Learning Joint Interventional Effects from Single-Variable Interventions in Additive Models
Authors: Armin Kekić
Sergio Hernan Garrido Mejia
Bernhard Schölkopf
Venue: ICML 2025 poster (2025)
Keywords: Causality
Treatment Effect
Causal Representation Learning
Primary area: general_machine_learning->causality
TL;DR: We show that joint interventional effects are identifiable under confounding from observational and single-variable interventions when the contributions of each action to the outcome are nonlinear but additive.

Abstract: Estimating causal effects of joint interventions on multiple variables is crucial in many domains, but obtaining data from such simultaneous interventions can be challenging. Our study explores how to learn joint interventional effects using only observational data and single-variable interventions. We present an identifiability result for this problem, showing that for a class of nonlinear additive outcome mechanisms, joint effects can be inferred without access to joint interventional data. We propose a practical estimator that decomposes the causal effect into confounded and unconfounded contributions for each intervention variable. Experiments on synthetic data demonstrate that our method achieves performance comparable to models trained directly on joint interventional data, outperforming a purely observational estimator.

Learning Joint Interventional Effects from Varibbe Interventions in Additive Models
Armin Kekić! Sergio Hernan Garrido Mejia Bernhard Schölkopf 33
Abstract
of marketing interventions would be prohibitively expensive and time-consuming, the number of required test con- ditions grows exponentially with each additional channel. This raises a crucial question: Can we learn about joint effects from simpler experiments?
Estimating causal effects of joint interventions on multiple variables is crucial in many domains, but obtaining data from such simultaneous interven- tions can be challenging. Our study explores how to learn joint interventional effects using only ob- servational data and single-variable interventions. We present an identifiability result for this prob- lem, showing that for a class of nonlinear additive outcome mechanisms, joint effects can be inferred without access to joint interventional data. We propose practical estimator that decomposes the causal effect into confounded and unconfounded contributions for each intervention variable. Ex- periments on synthetic data demonstrate that our method achieves performance comparable to mod- els trained directly on joint interventional data, outperforming a purely observational estimator.
We investigate whether joint interventional effects can be estimated using only observational data and experiments where intervene instance of the Intervention Generalization Problem (Bravo- Hermsdorff et al., 2023): predicting treatment effects in previously unseen interventional settings. Causal models encode additional structural relationships between variables that allow us generalize to settings which non-causal machine learning approaches assuming independent, identi- cally distributed (i.i.d)) data fail.
variable time. This is an
However, this problem is not solvable in its most general form. To achieve Intervention Generalization, we need to restrict the causal model class (Saengkyongam & Silva, 2020). The key question becomes: What model class restrictions enable this generalization while preserving broad applicability?
1. Introduction
Understanding the effects interventions is fundamental across many domains, from designing public health policies optimizing business operations and administering medical treatments. Particularly challenging and important joint interventional effects, where simultancous interventions multiple action variables influence a target outcome. Such scenarios are common in fields like epidemiology (Kekić et al., 2023a), e-commerce (Kunz et al., 2023; Schultz et al., 2023), and medicine (Prosperi et al., 2020).
In this study, we focus on causal models where each action contributes to the outcome variable in a nonlinear way and is subject to confounding. We show that when these com- plex individual effects combine additively to produce the outcome, the joint interventional effect is identifiable from observational and single-intervention data. This additivity assumption well-motivated across several domains-for instance. in pharmacology where many drug combinations exhibit approximately additive effects in the absence of spe- cific interaction mechanisms (Pearson et al., 2023), and in marketing analytics where additive models are widely used to understand how multiple advertising channels influence consumer behavior (Jin et al., 2017; Chan & Perry, 2017).
Consider company trying optimize their marketing strat- egy across multiple channels like social media, email cam- paigns, and display advertising. While the ultimate goal is to understand how these channels work together to drive sales, running experiments for every possible combination
Our main contributions are:
Max Planck Institute for Intelligent Systems, Tübingen, Ger- many "Amazon Research, Tübingen, Germany TTübingen AI Cen- ter, Tübingen, Germany *ELLIS Institute, Tübingen, Germany. Correspondence to: Armin Kekić <armin. @tw..mpp..e>>.
1. An identifiability result showing that joint interven- tional effects can be recovered from single-variable interventions and observational data in causal models with nonlinear additive outcome mechanism
Proceedings of the nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
practical estimator that decomposes the causal effect

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Learning Joint Interventional Effects from Single-Variable Variable Interventions in Additive Models
Y
Y
Y
A
Ak
A
Ak
A
A2
Ak
A
A2
Ak
C1
2
CK
C
Ck
C₁
C2
Ck
C1
C2
CK
p.M
p.((d((A())
p.M(do(Ak)
(M(do(A1,.
observational
single interventions
joint interventions
training
prediction
Figure 1. The Intervention Generalization Problem. The figure shows the different interventional regimes. Our goal is to estimate the joint interventional effect of the action variables ; Y, that is, E[Y do(ai, ... ax), (right). However, during training only have access to observational (left) and single-interventional data (middle). There are unobserved confounders {Ci..., , Ck] between the actions and the outcome variable Y. box around variable indicates that it was intervened on.
into confounded and unconfounded contributions for each intervention variable
Interventions in SCMs. Interventions on endogenous variables in SCMs are represented by the do(-) and modify how variables are assigned their values. perfect stochastic intervention do(X;) removes all incom- ing edges the intervened variable and replaces its struc- tural assignment with draw from intervention distribu- tion over X,. That is, we replace the structural assignment X == fx (Pag, U&) by X2 = Ün, where Us is a univari- ate distribution independent of the parents Pak. We use lowercase letters to denote specific realizations of these in- terventions, writing do(x;) to indicate that X; was set to the particular value 24. Such interventions transform the original model M into interventional model M(do that induces modified distribution over the endogenous variables, which we denote with superscript p.M(do(X.)), When no superscript is present, the distribution comes from the unintervened (observational) model M. In conditioning sets, we write {ø1.... dz; ... 2n) } to indicate that X; was intervened on while other variables have observational realizations.
3. Empirical validation demonstrating that our method achieves performance comparable to models trained directly on joint interventional data
2. Problem Statement
2.1. Background and Notation
We use boldface for vector-valued or sets of random vari- ables. We denote random variables with capital letters and realizations thereof in lowercase.
Definition 1 (SCM (Pearl, 2009; Peters et al., 2017)). An N -i-iimensional structural causal model is triplet M (G. S, Pu) consisting of:
joint distribution Pu over the jointly independent "exogenous" variables U {U1,... ... }N}.
directed acyclic graph G with N vertices,
2.2. Setting
set S (X, = (Paj. Uy).j 1, N) of struc- tural assignments, where each fy is a scalar-valued function and Pa, are the variables indexed by the set of parents of node j in G.
Let A {Ai,.., Ak} be set of trealment or action variables, C = (C,,,... Ck] be set of unobserved con- founders, and Y be outcome variable. The actions direct causes of the outcome, and allow for an arbitrary acyclic causal structure among the actions. For notational simplicity, we assume the actions are in topological order
such that for every u. the system {z, = (pa,, 4n)} has a unique solution. The SCM thus entails a joint dis- tribution over the "endogenous" random variables (X1,.., Nn).
2

