Title: Transferring Causal Effects using Proxies
Venue: NeurIPS 2025
Accepted: yes
Keywords: causal inference

domain shift

unmeasured confounding

proxies

identifiability

causal effect estimation
Abstract: We consider the problem of estimating a causal effect in a multi-domain setting. The causal effect of interest is confounded by an unobserved confounder and can change between the different domains. We assume that we have access to a proxy of the hidden confounder and that all variables are discrete or categorical. We propose methodology to estimate the causal effect in the target domain, where we assume to observe only the proxy variable. Under these conditions, we prove identifiability (even when treatment and response variables are continuous). We introduce two estimation techniques, prove consistency, and derive confidence intervals. The theoretical results are supported by simulation studies and a real-world example studying the causal effect of website rankings on consumer choices.

Transferring Causal Effects using Proxies

Abstract

We consider the problem of estimating a causal effect in a multi-domain setting. The causal effect of interest is confounded by an unobserved confounder and can change between the different domains. We assume that we have access to a proxy of the hidden confounder and that all variables are discrete or categorical. We propose methodology to estimate the causal effect in the target domain, where we assume to observe only the proxy variable. Under these conditions, we prove identifiability (even when treatment and response variables are continuous). We introduce two estimation techniques, prove consistency, and derive confidence intervals. The theoretical results are supported by simulation studies and a real-world example studying the causal effect of website rankings on consumer choices.

1 Introduction

Estimating the causal effect of a treatment or exposure X on an outcome or response Y of interest is a core objective across the social and natural sciences [1-4]. However, a major obstacle to such causal inference from observational (passively collected) data is the existence of unmeasured confounders U , i.e., unobserved variables that influence both the treatment and the outcome. For example, suppose that we wish to study the causal effect of a new experimental drug X on survival Y . Since such drugs tend to be administered only in severe cases, i.e., to patients in poor health with an already dire prognosis U , untreated patients will generally have better health outcomes. Naïvely comparing average survival between treated and untreated patients can thus lead to erroneous conclusions [5]. For accurate causal inference, it is therefore important that confounders are measured and correctly adjusted for [6].

Since it is difficult to rule out that some confounders remain unobserved, randomised controlled trials (RCTs) are considered the gold standard for causal inference [7, 8]. In an RCT, the treatment assignment is randomised, thereby eliminating any influences from potential confounders. However, for most settings of interest, RCTs are infeasible due to ethical or practical constraints. Much research has thus been dedicated to devising specialised methods for causal inference from purely observational data. An important step in this endeavour is to establish identifiability, i.e., to show that the causal estimand can be expressed in terms of distributions that only involve observed quantities [9]. To this end, existing methods leverage additional structure and assumptions, often in the form of other observed variables, such as instruments [10], mediators [11], or proxies [12, 13] (sometimes called negative controls [14]), which have certain causal relationships to the treatment, outcome, and confounder. In the present work, we propose a novel proxy-based approach.

E

U

W

X

Y

(a) Source domains

e

T

U

W

X

Y

(b) Target domain

1.1 Related work

The intuition behind proximal causal inference is that, even if some confounders U are unobserved and thus cannot be directly adjusted for, observed proxy variables W that are related to U may provide sufficient information to correct for the unobserved confounding [15]. In contrast to approaches rooted in latent variable modelling [e.g., 16-18], proximal causal inference methods typically do not aim to estimate the confounder explicitly but instead seek to devise more direct estimation methods. In contrast to literature on transportability and data fusion [e.g. 19-21] where identifiability is determined based purely on graphical criteria, proxy-based methods often rely on additional assumptions concerning the informativeness of the available proxies.

Single noisy measurement of the confounder. In the simplest case, we observe a single proxy W of U which is independent of X and Y given U (e.g., resulting from non-differential measurement error; see inset figure), and both U and W are discrete [22-24]. In this case, the confounding bias can be reduced (compared to no adjustment) by adjusting for W if the confounder acts monotonically [25]. Further, if the error mechanism,

U

W

X

Y

i.e., the conditional distribution P ( W | U ) is known, the causal effect is fully identified via the matrix adjustment method of Greenland and Lash [26]. However, in practice, P ( W | U ) is typically unknown, and proxies may causally influence the treatment or the outcome.

