deterministic approach”. Now, for the randomized approach, are the authors trying to identify one potential outcome (one W) (as in constrast with the prior section which requires the identification under all W)? This point needs to be made clear as in the hypothesis testing literature, the distinction between the deterministic approach and randomized approach lies in the fact that the tests are now allowed to be performed sequentially to better identify the unknown true hypothesis. It is unclear whether learning the transitive closer of the graph factors in the improved complexity of the algorithm. \n\n10. The transitive closer of the causal graph is known to be not as useful in inference. Additionally, it is unclear whether the graph is even well-specified under the presence of unmeasured confounding. The authors need to clarify this point.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)