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Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
and write the causal structure as a fully connected DAG.
in the nouparametric setting, where only knowledge about the causal structure and probability distributions in different interventional settings are given. Such methods are agnostic to the functional form of the causal mechanisms and the distribution of the exogenous noise variables. The founda- tional work in this area focuses on identifying interventional (3) effects from purely observational data (Tian & Pearl, 2002; Shpitser & Pearl, 2006). The multi-treatment setting, where effects of multiple simultaneous treatments must be esti- mated from purely observational data, presents additional challenges (Miao et al., 2023; Zheng et al., 2025). This has been generalized through g g-identification theory (Lee et al., 2019), which determines whether target interventional effects can be recovered from a given combination of ob- servational and experimental data. However, as we show in Section 4, nonparametric approaches are insufficient for our setting. This motivates our focus on parametric assumptions to identify the joint interventional effect (6).
We can write the structural assignments as follows:
Y= = S(A,, ; Ak, CK,,,CKKUU
(I)
A&- = (c(A1, .., A,- Ck.,, Ck-1, Vie)
(2)
Cx= = W
for kE E [L,,,,K] K}
where (U,V,,.. ... VR, Wi..., } are mutually indepen- dent exogenous noise variables.
We are given a dataset of observational i.i.d. samples
Doba ~ (Y,A)
(4)
and K datasets of i..dd samples for single-variable interven- tions on each action variable
(D ( int
(Y:A) 1k=1
(5)
Bravo-Hermsdorff et al. (2023) present a factor graph approach for the Intervention Generalization Problem, investigating identifiability of joint interventional effects given specific factorizations of observational and inter- ventional probability distributions. Similarly, Jung al. (2023) present graphical conditions for nonparametric identification of joint interventional effects -which they term Multiple Treatment Interactions- and employ double machine learning techniques for estimation from marginal interventional data. Related approaches study causal effects in novel action-context context pairs when actions and contexts are categorical variables (Ribot et al., 2024) or and treatment effect estimation for unobserved subgroups in causal mixture models (Mazaheri et al., 2024).
Here, each interventional dataset D consists of samples generated under perfect stochastic interventions on As.
Our objective is to estimate the joint interventional effect
E [Y| do(ai,...,. .x)]]
(6)
Identifiability. For given class of causal models, we say that causal effect is identifiable from a set of proba- bility distributions if it can be uniquely determined from these distributions. In our setting, we say that the joint interventional effect (6) is identifiable from observational and single-intervention distributions if it can be uniquely determined from (Y.A) and (P>Y.do(Aa)) JR=1- Note that this is equivalent to saying that the joint interventional ef- fect (6) is identifiable from infinite samples from P.M
The most closely related prior work is that of Saengky- ongam & Silva (2020), who employ parametric assumptions on the exogenous noise variables. They show that gener- alization from single-intervention and observational data to the joint interventional effect is possible for continuous variables when the exogenous noise is Gaussian and addi- tive. In contrast, we consider parametric assumptions on the causal mechanism (1) that relates the action variables to the outcome variable, assuming that it is additive. While this restricts the confounding between the actions, this allows us to treat any distributions for the exogenous noise as well as both discrete and continuous variables- see Appendix A.
(Y.A)
and (P:M(do(Aa))) IK=1. (Y,A) 1 i.e. when the datasets Dobs and {D+
int J are infinitely large.
Conversely, the joint interventional effect is not identifiable from longle-interventiona and observational distributions within a given model class if there exist two distinct causal models that entail the same single-ineerenttioall and ob- servational distributions but different joint-interventinaal distributions.
Additive models have a long history in statistics and ma- chine learning. They were first introduced for regression (Friedman & Stuetzle, 1981) and later extended to General- ized Additive Models (GAMs) (Breiman & Friedman, 1985; Hastie & Tibshirani, 1990), where the additive contributions from each input variable are transformed through a nonlin- ear link function. More recently, Neural Additive Models have combined the interpretability benefits of GAMs with
3. Related Work
Understanding causal effects from data with limited exper- imental control is a fundamental challenge in causal infer- ence. There is a rich literature that studies such problems
'This means that each action variable A can potentially de- pend all preceding actions {A,,.., A and confounders (C1..., Ck-1).
3

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Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
the flexibility of neural networks (Agarwal al., 2021). The concept of additivity has also enabled novel forms of generalization-Lachhee
where V2, W Bernoulll(()) with 0<p<1. These two models induce the same observational distribution over the observed variables; that is, PY, A1,AA) A2)
et al. (2023) showed that non- linear additive ground-truth decoders allow generalization to unseen combinations of latent factors, which they termed Cartesian- Curtesian-Product Extrapolation. While additive noise as- sumptions are common in causal inference (Hoyer et al., ( (Y, A2). However, they induce different distributions when A and A2 are jointly intervened:
(1, They also lead to the same single-variable interventional distributions; P.M(do(A:)) A1, A2)
p.M(do(Aa)) (Y, A1, A2) and p.M(do(A2)) (Y, A1, A2) =
2008), this is, the best of our knowledge, the first work that investigates additivity of causal mechanisms with re- spect to the parent variables.
p.M (Y=1) =
The complementary problem of generalizing from joint inter- ventions to single intervention effects was studied by Jeunen et al. (2022) and Elahi et al. (2024), and generalization to unseen interventions without identifiability was explored for stationary diffusion models (Lorch et al., 2024).
* x = - (Y=-))
(7)
Hence, two estimators trained on observational and single- interventional data from .M and M would arrive at identical predictions for the joint interventional effect (6), despite the true effects being different in these two models.
Intervention Generalization is akin to other types of gener- alization, which is aided by the structure encoded in causal models such as the Causal Marginal Problem (Gresele et al., 2022; Garrido Mejia et al., 2022; 2024). There, instead of generalizing to a new interventional regime, the goal is to learn about the joint behavior and causal structure of vari- ables that have only been observed in subsets, but never jointly. Another example is Out-of- Variable Generaliza- tion (Guo et al., 2024), where some variables have never been observed in training.
5. Assumptions
In the previous section, we demonstrated that joint inter- ventional effects cannot be identified from single-variable interventions in general. However, by imposing certain restrictions on the causal model class, we can achieve iden- tifiability. In this section, we introduce two key assumptions that together enable the identification of joint interventional effeets from single-variable interventions.
A key aspect of our method is the ability of causal mod- els to combine information from different data sets. This aligns with Causal Representation Learning, where many approaches use datasets or observation pairs that differ by interventions in the latent variables (Liang et al., 2023; Brehmer et al., 2022; Zhang et al., 2023; von Kügelgen et al., 2023; Buchholz et al., 2023), samples from scientific simulations (Kekić et al., 2023a;b), and multiple views and modalities (Yao et al., 2024).
Assumption 1 (Intervention Support). The distributions of the action variables have identical support across all inter- ventional regimes. That is,
supPp (A) = supp puppp.ace MadolA (A)
(A)
=suppp (A) for any k E {1, K}.
(8)
Assumption 2 (Additive Outcome Mechanism). There pair-wise confounding between the actions and the outcome. The outcome is generated by an additive combination of sep- arate nonlinear functions for each action and its associated confounder. The structural assignments can be written as:
4. General Non-Identifiabiiity
In general, the setting described in Section 2.2 is not iden- tifiable. We can show that two distinct SCMs can induce identical observational distributions and single-variable in- terventional distributions, but exhibit different behaviors under joint interventions.
Y Σ fa(As, C2)+U +U
(9)
A (,,, - Ak-1, Ck, V%)
(10)
Cs Wn
(11)
Example 1. Consider the following two SCMs over binary variables:
for l E {1, [1,...,K K}
where {U, V1. VK. W1,... WK) are mutually indepen- dent exogenous noise variables.
M:
M:
"These probability distributions are shown in Tables to 3 in Appendix D.
Y: = ACAUU
Y = AMACAU
A2 = AAACVV2
Ag = AACAVA
"We denote the support of random variable A under the proba- bility distribution P(A) as suppp.t, (A). Where M is the corre- sponding causal model; in this case, the SCM for the observational regime.
A = C
A C
C==W
C==
4