Learning from multiple proxies. As shown by Kuroki and Pearl [12], P ( W | U ) and thus the causal effect of X on Y can sometimes be identified if a second discrete proxy Z of U is available (see inset figure without the dashed edge), provided that the proxies are sufficiently informative (formalised via invertibility of certain matrices of conditional probabilities). Miao et al. [27] extend this result to the more general setting in which W

Z

U

W

X

Y

can influence Y (where P ( W | U ) is no longer identifiable) and establish conditions for identifiability when the confounder and proxies are continuous. For the latter, the matrix adjustment method is replaced by an approach based on bridge functions, obtained as the solutions to certain integral equations [27]. Subsequent work provides extensions to longitudinal settings [28, 29] and efficient estimation methods based on semiparametric models [30], kernel methods [31], or deep learning [32].

Proxies for domain adaptation. Alabdulmohsin et al. [33] use proxy methods for single-source unsupervised domain adaptation, i.e., to predict Y in a new target domain where only X is observed. To this end, they rely on the latent shift assumption, which posits that the domain shift is entirely due to a shift in the distribution of a discrete latent confounder U of X and Y , and leverage source observations of a noisy measurement W of U and a concept bottleneck variable [34] that mediates the effect of X on Y to identify the optimal target predictor. Tsai et al. [35] generalise these results in several ways, most notably showing that the concept bottleneck is not needed if multiple, sufficiently diverse source domains are available (see Fig. 1a) and both W and X are observed in the target domain. Prashant et al. [18] consider domain generalisation under the latent shift assumption via approximate identification of the unobserved confounder in the presence of either proxies or multiple domains.

E

U

W

X

Y

Z

E

U

W

X

Y

Z

E

Z

U

W

X

Y

Learning from multiple domains. Causal approaches to domain generalization [e.g., 36-40] typically seek to discard spurious features and learn one or more domain-invariant predictors for Y , rather than the target interventional distribution Q ( Y | do( x )) . In contrast to work on combining experimental and observational data to improve statistical efficiency [e.g., 41, 42], we assume no access to observations of the outcome Y in the target domain.

1.2 Overview and main contributions

In this work, we consider the problem of estimating the causal effect of X on Y in the presence of a discrete unobserved confounder U , given a proxy W of U and data from multiple domains. Our main focus is on the setting illustrated in Fig. 1, which also relies on the latent shift assumption [33] (though see Fig. 2, § 3.1, and Remark 9 in Appx. B for a discussion of the relaxation of this assumption) and is similar to that considered by Tsai et al. [35]. However, a key difference is the estimand of interest. Whereas Tsai et al. seek to estimate the conditional mean E [ Y | X = x ] under the target distribution Q , our focus is on the interventional distribution Q ( Y | do( x )) . Moreover, we only need to observe W in the target domain, as shown in Fig. 1b; we thus need to transfer information from the source domains to the target domain. While our goal is more aligned with proximal causal inference than with domain adaptation, a key difference to the setting studied, e.g., by Miao et al. [13] is that in our case, the causal effect of X on Y may differ across domains due to shifts in the distribution of U (even though the joint causal effect of X and U on Y is invariant [e.g., 36]). For ease of presentation, we focus on a setting in which all variables are discrete. We discuss some extensions to continuous variables in § 3.1 and § 7.

The remainder of the paper is organised as follows. First, we formally state the considered problem setting (§ 2) and establish sufficient conditions for identifiability (§ 3). We then propose two estimators based on alternative parametrisations of the model and study their statistical properties (§ 4). Empirically, we investigate our estimators in simulations (§ 5) and present an application to hotel ranking data (§ 6). We conclude with a summary and an outlook for future work (§ 7).

We highlight the following main contributions:

We prove that, given only observations of W in the target domain, the interventional distribution Q ( Y | do( x )) is identified (Thm. 1) if the available source domains are sufficiently diverse (Asm. 1). Furthermore, we relax the latent shift assumption by allowing for additional covariates and edges, for example (Fig. 2 and Thm. 8).

We propose two consistent estimators, one of which is asymptotically normal and comes with confidence intervals (Props. 4 and 5).

We empirically verify that our estimators compare favourably to naive and proxy-adjustment baselines (Fig. 4) and that the derived confidence intervals have valid coverage and decreasing length as sample size increases.

2 Intervention distributions under domain shifts using proxies

Data generating process. Formally, we assume that the data are generated by a structural causal model (SCM) [43, 44] C := ( S, P N ) with the following collection S of assignments:

The noise variables N := ( N E , N U , N W , N X , N Y ) follow the joint distribution P N and are jointly independent. The SCM C induces the causal graph G shown in Fig. 1 and a distribution over the random variables ( E,U,W,X,Y ) , defined as the push-forward of P N via (1), which we denote by P C or P . As proved, e.g., by Lauritzen et al. [45], this induced distribution is Markov with respect to (w.r.t.) G . (Since our work does not consider counterfactual statements, one could also use causal graphical models and assume that the observed distribution is Markov w.r.t. the graph in Fig. 1(a).)