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Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
Assumption is needed for technical reasons in the identi- fiability proof. In order identify the joint effect (6), we need to transfer information from the other interventional regimes (4) and (5) to the joint case (6). Assumption I en- sures that the actions cover the same range for the inputs of the outcome mechanism (1), Such assumptions on the support of variables are common in multi-dataset settings in Causal Representation Learning (Varici al., 2024) and can be seen as version of the positivity assumption in causal effect estimation (Hernan & Robins, 2010, Chap-
B|y "1,-,, ag)
Σ Ec C+) (14)
These decompositions correspond to the terms fx in the additive outcome mechanism (9). In each term, the expectation over the confounder Cs is taken with respect to a measure that depends on whether the corresponding action variable Aş was intervened on. When As is not intervened on, the confounding can introduce an additional dependence the other action variables, entangling their influences.
ter 3.3). In practice, identical intervention support can be ensured through choosing the appropriate experimental set- tings. When this is not possible, support mismatches can be mitigated by exploiting domain knowledge about the function classes in the structural assignments (Kekić al., 2023a); in our case H
However, these decompositions still allow to learn a repre- sentation that enables us to generalize from the observational and single-interventional setting to the joint-interventional effect. We define K estimator functions
Assumption 2 restricts the causal model class regarding how the actions influence the outcome. While the effects of each action A& on the outcome Y can be nonlinear and complex, we limit how the actions and confounders interact in the outcome mechanism (1). We assume that both the contributions of each action-confounder pair (Ag, C+) and the exogenous noise U are additive. Note that for the other structural assignments do not introduce such constraints for the exogenous noise or functional form. The resulting causal structure is illustrated in Figure (left).
(K,, Lu ,.... ,KK, R&),
k E (I,...K} (15)
where Rs E 10 indicates an intervention on A₃- Each Rg can be thought of as selecting one of two functions
{ ( (a), ... ag), if R2=0 fi (a1..... ox), if R =1
fala...,, ak. R&)
(16)
where fint represent the terms in the decompositions (12)- (14) where the corresponding action A is intervened on, and fobs are the factors with A observational. We then define an overall estimator
6. Theoretical Results
K
6.1. Identifiability of Joint Interventional Effect
flai..., ax, RI. Rk) Σ Se(o,..... .x Ra))
In this section, we show that in the causal model class with nonlinear additive outcome mechanism, outlined in Sec- tion 5, we can achieve Intervention Generalization.
k=1
(17)
to represent all regimes, depending on the setting of the indicator variables Ri,..., Rg:
Theorem 1 (Identifiability). Under the assumptions in Sec- tion 5, the joint interventional effect (6) is identifiable from single-variable interventions and observational data in the infinite data regime.
When R R1 ..., = the function f is an estimator for the joint interventional regime.
R1=0,... R .... Rx=1 corresponds to the single- intervention setting of M
Proof Sketch We first note that we can decompose the joint interventional effect (6), which we want to estimate, as well as the conditional expectations, for which we have data, as
-i=0,....K--- is the observational setting.
In the outcome mechanism (9), each term fa depends only on one action Ag. 4 In contrast, each model factor fx has to take all actions into account due to the entanglement introduced through confounding.
Y| do(aa,..., ... ak)]
Ba Box [fa C+)
(12)
Now if we fit the estimator f in the observational and the single-interventionnal regimes, that is,
E| an ..... do(a,),...,aa'
Ec Cj)l
Mai. ak -1=0,... ..... Rx = E|Y [a,,... ... [x]
+ +I Ec, B Ck)
(18)
for j E (1,..., K}
(13) ff also depends on the confounder Ck.
5

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Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
,., Rx 0k= [a in OK, R1=0,... .., Ry=,,.,,
Corollary 1.
Let B be a partition the index set (1,-. K} such that the ground-truth SCM M has an outcome mech- anism which is additive with respect to B. Then the joint interventional effect (6) can be identified from observational
=E|Y| @1, do(a)), ... ak, axl, for j {1,..,, K}
(19)
we can show that the estimator also identifies the joint inter- ventional effect:
data
Dobs P.M
(25)
(Y.A)
f(a1, aK, aK, R1=1,.., Rx= =| do(ai; , ..,,,aa) ]]
(20)
and [B] interventional datasets
The full proof is shown in Appendix A.
{Dim int
((YA)) (Y,A)
(MMido(An))) BEB
BEB.
(26)
Note that, while our approach assumes that the action vari- ables are direct causes of the outcome and that there is no confounding between the actions, we do not assume a par- ticular causal structure between the actions. The approach present here is agnostic to causal relationships among the actions and does not use this information infer the joint interventional effect (6).
The proof is given in Appendix C.
Hence, we can trade off assumptions of additivity on the outcome mechanism for joint experimentation on actions in two complementary ways. First, if it is unclear whether the outcome mechanism decomposes per variable within some subsets of actions, can choose not to make the additivity assumption on these subsets and instead collect joint interventional data on them. Second, our framework al- lows for shared confounding among action variables within the same partition subset, provided we collect joint inter- ventional data on that subset. In other words, while our base assumption requires pair-wise confounding (each con- founder Cs affects only action Ag and the outcome), Corol- lary permits more complex confounding structures within partition subsets-for instance, a single confounder could affect multiple actions within the same subset-as long as we have joint interventional data for that subset. This flex- ibility makes our method applicable to a broader range of real-world scenarios where strict additivity simple con- founding structures may not hold across all variables, while still maintaining identifiability guarantees through strategic experimental design.
6.2. Mixed Interventional Effects
Moreover, can extend our results to any combination of intervened and observational actions:
Proposition (Identifiability of Mixed Interventional Ef- fects). Let Aint U Aobs = (AI...,AA,, be a partition of the action variables into intervened and observational actions. Under the assumptions in Section 5, the effect
E|y (aoaaat): aubs]
(21)
is identifiable from single-variable interventions and obser- vational data in the infinite data regime. The proof is given in Appendix B.
6.3. Additivity with Respect to Partition
Theorem implies that for an additive outcome mecha- nism (1), the number of interventional datasets required for identification the joint effect (6) grows only linearly with the number of actions. We can show that even when (I) is only additive with respect to the effect of subsets of actions, identification is possible as long as we have joint interventional data on each subset.
7. Experiments
7.1. Setting
Synthetic Data-Generating Process. We sample a struc- tural causal model with five actions and confounders and causal relationships as shown in Figure 3. The structural as- signments are second order polynomials with randomly sam- pled coefficients. The exogenous noises are Gaussian, uni- form or logistic. The corresponding parameters are drawn at random before each experiment run. The dependencies between actions are probabilistic, with each potential edge (22) having probability 0.3 of being active. We sample 100 SCMs, where for each run sample Note- Nusut and Njiut (23) data points for the observational, the single-interventiona. and joint-nnterventional datasets. We split each dataset into 80% training- and 20% test data. For evaluation, all models are tested on the joint interventional test dataset. Further de- tails about the experimental setup are given in Appendix E.
Definition 2 (Additivity w.r.t. a Partition). Let 2 be a par- tition of the index set {1, ..., K}. For an SCM of the form discussed in Section 2.2, we call the outcome mechanism (1) additive with respect B, if we can write the structural assignments as:
Y Σ JB(AB CB) UU BEB
A = ga (A Ar-1. CB()) Vv)
C == Wn
(24)
for k E {I,..., K}.
where Bkk) E B the partition that contains index k.
6

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Learning Joint Interventional Effects from Single-Variable Variable Interventions in Additive Models
Training Regime
method
joint int. data
only obs. data
0.40
0.40
Average RMSE
(a)
Average RMSE
(b)
Average RMSE
1.5
(c)
0.35
0.35
1.0
0.5
0.30
0.30
method
pooled data
100
10
10*
10a
10c
107
joint intervention data
only observational data
obss
Ntot. Njnn
Figure 2. Experiments Synthetic Data. Average root mean squared (RMSE) for predicting the joint interventional effect E|Y do(ai,..., as)], averaged over 100 experiment runs. Each run uses a randomly generated ground truth SCM. We compare three approaches: (i) Our Intervention Generalization method, training the estimator (17) observational and single-intervention data (Section 6). (ii) An estimator trained directly on joint interventional data (topline). (iii) A naive estimator trained on pooled dataset of all observational
single-inted interenntereenionaall data. (iv) An estimator trained solely on observational data. The error bars show the standard error of the mean. (b) Prediction error of the joint interventional effect (6) under varying ratios of observational and single ingleeinterventional data for a fixed number total data points. (c) Prediction error (6) with increasing total number of data points. The full experimental details are given in Appendix E.
real-world applications (Kunz al., 2023).
The full experimental details and additional results are given in Appendix E.
A
A2
A3
A4
As
7.2. Results
C
C2
C3
C4
C5
The mean root mean squared error (RMSE) over all sam- pled SCMs in each of the four estimation methods shown in Figure 2(a). We observe that our method achieves an error that is close to the minimally achievable error of the topline model that is directly trained joint interventional data. Both our approach and the topline benchmark signifi- cantly outperform the naive observational-only model and the estimator trained on pooled data. The results empirically validate the effectiveness of our Intervention Generalization technique leveraging single-interventie data to predict -nterrenteeeenn effects.
Figure 3. Causal Graph in Synthetic Experiments. Dashed edges between actions represent probabilistic dependencies that may or may not exist in each sampled SCM.
Models and Benchmarks. We train third order polyno- mial estimator functions (17) as outlined Section We compare that model to three baselines.
(i) model that is directly trained on joint interventional data. That is, we directly fit E| do(aj aa)]] This comparison represents the minimal error that our method could achieve.
A practically relevant question is: How much single inter- ventional data is necessary to obtain a good estimate of the joint effect (6)? Typically, interventional data is much harder to obtain than observations the unintervened system. We empirically test the behavior of the Intervention Generaliza- tion approach with varying shares of single-inters iangle-interventionall and observational data, shown in Figure 2(b). That is, the total number of data points Noo: Nobee + KN is fixed, and the ratio Naint/ /Nob is changed. The precise value of
ii A naive estimator directly trained on the pooled single- interventional and observational data.
(iii) A model that only considers the observational data. That is, we use model fit to E|Y a..... az] to predict E[Y do(a,.... as)]. This a typical approx- imation made the absence of interventional data in
'The additional error incurred through making this simplifying assumption quantifies Cousal (Vankadara et al., 2022).

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Learning Joint Interventional Effects from Single-Variable Variable Interventions in Additive Models
the optimal ratio depends the overall number of data points and the number of actions in the SCM. We observe two trends: (i) The optimal ratio tends to be between 0.1 and 0.9. Thus, having more observational data is favorable. This aligns well with real-world scenarios where observa- tional data is typically more abundant and easier to collect. (ii) The error curve flattens as Noot increases. Additional experimental runs for more settings are shown in Figure B in Appendix E.1.
One avenue for generalizing the results of this work is the function class of the outcome mechanism (1). At the most general end of the spectrum is the additive structure used in the Kolmogorov-Amold representation theorem (Kol- mogorov, 1957; 1956), which can represent any continuous function from a compact set in IN to IR. However, with such broad representational power, we are unlikely to main- tain identifiability. Moving toward more restricted classes, Generalized Additive Functions (Hastie & Tibshirani, 1990) and Post-Nonlinear Models (Zhang & Hyvärinen, 2009) represent promising intermediate points that could balance expressiveness with structural constraints. The latter have al- ready demonstrated utility in causal structure learning. Such intermediate function classes could extend our approach to scenarios where strict additivity may not hold.
The theoretical results in Section 6 establish that the joint interventional effect (6) can be identified from single- interventional and observational data. However, from those results we cannot infer the sample efficiency of the estima- tion procedure in the proof of Theorem Figure 2(c) shows the error as function of the number of total data points Noot- We compare this against estimator that is directly trained on dataset of joint interventions of size Njint- We find that the Intervention Generalization approach takes over an order of magnitude more data to reach similar accuracy as the model directly trained on joint interventions. Hence, the benefit of requiring only single interventions comes at the cost of decreased sample efficiency and has be weighed against the difficulty of obtaining joint interventional data. Figure A in Appendix E. shows the convergence for SCMs with varying numbers of actions K.
The precise confounding structure can be difficult to assess and often there are additional covariates to account for. An- other area for investigation is thus the adaptation of our estimation technique to more complex scenarios. These in- clude settings with additional non- non-intervened covariates or under more general confounding structures between action variables.
While identifiability guarantees that the joint effect can be estimated without joint interventional data, we have ob- served that this estimate can be inefficient in terms of sample complexity compared to directly training on joint interven- tions. Therefore, another direction of future work could be to investigate finite sample guarantees of Intervention Generalization.
8. Discussion and Outlook
We have shown that by constraining the outcome mech- anism (1) to additive model class, we can success- fully identify joint interventional effects from single- interventional and observational data. Our constructive identifiability proof provides a practical estimator for the joint interventional effect (6). The estimator function de- composes into terms for the confounded and unconfounded contribution of each action to the outcome.
A Acknowledgements
We would like to thank Simon Buchholz, Amartya Sanyal and Frederik Träuble for their valuable contributions through interesting and helpful discussions about our ap- proach.
While additivity offers mathematical tractability, careful consideration is needed before applying this assumption. In ecological systems, for instance, Brook et al. (2008) showed that environmental threats often interact synergis- tically rather than additively, where the combined impact of multiple stressors exceeded the sum of their individual effects, leading to systematic underestimation of extinction risks when additive models were used.
Software and Data
The code used for the experiments is available at sethau.oonkkkkkiieeeeeeiinnnnnnn
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Our identifiability result based on additivity constraints points to broader theoretical question: What is the mini- mal set of assumptions required for identifiability, and can we find more general causal model classes that still per- mit identification while relaxing our current constraints? Identifying such classes would expand the practical appli- cability of our approach while maintaining its theoretical guarantees.
8

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Learning Joint Interventional Effects from Single-Variable Variable Interventions in Additive Models
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Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
A. Proof of Theorem 1
A.1. A Note on Variable Types
For notational simplicity, we present all proofs using continuous variables and the Lebesgue measure. However, our results hold more generally for discrete variables or a mix of discrete and continuous variables. The proofs can be adapted by replacing the Lebesgue measure with appropriate measure for each variable type-the counting measure for discrete variables and the Lebesgue measure for continuous variables. All integrals would then be interpreted with respect to these measures, effectively becoming sums for discrete variables.
The remaining ussumptions and proof steps carry through with these modifications, as the key properties we rely on- on-such as the law of total probability and the independence relations in the causal graph-hold under general probability measures.
A.2. Lemmas
Before we prove the main proposition, we introduce two useful lemmas.
Lemma 1. Let M be an SCM as defined above. Then,
p Midol(,)) (C& (l,...,oo((.)).
= p(ck @1, :j. ak),
(27)
for all k E (2,-,, K), and (1,.., 1)
Proof (Lemma 1) Using Bayes" rule on the left-hand side of Equation (27) we have,
Midoles) an,-, do(a)),..
(28)
open path between Cs and A1 da (A,),
p Mido(a;)) (ck a,,-., do(ag), 0K ak-11)
M(do(as)) (ax oi, 1k-1, M (ax @l, do((a)))))))))) a (1--1)
(29)
causal mechanism
root made
p Mido(a;))
p(ak
0ly." O...., ag-1, (k) Midola,))) ai,... do(a)).... (1k-1) p(cx)
(30)
= =(akk aj.".,a..... ag ak-1, M
(31)
Midiola,))aa 11,,,,,,, ak-1)
now focus on the denominator,
M(do(as)) (ax "i,,..,
(32)
M(do(a,)) (ax: Ck ai.... ai...... do(a,), ..., ak-1) des
(33)
I Midooss) (ax ai, /o,,..,o(aa)) ... ak-,, ox) p ((dolss)) al, do(a),, ... ax- de:
(34)
/ plax "i.", a1 Midola,)) (ex) der
(35)
= / plax ai..., aj. a - p(c))(())))))) des
(36)
/ p(ak Oi.... aj. a p(ck Q1, aj. ak a1-1) dok
(37)
p(ak @o...., aj. ak-1)
(38)
Using Equation (38) on Equation (31), and using Bayes' rule we obtain,
p(ck (1, , Oj: ak ak-1) 1)
p(a (1,,., aj a 1, Ck)
p(ck a1,... aj... 0k),
(39)
p(az a1,.. aj ax-1)
as required.

---

Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
Lemma 2. The following identities hold:
a) p p (c..., CK do(al,.... ak)) = II p((+).
b) (M(do(s))) (ct,.., Gp a1; ( dd((a)))))
p(c;) Mazz p(ck ;
c) p(c Ck ,1,- ax) = II p(ck (11--.,0k))
Proof (Lemma 2)
a) Given the causal graph, and since all action variables A are intervened on, the only way in which could introduce dependencies between the confounders is through conditioning the collider Y. Hence, C TL Cj do(A), ... .A), for all j E {1,... K}. Thus, p(c1..... . | do(a1.... ax)) = III p(ck do(ai,..., ax)). Furthermore, since intervention cuts the dependence to the action variables, and Y is not conditioned on, we have Ck II. d Ak) for all K, giving us the first identity.
b) We have C. TL C2 A1, ... (o(A)) As for all i, [L, since the conditioning blocks all paths between the confounders. Either Ag block the outgoing path from C₁ for unintervened actions, there is no outgoing edge (other than to Y) for Cj.
Additionally, we have Cj TL do(y)), Ak. Hence,
p (c,,,,,
cx aa 1i,,..,,
do(ag).....
(40)
Midola
Midofa,)) (ck 1 ,..,..., do(as).,
(41)
p(c,) p M(dolas)) 01,
| do(a))) ak)
(42)
kéj
p(e,) II
p(ck 01 (1,"..ak)) ax);
(43)
ktj
where in the last line we use Lemma 1.
c) Follows from an argument analogous to b).
A.3. Proof of main result
Proof (Theorem 1) First note that
a)
E| do(aa,.... ... ak)l / y MyM do(ai, ... ak)) dy
(44)
f yp M) (y do(a),... ax)x C1, ex) dy
M(dofsn. (cI, ck do(a), ak)) dei ... dex
(45)
E|Y ax).
=
x p Mittoor..ax (ci, cx do(a), ax)) dei dex
(46)
=s ... / Σ falax:cc) ((loolar-ak)))a...... (q. CK do(ai.... ak)) de dek
(47)
12