Generalization. For sake of accessibility, we focus our presentation on the setting from Fig. 1. However, our identifiability results directly apply to a broader class of causal models (that may contain additional covariates Z ), whose induced causal graphs G satisfy the d-separation [9] statements

where G X and G X denote the graphs obtained by respectively removing the edges coming into X and going out of X from G , see Fig. 2 for examples and Appx. B.4 for a more detailed discussion. We believe that our estimators and inference results can also be generalised to these settings analogously.

Domains. We consider k E source domains. The variable E is a categorical domain indicator taking values in { e 1 , . . . , e k E } . The labels e i are arbitrary and serve only to distinguish domains. We assume no structure or prior knowledge about the domains. The target domain corresponds to E = e T (not necessarily contained in { e 1 , . . . , e k E } ) and is described by the distribution P C ;do( E := e T ) , which we denote by Q C or Q .

Support. Throughout, we assume that E , U , W , X , and Y are discrete (though see the discussion after Thm. 1). We denote the support of a discrete random variable V by supp( V ) = { v 1 , . . . , v k V } , so, e.g., supp( X ) = { x 1 , . . . , x k X } . We assume that P C has full support over ( E,U,W,X,Y ) , i.e., supp(( E,U,W,X,Y )) = supp( E ) × supp( U ) × supp( W ) × supp( X ) × supp( Y ) . The probability mass functions (pmfs) in the source and target domains are denoted by lowercase letters p and q , respectively, so, e.g., p ( y | e, x ) = P ( Y = y | E = e, X = x ) or q ( w ) = Q ( W = w ) .

Matrix notation. We write marginal pmfs as column vectors, P ( A ) := ( p ( a 1 ) , . . . , p ( a k A )) ⊤ , and conditional pmfs as matrices, P ( A | B ) = ( p ( a i | b j )) ij , with the i th row given by P ( a i | B ) and the j th column by P ( A | b j ) . For additional discrete random variables C and D , we write P ( a | B,c ) := ( p ( a | b 1 , c ) , . . . , p ( a | b k B , c ) ) and P ( a | B,C,d ) := ( p ( a | B,c 1 , d ) ⊤ , . . . , p ( a | B,c k C , d ) ⊤ ) . In the target domain, we use Q instead of P . The distributions induced by the last four structural assignments of the SCM C in Eq. (1) are thus described by the matrices { P ( U | E ) , Q ( U ) } , P ( W | U ) , P ( X | U ) , and { P ( y | U, W, x ) } x ∈ supp( X ) ,y ∈ supp( Y ) , respectively.

Data. As illustrated in Fig. 1, we assume that we obtain a sample of n src independent and identically distributed (i.i.d.) source realizations ( E 1 , W 1 , X 1 , Y 1 ) , . . . , ( E n src , W n src , X n src , Y n src ) from P O := P C ( E,W,X,Y ) and n tgt := n -n src i.i.d. target realizations W n src +1 , . . . , W n from Q .

Estimand. The goal is to estimate

which, by slight abuse of notation, we call the causal effect from X on Y (in the target domain e T ). Since the domain is allowed to affect the hidden confounder, the causal effect in the target domain is, in general, different from p ( y | e, do( x )) when e ∈ { 1 , . . . , e K } is a source domain.

3 Identifiability

We now show that (2) is identifiable from P O and Q W . More formally, under suitable assumptions, we prove the following statement: if C 1 and C 2 are of the form of (1) such that P C 1 O = P C 2 O and Q C 1 W = Q C 2 W , then q C 1 ( y | do( x )) = q C 2 ( y | do( x )) . The key idea of the proof (inspired by ideas presented by Miao et al. [27, Section 2]) is to consider the covariate adjustment formula [6, 46, 47]

Clearly, U is unobserved, so we cannot exploit this equation directly. Instead, we will rewrite Q ( y | U, x ) , so that it depends on U only through a factor P ( W | U ) . Since, by invariance or modularity, P ( W | U ) = Q ( W | U ) , the dependence on U vanishes after marginalising w.r.t. U as in Eq. (3). To do so, we use invertibility of P ( W | U ) , which is guaranteed by the following assumption:

Assumption 1. For all x ∈ supp( X ) , we have rank( P ( W | E,x )) ≥ k U .