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Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
Lemma 2a)
/ /
(I Salas: ok)
Σ
II
p(ck) dei-.. dck
(48)
&
= Σ / fu((o).))) cx)k) dex
(49)
Bowploon [Ta(ax, Cx)l
(50)
Second,
b) For every j KK we have
E|y a1, do(ay)..... ax]= = / y UpM(dalas))) (y a1, do(ay), a.. ax) dy
(51)
= 1 - / up UpMiidles)()) (y a1,- dn,-,) do(a)) (olaa))--.00 0k -,, CK) dy
priddllas) (c), cK 0,,,,, do(a,); ... ox) de ... dek
(52)
1 / up(y aumm,ay an.,, ak Cl..., ck) dy
p Mido(ss)) (cI... CK ai,.., ai, do(a,), ... ak) de ... dek
(53)
EIY (1,, "y: ak cI,,,.c.]] cx]
p Midolas)) (ct, CK al....., do(a;), ... .x) dei ... dek
(54)
- - ... / Σ (s) (Mdoles)) CK di,..., dola.,, ... ax) dej ... dex
(55)
Lemma 2h)
{ Σ [ p((s)
II p(ce ax) dei ... dex
(56)
Kij
| Blans)plss)de +
Σ
Ja(an,ca)p(c an...., ax) des
(57)
kzj""
Σ
(n (x),
(58)
And finally,
c)
E a1 ,., ax] = f yply I ll,...,,.k dy
(59)
=
y uply laa,..,, ak C,, c.) dy p(c cx I ll,...,, deg dek
(60)
= =f- E any... ax -,, cx) cx)_pcxkppp..pppp...pppp p(c), CK ai, ax) des ... dek
(61)
=fm.. ... / E falax: ca)) plei,...ck cK ai,...., ax) de dck
=
(62)
Lemma 2c) f ... / Σ Ialen II
plee ak, ak) des ... dex
(63)
> / fa(ex.c))p(e) an..., ax) des
(64)
13

---

Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
ac.-p(Calea...x) Cx)]
(65)
We learn K functions
Pa (at, oK, Rx), where Rg E {,, 1 is an indicator for whether Aş was intervened on. Rg can be thought of as selecting one of two functions
k: E (
(66)
fightr (a1,..., (an,
if Rs = 0
falaa...., aK Rx)
(67)
if Rg 1
where fohs and fint are universal function approximators.
We define
f(a1,-., ak, R1,.., Rk) Σ
aa,, Rs)-
(68)
k=1
Since are in the infinite data regime and have universal function approximators can fit such that
f(a1 , ak : R4 00, ... Rx 00 -E 01, [
(69)
fla OK R- =x= = EF a1.... do(..dd))))...... for E fl...,, K}.
(70)
From the definition of Equation (68), we have for each j E , K}
fa,,,.,, ak Ri OQ-0,... 1 -k= - (ai, (a,,,.,, aK, R -, ... Rg-0, Rx=0)
(71)
= f ag ak R,=1) fa f. Ry=0). RK,
(72)
After training the estimator we have
fla aK R1 -1. RI Rx=0) - fa...... OK, R1=0,... Ry=0,... Rx E ,,, ax]
(73)
EE 01 "
(74)
(58) Ee Ec C₁]] (cc..p(c (aj, Cj)l
(75)
where in the last step we have plugged in the decomposition of the expectation in the single- intervention (58) and observational (65) setting.
Combining the definition of the estimator (68), and Equations (72) and (75), we get an expression for the joint interventional effect:
K fai,,.., ak, R1=1,... Rx=1) = Σ (a), .,, R,,
(76)
Σ (Berolic) Ce [aa,,,,, R R,-
(77)
K
K Σ Ec C,) Σ ax,
K
Ec, Bc Bc, Cj]l
(78)
(68)
do(ai, ax)
E a1, axl + flai, ak, Ri -0 Rx -0)
(79)
680
=| do(aj,..., ak)]
(80)
14

---

Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
B. Proof of Proposition 1
Proof (Proposition 1) As the proof of Theorem we can decompose the interventional effect as
E Y
do(aim)): aohs]
Σ Ecg-n,c,l(tlle,c..)) C,]+
Σ
Eo C+)]
(81)
EAins
AgEApba
We learn K functions and fit them the observational and single-interventionn expectation (Equations (66) to (70)).
Let
1f As E Ait
R
for all &s f1,... K}.
(82)
Of E Aohs
Then our estimator identifies the interventional effect (21), since
f(a (KK Ri, Rk)
(83)
(67),68)
Σ felai, aK, Rx= +
Σ (K,
(84)
AgEAi
Σ akk ak, (..KKKKK-)) Rk=0)+
Σ
ak, RK,R::0))
As Anbs
ArEAm
ax;
Σ
ax, R,=1) [R
(85)
AeAija
A₁ A5EAint
K
fel ak. R
Σ
aa, ((ai,,,,k ak Ry=1) -,a-----ffaaa..-- ,K, Rg Ry-
(86)
án AjiAim
2/ E[V| [
Ee,
(65)
Ey Ck)
Σ
(Eo, (Eo,noc)lls(s), Cj)l - Ec, Be,po(Chlo1n.000)5500 Cj)l)
(87)
J. Ayt
Σ C))
Σ
Bewnp(Cc (s Ck)
(88)
Ag Aue
(81)
E (.(jt))) aobs]
(89)
C. Proof of Corollary
Proof (Corollary 1) Similar to the proof Theorem decompose the outcome expectations in the joint interventional regime and the settings for which we have data as
Y do(aj,... ... ak)] Σ ECg p(C)) [fB(aa, Cz)]
(90)
BEB
Y do(ap), a. = -EC P(Cx) [Sa(an, Ca)l
+
Σ
BC (f Cg)l
BB e B
(91)
B
E ai,... axl = Σ
ECg (CC on... of Cs)].
(92)
B
15

---

Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
where a-g denotes all actions that are not in subset B.
We define B estimator functions
fa(ai.... ............. aK Rg), Σ BE
(93)
where Rg E {0, 1} indicates whether the actions the subset were intervened on. Then, following the analogous steps in Theorem 1, the joint interventional effect (6) is identified through
Σ
fb(a1...., ak Rg=1).
(94)
BEB
D. Probability Distributions for Example
Table Observational distribution: PM A1, A2) = PM, A1, A2)
P.M (1, A2) A2)
Y =
Y ==1
A = 0, A₂ =
(1 1-pp)
0
A = A, 0
r(1 p)
0
A = A2 =1
A 1, A2
p²
p
Table 2. Single-intervention distribution: (MM(o(AA)) A2) (MM
P.M(do(A.)) (Y, A2)
Y ==1
A₂ = 0
1
0
do(A1 = 0)
A2 =
0
0
Ag = 0
(1
p)
0
(1 1)
A2 = 1
0
P
Table 3. Single- interveittrrvnii distribution: p.M(do(A2)) (Y, - M(do(A))) (Y, A1)
[M((o((A)) A1)
Y=0
Y=1
A = 0
(1-p)
0
do(A2 0)
A =
p(1 (1)
p²
A 0
(1 - p)
do(A2 1)
A 1
p(1 p)
n²
E. Experiments
Table Standard deviations the exogenous noise variables the synthetic experiments.
Exogenous noise variable
std
U
0.1
Vi,.., V/
0.1
Wi,... Ws
0.5
16