Intuitively, Asm. 1 states that the proxy W is sufficiently informative about the confounder U in the sense that the domain shifts in U induce sufficient variability in the conditional distribution of W | X = x . As we show in Prop. 6 in Appx. B.1, Asm. 1 implies that the proxy W can be transformed in such a way that the matrix P ( W | E,x ) has linearly independent rows. We therefore assume this without loss of generality (see Appx. B.1 for a more detailed treatment). Finally, this implies that the matrix P ( W | E,x ) P ( W | E,x ) ⊤ is invertible [48], so its right pseudoinverse P ( W | E,x ) † exists. 3 Similar assumptions about the rank of conditional probability matrices are common in proxy-based identifiability results [e.g. 12, 13, 35].

We are now able to state the main identifiability result. Its proof, together with other proofs of this work, can be found in Appx. A.

Theorem 1. Under the data generating process described in § 2 (ignoring the paragraph 'Generalization') and assuming Asm. 1, we have for all x ∈ supp( X ) and y ∈ supp( Y ) :

Therefore, if ( E,W,X,Y ) is observed in the source domains and W in the target domain, the causal effect of X on Y in the target domain e T is identifiable.

Remark 2. If, within our setting, Asm. 1 does not hold, identifiability is, in general, lost (see Appx. B.2 for a counterexample). Thus, in this sense, Asm. 1 is necessary for Eq. (4) to hold.

3.1 Extensions of the identifiability result

Relaxing assumptions of Thm. 1 and special cases. The result still holds (with an analogous proof) if X and Y are not discrete but continuous. In this case, P ( y | E,x ) is a vector of evaluations of a conditional probability density function (pdf), for example. Furthermore, we do not assume faithfulness, and Thm. 1 still holds if there is no edge between W and Y . Moreover, the latent shift assumption [33] can be relaxed to also allow for (unmediated) covariate shift via E → X , see Remark 9 in Appx. B.4 for a characterisation of the necessary graphical assumptions.

Incorporating covariates/observed confounders. In practice, we often observe additional covariates that we want to include in the model. For the case of observed confounders Z of X and Y , we show that the (target-domain) conditional causal effect q ( y | do( x ) , z ) given Z = z is identified if Z is observed in both the source and target domains and subject to an assumption similar to Asm. 1 for the matrix P ( W | E,x,z ) , see Asm. 2 and Thm. 8 in Appx. B.4 for details.

(Conditional) average treatment effects. Thm. 1 directly implies that the (target-domain) average treatment effect (ATE), given by E [ Y | do( X := 1)] -E [ Y | do( X := 0)] under Q , is identifiable, too. Moreover, due to the identifiability of the causal effect conditional on observed confounders Z = z , we can also identify the (target-domain) conditional average treatment effect (CATE), given by E [ Y | do( X := 1) , Z = z ] -E [ Y | do( X := 0) , Z = z ] under Q .

Continuous proxy. In Appx. B.5, we discuss an extension to the case where the proxy W is continuous. The proposed approach is based on the idea that suitable discretisation can yield a discrete proxy variable that satisfies Asm. 1, see Asm. 3 and Prop. 10 for details.

4 Estimation and inference

Having established the identifiability of q ( y | do( x )) , we now construct estimators for this causal effect. Specifically, we propose two estimators based on different decompositions of the causal effect,

denoted by ̂ q C,n ( y | do( x )) and ̂ q R,n ( y | do( x )) , respectively. The subscript n indicates the sample size used in each case. By a slight abuse of notation, we use the same symbols to denote the estimators and their values evaluated on a specific dataset. Code implementing both estimators, along with the experiments, is available on https://github.com/manueligal/proxy-intervention .

4.1 Causal parametrisation

As discussed in § 2, the structural assignments in Eq. (1) can be represented by the matrices P ( U | E ) , Q ( U ) , P ( W | U ) , P ( X | U ) and { P ( y | U, W, x ) } x ∈ supp( X ) ,y ∈ supp( Y ) . For our first approach we explicitly parametrise these matrices, thereby capturing the underlying causal mechanisms. Prop. 3 shows that the estimand q ( y | do( x )) can be expressed in terms of a subset of these (unknown) matrices.

Proposition 3. For all x ∈ supp( X ) and y ∈ supp( Y ) we have 4

We now proceed by maximum likelihood estimation. Consider a parameter θ whose components are the entries of P ( U | E ) , Q ( U ) , P ( W | U ) , P ( X | U ) and { P ( y | U, W, x ) } x ∈ supp( X ) ,y ∈ supp( Y ) . By an abuse of notation, we denote the entries of θ by the corresponding probabilities with a subscript θ , i.e.,

The entries are restricted to be non-zero, each column of P ( U | E ) , Q ( U ) , P ( W | U ) , and P ( X | U ) sums to 1, and similarly ∑ k Y i =1 p ( y i | u, w, x ) = 1 for all u ∈ supp( U ) , w ∈ supp( W ) , and x ∈ supp( X ) . To ensure these constraints are satisfied without explicitly enforcing them during optimisation, we use real-valued parameters (representing logits) and transform them into valid pmfs using the softmax function.

Naturally, the joint distributions of the observed variables P W,X,Y | E in the source domains and Q W in the target domain are also parametrised by θ . Specifically, for all i ∈ { 1 , . . . , k Y } , j ∈ { 1 , . . . , k W } , s ∈ { 1 , . . . , k X } , and l ∈ { 1 , . . . , k E } , we have that

where both of these equations hold for the true (conditional) pmfs. The (conditional) likelihood over the observed variables is thus given by

where n ( y i , x s , w j , e l ) denotes the number of observations for the tuple ( y i , x s , w j , e l ) in the source domains and n ( w j ) denotes the number of observations of w j in the target domain.

For all x ∈ supp( X ) and y ∈ supp( Y ) , we define the function

where the entries of these matrices are components of θ . We denote by θ 0 the true value of θ , so by Prop. 3 we have that g x,y ( θ 0 ) = q ( y | do( x )) . Therefore, if we denote by ̂ θ n a maximizer of

the likelihood function in Eq. (7), we define the plug-in estimator of the causal effect in the causal parametrisation for all x ∈ supp( X ) and y ∈ supp( Y ) as ̂ q C,n ( y | do( x )) := g x,y ( ̂ θ n ) .

Even if the estimator ̂ θ n does not converge, its induced probabilities over the observed variables concentrate around the true values, and the estimator ̂ q C,n ( y | do( x )) is consistent.

Proposition 4. Suppose that Asm. 1 holds. The estimator q ( y | do( x )) is consistent, that is,

4.2 Reduced parametrisation

Although q ( y | do( x )) is identifiable, not all the components of the vector θ that we optimise in the causal parametrisation can be identified from the observed data - the estimator is based on an overparametrisation of the problem. Our second approach estimates only the components necessary to compute Eq. (4). To do so, for all x ∈ supp( X ) and y ∈ supp( Y ) , we consider one long parameter vector η x,y describing all entries of 5 Q ( W,e T ) , q ( e T ) , P ( W,x,E ) , P ( y, x, E ) , and P ( x, E ) (see Appx. B.3 for details). Given an η x,y , we can then not only form the matrices Q η x,y ( W,e T ) , q η x,y ( e T ) , P η x,y ( W,x,E ) , P η x,y ( y, x, E ) and P η x,y ( x, E ) but also Q η x,y ( W ) , P η x,y ( W | E,x ) , and P η x,y ( y | E,x ) , which we use to define an estimator based on Eq. (4). The components of η x,y are probabilities, so we can estimate their true value by empirical frequencies 6 , that is, ̂ η x,y,n := 1 n ∑ n i =1 η i x,y , where, for all i ∈ { 1 , . . . , n } ,

Then, for example,

For all x ∈ supp( X ) and y ∈ supp( Y ) , we define the function

and define the estimator of the causal effect in the reduced parametrisation as

The following proposition proves consistency of ̂ q R,n ( y | do( x )) and provides confidence intervals.

Proposition 5. Suppose that Asm. 1 holds. For all x ∈ supp( X ) and y ∈ supp( Y ) , the estimator ̂ q R,n ( y | do( x )) defined in Eq. (12) is a consistent estimator of q ( y | do( x )) . Furthermore, where

with ̂ Σ x,y being the sample covariance matrix of ( η 1 x,y , . . . , η n x,y ) .

̸

̂ C,n .

Causal

Reduced

2.5

5.0

7.5

10.0

2.5

5.0

7.5

10.0

0.0

0.2

0.4

0.6

0.8

log(

k

(

P n

(W|E,x)))

Absolute error

log(

k

(

P(W|E,x)))

2

3

4

5

6

7

By Prop. 5, pointwise asymptotic confidence intervals for q ( y | do( x )) at level (1 -α ) are given by

where z β denotes the β -quantile of a standard normal distribution.