---

Learning Joint Interventional Effects from Single- Variable Interventions in Additive Models
Sampling SCMs.
We sample SCMs of the form
== Σ fa(AA, Cx)
(95)
A (Ck. ((()))
for 1,... for 1,..,
(96)
C W4
(97)
where the functions fk. 9s are third order polynomials, and Apa(k) are the actions that are also parents of action Ag. Each potential edge A, - As with k> > j is drawn at random from Bernoulli(Peage) where the edge probability Pedge is set to 0.3 in our experiments. The distributions of the exogenous noise variables {U, Vi..,, Vo, Wi,... Wo] } have zero mean are either Gaussian, Uniform or Logistic, which is chosen at random. The standard deviations are given in Table 4.
The polynomial functions f& and Ok are also randomly generated. For each function, we create a multivariate polynomial of second order with mixed terms. The process for generating these functions follows:
1. We consider all possible combinations of powers for the input variables, up to the second order. For a function with n input variables, we consider all non-nege non-negative integer power combinations (pi....,Ph Pr.) such that Σ Ps S
2. For each of these power combinations, we generate a random coefficient drawn from a normal distribution with mean 0 and a small standard deviation a (in our case, o = 0.1).
3. To ensure that the scales of the variables do not grow exponentially along the topological order, normalize the coefficients. This is done by dividing each coefficient by the sum of the absolute values of all coefficients.
The resulting polynomial function for each S& takes the form:
fa(An, = An
(98)
4
where i j+j S 2 (since we're using second-order polynomials), and Ei, la:yl due to the normalization. Similarly, each Ih is a polynomial function C₁ and the parent actions Apa(k), with the same properties of being second-order and having normalized coefficients. For a gk with m parent actions, the function takes the form:
gh (Ck, Ap...... Apm..
Σ fa
iigin...im
Che A ... Pm
(99)
where Ey ij S 2 and Eioii......... Ji
In order to satisfy Assumption 1 on the support of the interventions, we sample single interventions and joint interventions on the action variables to match the observational distributions. That is, sample intervention values from normal distribution:
A
Wpr, o
(100)
where iig and og are the empirical mean and standard deviation of A₃ the observational data, respectively. For joint interventions, the intervention values are sampled independently for each action variable, following the same distribution as in the single intervention case.
Estimators. For each interventional setting
E ai...., agl
(101)
E ai...,, do(a))....,.a ux], for E (1,..., K}
(102)
we fit a third order polynomial estimator. Hence, the estimators
f(a1,-,,KK aK; =1-0... Rx Rx=
(103)
f(a,,., ak, R1 =0, ... R2 ... RK=0), for E K},
(104)
17

---

Learning Joint Interventional Effects from Single-Varisn Variable Interventions in Additive Models
represented by K polynomials. Hence, the estimator functions for the effect of each action Sx represented implicitly and can be recovered by adding and subtracting the corresponding terms. For example,
fj(at.... 0K, R2 0) f(aa,,.. ... ak Ri -R, Ry-1, ... RK= f(a1,..,, ag, =0, -k=
(105)
We regularize the estimators using Ridge regression and find the optimal regularization parameter through 3-fold cross validation for each estimator.
Evaluation. We evaluate all models the test dataset of the joint interventional data. We report average root mean squared errors and the corresponding standard error of the mean.
Experimental settings.
In Figure 2(a), we generate Nobs Nsuut Nobs 10 samples for the observational, single interventional and the joint interventional dataset. For the observational baseline, train estimator the observational samples. For the joint interventional baseline, train estimator the joint interventional samples. Then we train one estimator on pooled observational and inngle-interentiioaal data.
Figure 2(b) shows the prediction error for varying ratios of ingle-interventiona. and observational samples. The total number of data points is kept constant at Not (K 1) 10 samples. The observational and joint interventional baselines represent estimators trained Noo samples from their respective datasets.
For each point in Figure 2(c) the Intervention Generalization method is trained on Noot = Nobs + K sint data points, where the observational and single-interventions data sets have the same number of samples, that is, Nobs Nsint- We compare its prediction error for the joint interventional effect estimator trained No joint interventional samples.
F. Additional Results
Training Regime
method
jint
K = 2
=
3
K=
0.40
0.8
Average RMSE
0.22
(a)
Average RMSE
(b)
Average RMSE
(c)
0.35
0.6
0.21
0.30
0.20
0.4
0.25
103
104
10s
106
107
103
104
105
10
107
103
104
105
10
107
Noot Njnt
Ntat Njint
Noot Njnt
Figure A. Convergence the Intervention Gemeralization Method. Average root mean squared error (RMSE) for predicting the joint interventional effect E do(as,..., os)] )] for different numbers of total data points Neos. Each data point in the plot is averaged over 100 randomly sampled SCMs. The columns show different numbers action variables K.
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Learning Joint Interventional Effects from Single-Variable Variable Interventions in Additive Models
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Figure B. Experiments with Varying Ratios of Tinnle-Intervettionnall and Obscrvational Data. Average (RMSE) for predicting the joint interventional effect E[y do(aj,...,, aa)] for different ratios /Nine/ooee while keeping the total number of data points No.: Nobs Nsint constant. Each data point the plot averaged over 100 randomly sampled SCMs. The observational and joint interventional baselines represent estimators trained on Nos samples from their respective datasets. The columns correspond to diffcrent total numbers data points Neot, while the rows show different numbers of action variables K.
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19