Both ̂ q R,n ( y | do( x )) and the bounds of the confidence interval can lie outside [0 , 1] because the data-generating process adds further constraints on the combination of P ( y | E,x ) , P ( W | E,x ) , and Q ( W ) that are not reflected by the estimator. To reduce bias and variance, we thus clip the estimate ̂ q R,n ( y | do( x )) and the confidence intervals to [0 , 1] .

5 Simulation experiments

5.1 Data generation

The data generating mechanism can be described using the matrices P ( U | E ) , Q ( U ) , P ( W | U ) , P ( X | U ) and { P ( y | U, W, x ) } x ∈ supp( X ) ,y ∈ supp( Y ) . Werandomly and independently generate M ∈ N different sets of such matrices. For each choice, we generate N ∈ N i.i.d. data sets with sample size n ∈ N . Appx. C.1 provides details on how we choose the matrices and the size of the causal effects.

5.2 Point estimation

We first compare the point estimates obtained from the two proposed estimators. To do so, we fix values x ∈ supp( X ) and y ∈ supp( Y ) and compute, for several data sets, the absolute estimation errors | ̂ q R,n ( y | do( x )) -q ( y | do( x )) | and | ̂ q C,n ( y | do( x )) -q ( y | do( x )) | .

In our setting, we can assess Asm. 1 by considering the condition number κ ( P ( W | E,x )) . When κ ( P ( W | E,x )) is large, the rows of P ( W | E,x ) are almost linearly dependent, indicating an almost-violation of Asm. 1. By Eq. (4), we thus expect that the performance of ̂ q C,n ( y | do( x )) and ̂ q R,n ( y | do( x )) is sensitive to large conditioning numbers and that for larger conditioning numbers, we require larger sample sizes. We validate this empirically in Fig. 3. It shows that the absolute estimation errors of both estimators grow with κ ( P ( W | E,x )) . It also shows that κ ( P ( W | E,x )) ≈ ˆ κ ( P ( W | E,x )) so we can estimate κ directly from the data. Whenever ˆ κ ( P ( W | E,x )) is large, we can flag the corresponding estimates as potentially unreliable.

The results suggest that both estimation procedures perform similarly in terms of absolute estimation error, with an average value equal to 0.040 for the causal estimator and 0.058 for the reduced estimator, with the reduced estimator being faster to compute (see Appx. C.5 for a runtime analysis). Figs. 7 and 8 in Appx. C.3 show the same results for n = 1000 and n = 100000 supporting the theoretical consistency result. If we add another column to P ( W | E,x ) , that is, another source domain, the reduced estimator seems to perform slightly better than the causal estimator, see Fig. 9 in Appx. C.3.

We further compare the two estimators against other baseline estimators. In particular, we consider an oracle which estimates the causal effect from n i.i.d. realizations of the target intervention

0.0

0.1

0.2

0.3

Oracle

Causal

Reduced

NoAdj

NoAdj*

WAdj

WAdj*

Method

Absolute error

distribution (using data not available to other methods); the no-adjustment estimator, which estimates the distribution Q C , do( x ) Y by P C Y | x (NoAdj) or Q C Y | x (NoAdj ∗ ; using data not available to other methods); and the W -adjustment estimator, which uses the adjustment formula in Eq. (4) but with W instead of U (even though W is not a valid adjustment set) using either pooled data from the source domains (WAdj) or from the target domain (WAdj ∗ ; using data not available to other methods). The formal definitions of these estimators can be found in Appx. C.3. Fig. 4 shows the comparison in terms of absolute error. The causal and reduced estimators outperform the baselines different from the oracle, which has the lowest error.

5.3 Coverage of confidence intervals

Prop. 5 proves asymptotic normality of (the unclipped version of) ̂ q R,n ( y | do( x )) and provides asymptotically valid confidence intervals. We show that the empirical coverage is close to the nominal value and that the sample median confidence interval length, as suggested by consistency, approaches zero with growing sample size (see Appx. C.4 for details). For comparison, we have also implemented bootstrap confidence intervals based on the normal approximation method [49], see also Appx. C.4 for details. Fig. 10 shows that the empirical coverage of the bootstrap confidence intervals is slightly closer to the nominal level but comes at the cost of longer confidence intervals.

6 Application: Hotel searches

We consider the Expedia Hotel Searches dataset [50], in which each observation corresponds to a query made by a user of Expedia's webpage. It contains information about the user, the search filters, the hotels displayed, and whether the user clicked on or booked any of the shown accommodations. We are interested in studying the causal effect of the position of the hotel in the list shown to the user ( X ) on whether the user clicked on the hotel to obtain more information ( Y ). We regard it as plausible that there is an unmeasured confounder U of X and Y (e.g., features of the hotel, such as the distance to the city centre or the additional services provided), but that some of this information is reflected in the price ( W ), which we treat as a proxy. Domains E are comprised of searches in which a given hotel appears. Tab. 1 provides an overview of the involved variables and their role relative to the studied setting. As source domains, we use those hotels that have at least 2000 observations (except for the ones chosen as target domains, see below); this results in 25 hotels. If several of such hotels appear in the same query, this yields several observations (introducing a small dependence between the observations).

The dataset can be viewed as containing both observational and interventional data. For some searches (the ones we use for our estimators), hotels are sorted according to an existing algorithm (unknown to us). For all other searches, hotels are ranked uniformly at random. The latter part allows us to compute a 'ground truth' or 'oracle' causal effect (containing a point estimate and Wald confidence interval) that we can compare against. Ursu [51] uses this dataset to study the effect of

0.0

0.1

0.2

0.3

0.4

0

5

10

15

Target Domain ID

qn

(Y=1|do(X=1))

Reduced

NoAdj*

NoAdj

Oracle

WAdj*

WAdj

the ranking in the customer choice, but only considers the part of the dataset in which hotels are ranked uniformly at random. This is different from our setting, since we estimate the causal effect q ( Y = 1 | do( X = x )) in a target domain (the hotel of interest) from the observational data.

In the first experiment, we estimate q ( Y = 1 | do( X = 1)) . As target domains, we choose 18 hotels among those hotels that appear in at least 1500 different queries in the randomized dataset. This ensures that we can obtain reasonable estimates using the oracle, which we consider the ground truth in this experiment. In total, we obtain ca. 64 000 and 50 000 observations from 8400 and 4000 queries for the source and target domains, respectively. Fig. 5 compares the confidence intervals obtained by the reduced estimator with the oracle ones. As a comparison we also show the result of the no-adjustment and W-adjustment baselines (using both versions: pooled and target), see Appx. C.3, including Wald confidence intervals for the no-adjustment estimator (all confidence intervals in this experiment are at level 0 . 95 ). Generally, both estimators overlap with the oracle confidence intervals. The reduced estimator yields point estimates that are closer to the oracle estimates (average absolute error of 0.044 (Reduced), 0.051 (NoAdj), 0.080 (NoAdj*), 0.053 (WAdj) and 0.075 (WAdj*)) and confidence intervals that are on average shorter than the ones from the no-adjustment baseline (median length of 0.14 versus 0.17). In a second experiment, we estimate the causal effect q ( Y = 1 | do( X = x )) for different values of x ∈ { 1 , . . . , 20 } in a fixed target domain to study the causal effect of the different positions on the clicks of the users. Again, the confidence intervals produced by the reduced estimator overlap with all oracle ones, see Appx. D for details.

7 Summary and future work

We consider the causal domain adaptation problem of estimating a confounded causal effect in an unseen target domain using data from source domains and an observed proxy variable. We provide two estimators, prove consistency and, in one case, provide asymptotically valid confidence intervals. Experiments on simulated and real data show that these estimators outperform existing methods and support the theoretical claims. We consider the case of a continuous U as an interesting setting, too; here, the ideas discussed by Miao et al. [13] and Tsai et al. [35] could prove helpful.

Acknowledgments

The project that gave rise to these results received the support of a fellowship from 'la Caixa' Foundation (ID 100010434). The fellowship code is LCF/BQ/EU23/12010098.

References

Joshua D Angrist and Jörn-Steffen Pischke. Mostly harmless econometrics: An empiricist's companion . Princeton University Press, 2009. [Cited on p. 1.]

Stephen L Morgan and Christopher Winship. Counterfactuals and Causal Inference: Methods and Principles for Social Research . Cambridge University Press, 2014.

Guido W Imbens and Donald B Rubin. Causal inference in statistics, social, and biomedical sciences . Cambridge University Press, 2015.

Miguel A Hernán and James M Robins. Causal inference: What if . Boca Raton: Chapman & Hall/CRC., 2020. [Cited on p. 1.]

Edward H Simpson. The interpretation of interaction in contingency tables. Journal of the Royal Statistical Society: Series B (Methodological) , 13(2):238-241, 1951. [Cited on p. 1.]