Reviews:
[{"review_id": "smxbv1FwAF", "reviewer": "Reviewer_dotZ", "summary": "This paper addresses the challenge of unobserved confounding in multi-treatment (joint intervention) causal inference. It introduces a method leveraging observational data and experimental data where interventions are applied to only one variable at a time, under the assumption of a nonlinear additive outcome and single-cause confounding.\n\n## Update after rebuttal:\nI thank the authors for their response. While the theoretical results are well presented, the real-world impact is not entirely clear and the untestable assumptions require strong justification. I am maintaining my score of 3.", "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": "# Response to Reviewer dotZ\n\nThank you for your thoughtful review and constructive feedback on our paper. We appreciate the time you've invested and would like to address your key concerns.\n\n## On the strong assumptions of our method\n\nWe acknowledge that our model makes strong assumptions, particularly regarding the additive outcome mechanism and the single-cause confounding structure. These assumptions indeed limit the generalizability to certain real-world scenarios where shared confounding or more complex outcome mechanisms might be present.\n\nFrom general identifiability theory [1,2], we know that the intervention generalization problem is non-identifiable in the non-parametric case (considering only the causal structure). Without additional assumptions, given single-interventional and observational data, the joint-interventional effect cannot be uniquely determined in general.\n\nThe key question becomes: \"What minimal set of assumptions enables identification in this setting?\" Our work proposes one such set of assumptions that makes the problem identifiable. We view this as a starting point rather than a complete solution to all real-world scenarios with confounding.\n\nImportantly, our Corollary 1 provides a way to trade-off assumptions on additivity for additional experimental data. If for practical applications it is unclear whether a subset of actions contributes additively, we can choose not to make that assumption on this subset. This then comes at the cost of having to collect joint interventional data for this subset. We think that allowing for this trade-off between additivity and joint interventional data makes our method applicable to a broader range of real-world cases than full additivity would allow.\n\nIn many domains where additive models are already commonly used (e.g., marketing analytics [3], certain pharmacological interactions [4]), our approach could provide practical value despite its limitations. We hope future work will find more general solutions with weaker assumptions that cover a broader range of real-world scenarios. (See also our response to Reviewer tbRD.)\n\n## On the multi-treatment causal inference literature\n\nYou make an excellent point about the connection to multi-treatment causal inference literature. We thank you for suggesting these important references. Our work is indeed related to this literature, and we will add a discussion of these connections in our Related Works section. The papers you suggested (Miao et al., 2023; Wang and Blei, 2020; Zheng et al., 2021) provide valuable context for our work and would strengthen our paper's positioning within the broader causal inference literature.\n\n## On the limitations of synthetic experiments\n\nOur main goal in the experimental section was to show that, while we have shown that the problem setting is identifiable, the joint-interventional effect is practically estimatable from data with our proposed estimator.\n\nWe agree that our empirical evaluation using synthetic data has limitations in representing the full complexity of real-world causal structures. While we've tried to incorporate some level of complexity through randomly generated SCMs with varying parameters, this approach cannot fully capture all real-world nuances.\n\nThank you again for highlighting these important limitations. Your feedback will help us improve our research and better understand the practical applicability of our method.\n\n## References\n\n[1] Lee, S., Correa, J. D., and Bareinboim, E. \"General identifiability with arbitrary surrogate experiments.\" Uncertainty in artificial intelligence. PMLR, 2020.\n\n[2] Kivva, Y., et al. \"Revisiting the general identifiability problem.\" Uncertainty in Artificial Intelligence. PMLR, 2022.\n\n[3] Chan, D., & Perry, M. \"Challenges and opportunities in media mix modeling.\" Google Inc, 16, 2017.\n\n[4] Pearson, R. A., Wicha, S. G., & Okour, M. \"Drug combination modeling: methods and applications in drug development.\" The Journal of Clinical Pharmacology, 63(2), 151-165, 2023.", "claims_and_evidence": "The paper clearly establishes identifiability under the specified assumptions. Additionally, the assumptions required for identifiability are transparently stated and reasonably justified. Empirically, the authors convincingly demonstrate that their proposed estimator performs close to a model trained directly on joint interventional data as sample size increases. And it significantly outperforms baselines such as observational-only estimation or naive pooling of single-variable interventions.", "methods_and_evaluation_criteria": "1. The proposed method relies on strong assumptions, particularly an additive outcome mechanism and single-cause (pair-wise) confounding, where each confounder influences exactly one treatment variable and the outcome. Such strict assumptions limit the generalizability and practical applicability of the method, as real-world scenarios often involve shared confounding across multiple treatments  and complex outcome mechanism.\n2. The empirical evaluation is limited to relatively simple synthetic experiments, which may not adequately capture the complexities of real-world causal structures or realistic data generation processes. Investigations into robustness against violations of key assumptions can also clarify the practical utility and limitations of the proposed approach.", "theoretical_claims": "Look fine to me.", "experimental_designs_or_analyses": "See above.", "relation_to_broader_scientific_literature": "The paper positions itself at the intersection of causal identifiability theory, intervention generalization, and additive statistical modeling, combining insights from these areas to propose a novel method for estimating joint interventional effects from single-variable interventions.", "essential_references_not_discussed": "This work is related to multi-treatment causal inference literature. For example:\n- [1] Miao et al. (2023) Identifying effects of multiple treatments in the presence of unmeasured confounding\n- [2] Wang and Blei (2020) The blessings of multiple causes\n- [3] Zheng et al. (2021) Copula-based Sensitivity Analysis for Multi-Treatment Causal Inference with Unobserved Confounding", "other_strengths_and_weaknesses": "Overall, this paper is well written and clearly structured.", "other_comments_or_suggestions": "It seems Equation (2) is not compatible with the approach and graph.", "questions_for_authors": "None", "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": "nGih9EcFSK", "reviewer": "Reviewer_9nyV", "summary": "The paper proposes a method for estimating joint interventional effects using observational data and single-variable interventions within nonlinear additive models. It establishes identifiability results, showing that joint effects are recoverable without direct joint interventional data. The authors validate their approach empirically using synthetic data. However, there are some concerns about the novelty of the proposed method.", "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": "# Response to Reviewer 9nyV\n\nThank you for your thoughtful review and for engaging with our work. We appreciate your careful reading and feedback.\n\nWe would like to address your primary concern regarding the novelty of our approach and the necessity of our assumptions:\nYou mentioned that Assumption 1 might be sufficient for identifying the joint interventional effect using only observational data, referring to Definition 5 and Theorem 3 in \"On the validity of covariate adjustment for estimating causal effects\" (UAI 2010).\n\n**Critical clarification:** In our problem setting, the confounders ${C_1, \\ldots, C_K}$ are unobserved. This is a fundamental aspect of our work, explicitly stated in Section 2.2 where we define $C = {C_1, \\ldots, C_K}$ as \"a set of unobserved confounders.\"\n\nThe covariate adjustment approach you reference would indeed be applicable if these confounders were observed. However, when confounders are unobserved (as in our setting), covariate adjustment is not possible, and identification becomes significantly more challenging. This is precisely why we need both Assumption 1 (Intervention Support) and Assumption 2 (Additive Outcome Mechanism), along with single-variable interventional data, to achieve identification.\n\nWe believe this clarification addresses your main concern about the novelty and necessity of our approach. Thank you again for your review, and we hope this clarifies the contribution of our work.", "claims_and_evidence": "The authors convincingly support their claims regarding identifiability and practical effectiveness through clear theoretical results and robust empirical evidence. No significant problematic claims are evident, as each theoretical assertion is carefully supported by rigorous mathematical justification and experiments.", "methods_and_evaluation_criteria": "The method makes sense, but I think the assumptions and inteventional data are not necessary.", "theoretical_claims": "I did not check the proofs, but I believe they are correct, because it seems that the results have been verified by previous studies.", "experimental_designs_or_analyses": "The experimental design utilizing synthetic structural causal models (SCMs) is valid and well-executed. The analyses, particularly comparisons with relevant baselines, including observational-only and joint-interventional estimators, are sound and provide strong validation of the proposed approach.", "relation_to_broader_scientific_literature": "The problem is vital in the scientific literature.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "I apologize for my oversight in missing the statement that $C$ represents a set of unobserved confounders. I have updated my score accordingly. I suggest that the authors consider using different formats in Fig. 1 to clearly distinguish between latent variables and observable variables.\n\n#----------------------------------#\n\nI think Assumption 1 is sufficient for identifying the joint interventional effect with only observational data in the setting of the paper. According to Def. 5 and Thm. 3 of [1], the joint interventional effect is identifiable by covariate adjustment with adjustment set being {$C_1,\\cdots,C_k$}. The reason that the two models in Example 1 are with different causal effects is the violation of Assumption 1. Hence, there is no novelty in the current method, which additionally uses interventional data and Assumption 2. I am not sure that my comments are completely correct. If the authors could correct me, I am happy to increase my score.\n\n[1] On the validity of covariate adjustment for estimating causal effects. UAI2010", "other_comments_or_suggestions": "Fig.1 is too big.", "questions_for_authors": "See Other Strengths And Weaknesses", "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": "M4ygz8nWnV", "reviewer": "Reviewer_tbRD", "summary": "This paper presents a novel construction for identifiability of joint interventional effect from single variable intervention results, with additive assumptions placed on the causal mechanism. The method is tested on synthetic data and demonstrated effectiveness and on pare performance when compare with a model trained on joint interventional data.", "questions": "", "limitations": "", "rating": 2, "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": "Thank you for taking the time to review our paper and for your thoughtful comments. We appreciate your feedback and would like to address your concerns below.\n\n## On the need for parametric identification\n\nFrom g-ID theory [1], we know that our problem setting is non-identifiable in the non-parametric case (i.e., only considering the causal structure in Fig. 1). I.e., given single-interventional and observational data, the joint-interventional effect cannot be uniquely determined in general. The causal structure, together with the intervention sets, forms what's known as a \"thicket\" (Def. 6 in [1]), which precludes identifiability. We believe that such multi-cause settings are interesting across many domains (see citations in Introduction). Unfortunately, these settings are limited by non-identifiability in the intervention generalization setting without additional assumptions.\n\n## On the novelty of the additivity assumption\n\nWe don't claim the additivity assumption itself is novel. There is indeed a rich literature on additivity in statistics [2,3]. In causal inference, additivity of exogenous noise has been quite common. However, functional additivity, where the function is composed of a sum of nonlinear functions, to the best of our knowledge, has not been explored in causal inference. In particular, it has not been investigated as a parametric assumption in the intervention generalization setting.\n\nThis function class is particularly interesting here since it limits the complexity of interaction between actions in the outcome mechanism. This effectively prevents two distinct causal models from agreeing across single-interventional regimes while differing on the joint-interventional.\n\n## On the limitations of Assumption 2\n\nWe agree that Ass. 2 limits the generality of our results. However, as discussed above, the problem is hard and not solvable in the most general case. For many if not most real-world settings with confounding between outcome and actions, intervention generalization is impossible without additional assumptions.\n\nThe key question becomes: \"What parametric and structural assumptions do we have to make to achieve identifiability in the intervention generalization setting?\" In our work, we have shown that under our set of assumptions using the additive outcome mechanism, intervention generalization is possible, and we have provided a practical estimator for it.\n\n## On the practical applicability\n\nWhile intervention generalization may not be achievable in all real-world cases, the benefits for the cases where it is applicable are immense. We can reduce the required experimental conditions from growing exponentially with the No. of actions to a linear regime. This makes a significant difference in many real-world settings, where obtaining experimental data is often very hard or expensive.\n\nAdditionally, Cor. 1 provides a way to trade-off assumptions on additivity for additional experimental data. If for practical applications it is unclear whether a subset of actions contributes additively, we can choose drop the assumption on this subset. This then comes at the cost of having to collect joint interventional data for this subset. We think that allowing for this trade-off between additivity and joint interventional data makes our method applicable to a broader range of real-world cases than full additivity would allow.\n\n## On future directions\n\nOur set of assumptions may not be the most parsimonious, and we hope that future work will find more general solutions. One motivation for choosing the additive model class was its relation to more general model classes, namely Generalized Additive Models (GAMs) [4], which in turn are related to the general functions of the Kolmogorov-Arnold Theorem [5]. We hope that our work can serve as a stepping stone toward proving identifiability results for the GAM case.\n\n## On extensions to more general confounding between actions\n\nIf we add confounding between actions, Lem. 1 would not hold. However, we believe this is not a complete blocker for the problem. Our identifiability proof is agnostic to the causal structure between the actions (assuming it has a DAG structure). We suppose that addressing more general action confounding would require modeling the dependencies between the actions as well as the outcome mechanism to achieve intervention generalization. We hope to make progress in this direction in future work.\n\n## References\n[1] Lee, S. et al. \"General identifiability with arbitrary surrogate experiments.\" 2020.\n\n[2] Friedman, J. H. et al. \"Projection pursuit regression.\" 1981.\n\n[3] Breiman, L. et al. \"Estimating optimal transformations for multiple regression and correlation.\" 1985.\n\n[4] Hastie, T. J. et al. \"Generalized Additive Models.\" 1990.\n\n[5] Kolmogorov, A.N. \"On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables.\" 1956.", "claims_and_evidence": "The claims are in general correct with enough evidence.", "methods_and_evaluation_criteria": "The proposed methods are correct and evaluation seems to be reasonable.", "theoretical_claims": "Yes. the identifiable results seems to be correct.", "experimental_designs_or_analyses": "The experimental design and analysis seems to be fine, although (1) semi-synthetic experiment could be used for better representing its real-world effect (2) better motivation need for assumption 2 (Additive Outcome Mechanism) and (3) discussion should be made in case of violation of assumption 2 (e.g. feedback loop or no direct ordering in DAG as in \"Sachs, K., Perez, O., Pe'er, D., Lauffenburger, D.A. and Nolan, G.P., 2005. Causal protein-signaling networks derived from multiparameter single-cell data. Science, 308(5721), pp.523-529.\") See more in question section.", "relation_to_broader_scientific_literature": "This is an interesting and one of the frontier questions in the causal inference field, learning the joint effect from single causal effects. The key ingredient of this paper is the additive causal mechanism assumption, which is interesting but also quite strong and restrictive to real world scenarios.", "essential_references_not_discussed": "I think most of the relevant literatures are discussed in the paper.", "other_strengths_and_weaknesses": "The paper is well-motivated and provide clear contributions with experimental validations, the theory is in general correct but I am a bit less convinced its impact on real-world scenarios as (1) the additive assumption is nothing new and not very novel (one can claim this is novel as it is not placed on the mechanism or functional forms instead of parameters but sill this is not a major novel contribution) (2) the limitation of having confounder $C_{i}$ point to $X_{i}$ only seems to be restrictive and might not be applicable in real-world (how about have $C_{i}$ point to $X_{i}$ and $X_{j}$?), please refers to comments in \"**Experimental Designs Or Analyses**\" section. (3) experimental results are quite weak as it use purely synthetic data without constraining on its parameters (100 different SCMs helps but still quite weak), my suggestion is to a.) learn parameters from real-world data and b.)have discussion on when the identifiable results does not apply and conduct experiments under this to give us a better understanding on how applicable the results are in more realistic setting.", "other_comments_or_suggestions": "Please see my comments earlier.\n\n\n## update after rebuttal\n\nI thank the authors for the detailed and thoughtful responses. While the clarifications are helpful and address some of my concerns, I remain unconvinced about the real-world impact (in terms of application) of this research and empirical strength of the paper (with synthetic data only) under the current assumptions. Therefore, my score remains unchanged.", "questions_for_authors": "I still found the justification is weak for why the authors focus on parametric identification this case? The applicable of this method for real-world problem can be completely trashed if the data-generation process is not under this constrained form.\n\nCould you please answer some of my concerns in the comments 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": "wEF03tzIkg", "reviewer": "Reviewer_2XuD", "summary": "The manuscript studies the problem of intervention generalization in additive causal models. Specifically, the authors consider a data generating process in which an outcome variable $Y$ is an additive function of actions $A_1, \\dots, A_K$ and unobserved confounders $C_1, \\dots, C_K$. Given some set of observational data and single-intervention experiments, the method estimates a joint intervention effect $\\mathbb E[Y \\mid do(a_1, \\dots, a_K)]$. Several identifiability results are presented, along with experiments illustrating the utility of their proposal.\n\n## update after rebuttal\n\nI thank the authors for their reply and look forward to seeing the extended benchmarks should the paper be accepted. I will maintain my score of 4.", "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": "# Response to Reviewer 2XuD\n\nWe thank the reviewer for the valuable comments and the pointers to Caroline Uhler’s lab related research. We would like to address some of the reviewer’s concerns.\n\nOn the baselines: we agree with the reviewer that using the purely observational data the pooled data might not be a fair comparison due to their simplicity. However, using joint interventional data is the closest one gets to an “oracle” version, since then the problem is reduced to estimating a conditional function $P(Y \\mid do(A))$. Moreover we believe it would be unfair to use the methods of Bravo-Hermsdorff et.al., Jung et.al. and Saengkyongam & Silva as benchmarks, as their assumptions usually do not match ours.\n\nIn particular, we cannot apply Saengkyongam & Silva’s method because their method requires the noise to be Gaussian (indeed, our method is also applicable for discrete variables). Likewise, our setting with the additive outcome mechanism is not identifiable in the interventional factor graph model by Bravo-Hermsdorff, since the additive structure of the causal mechanism does not translate to an extra factorization of the joint probability in the different interventional settings.\n\nAbout Caroline Uhler’s lab related research: We thank the reviewer for the pointers. These are indeed quite related to our work. \n\n- In particular Ribot et.al. is also interested in causal effect estimation in novel scenarios when some action-context pairs have been observed, which in the language of our paper would amount to estimating joint causal effects from observing only certain subsets of the interventions. Likewise, their identification strategy relies on making a linearity assumption, which shows that a latent factor model is induced.\n- On the other hand, Bijan et.al. cluster variables with respect to interventions that come from a mixture distribution and then use the clustering to produce Synthetic Potential Outcomes to compute a previously unseen intervention. Their setting differ from ours in that we have several possible actions as opposed to a single treatment and we make an assumption on the functional form of the SCM in order to make the joint interventions identifiable.\n\nWe will add an extended comparison in the camera-ready version of the paper.\n\nOn the running example: We believe that having a running example is a good idea. However, there are two reasons why we don’t include it. First, lack of space in the initial submission, and second, including a running example on the camera-ready version would potentially require large changes to the paper, which would be challenging to do given the time constraints and difficult to review.\n\nWe appreciate the time and effort you have invested in providing constructive feedback, which will help us improve our paper.", "claims_and_evidence": "The paper is primarily theoretical, with the joint and mixed identifiability results taking center stage. The proof sketch helps walk the reader through the authors' reasoning on this point. The experiments are a little perfunctory (more on this below), but sufficient to illustrate the principle.", "methods_and_evaluation_criteria": "The data generating process for the experiments makes good sense, but the alternative estimators seem a little naive. Several intervention generalization papers were mentioned in the Related Work section, and I was expecting to see these return in the benchmarks. (I'm thinking specifically of Bravo-Hermsdorff et al., Jung et al., and Saengkyongam & Silva.) Are these methods inapplicable in this setting? If so, why? Caroline Uhler's lab has also done considerable work in this area but I don't believe any of those methods are cited (see below).\n\n-https://arxiv.org/abs/2405.19225\n\n-https://proceedings.mlr.press/v236/ribot24a.html", "theoretical_claims": "Theoretical results are exceptionally clear and well presented.", "experimental_designs_or_analyses": "As noted above, the experimental DGP is sound but comparisons are a little thin.", "relation_to_broader_scientific_literature": "This work makes a meaningful contribution to the intervention generalization literature, which has numerous applications in science and industry. One recommendation to help improve the manuscript – there are several potential use cases cited in the introduction, but these are basically never mentioned again after the first paragraph. Readers may benefit from a running example that grounds us as we move through theoretical and experimental results.", "essential_references_not_discussed": "See the aforementioned works from Caroline Uhler's lab. These may not be identical to the problem setting here, but they're certainly in the neighborhood.", "other_strengths_and_weaknesses": "The paper is clear and convincing. I think with just a couple of minor amendments – namely, grounding the discussion with a running example and extending the benchmarks to include alternative intervention generalization techniques – it could realize its full potential.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See comments 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": {"ethical_review_concerns": "N/A"}}]

Author rebuttal:

Meta-review: This paper studies the problem of estimating joint interventional effects from observational and single-intervention data under a nonlinear additive outcome model. The theoretical results are clearly presented, and the proposed estimator shows promising performance on synthetic data.

Reviewers agree that the problem is important and the paper is well written. However, there are concerns about the strength of the identifiability assumptions and the limited empirical validation. The method relies on an additive outcome mechanism and single-cause confounding, which may limit real-world applicability. Moreover, the evaluation is restricted to synthetic data without comparisons to more recent baselines. Overall, I recommend a Weak Accept.
Decision: Accept (poster)