James Robins. A new approach to causal inference in mortality studies with a sustained exposure period-application to control of the healthy worker survivor effect. Mathematical modelling , 7(9-12): 1393-1512, 1986. [Cited on p. 1 and 4.]

Jerzy Neyman. On the application of probability theory to agricultural experiments. essay on principles. Ann. Agricultural Sciences , pages 1-51, 1923. [Cited on p. 1.]

Ronald A Fisher. The Design of Experiments . Oliver and Boyd, Edinburgh, 1935. [Cited on p. 1.]

Judea Pearl. Causality: Models, Reasoning, and Inference . Cambridge University Press, 2009. [Cited on p. 1 and 4.]

Joshua D Angrist, Guido W Imbens, and Donald B Rubin. Identification of causal effects using instrumental variables. Journal of the American Statistical Association , 91(434):444-455, 1996. [Cited on p. 1.]

Judea Pearl. Causal diagrams for empirical research. Biometrika , 82(4):669-710, 1995. [Cited on p. 1.]

Manabu Kuroki and Judea Pearl. Measurement bias and effect restoration in causal inference. Biometrika , 101(2):423-437, 2014. [Cited on p. 1, 2, and 5.]

Wang Miao, Zhi Geng, and Eric J Tchetgen Tchetgen. Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika , 105(4):987-993, 2018. [Cited on p. 1, 3, 5, and 10.]

Marc Lipsitch, Eric Tchetgen Tchetgen, and Ted Cohen. Negative controls: a tool for detecting confounding and bias in observational studies. Epidemiology , 21(3):383-388, 2010. [Cited on p. 1.]

Eric J Tchetgen Tchetgen, Andrew Ying, Yifan Cui, Xu Shi, and Wang Miao. An introduction to proximal causal inference. Statistical Science , 39(3):375-390, 2024. [Cited on p. 2.]

Christos Louizos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling. Causal effect inference with deep latent-variable models. Advances in neural information processing systems , 30, 2017. [Cited on p. 2.]

Yixin Wang and David M Blei. The blessings of multiple causes. Journal of the American Statistical Association , 114(528):1574-1596, 2019.

Parjanya Prajakta Prashant, Seyedeh Baharan Khatami, Bruno Ribeiro, and Babak Salimi. Scalable out-of-distribution robustness in the presence of unobserved confounders. In International Conference on Artificial Intelligence and Statistics , pages 3763-3771. PMLR, 2025. [Cited on p. 2.]

Elias Bareinboim and Judea Pearl. A general algorithm for deciding transportability of experimental results. Journal of Causal Inference , 1(1):107-134, 2013. [Cited on p. 2.]

Judea Pearl and Elias Bareinboim. External validity: From do-calculus to transportability across populations. Statistical Science , 29(4):579-595, 2014.

Elias Bareinboim and Judea Pearl. Causal inference and the data-fusion problem. Proceedings of the National Academy of Sciences , 113(27):7345-7352, 2016. [Cited on p. 2.]

Sander Greenland. The effect of misclassification in the presence of covariates. American journal of epidemiology , 112(4):564-569, 1980. [Cited on p. 2.]

Raymond J Carroll, David Ruppert, Leonard A Stefanski, and Ciprian M Crainiceanu. Measurement error in nonlinear models: a modern perspective . Chapman and Hall/CRC, 2006.

Judea Pearl. On measurement bias in causal inference. In Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence , pages 425-432, 2010. [Cited on p. 2.]

Elizabeth L Ogburn and Tyler J Vanderweele. Bias attenuation results for nondifferentially mismeasured ordinal and coarsened confounders. Biometrika , 100(1):241-248, 2013. [Cited on p. 2.]

Sander Greenland and Timothy L. Lash. Bias analysis. In Modern Epidemiology , chapter 19, pages 345-380. Lippincott Williams and Wilkins, 2008. [Cited on p. 2.]

Wang Miao, Zhi Geng, and Eric J Tchetgen Tchetgen. Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika , 105(4):987-993, 2018. [Cited on p. 2 and 4.]

Andrew Ying, Wang Miao, Xu Shi, and Eric J Tchetgen Tchetgen. Proximal causal inference for complex longitudinal studies. Journal of the Royal Statistical Society Series B: Statistical Methodology , 85(3): 684-704, 2023. [Cited on p. 2.]

Guido Imbens, Nathan Kallus, Xiaojie Mao, and Yuhao Wang. Long-term causal inference under persistent confounding via data combination. Journal of the Royal Statistical Society Series B: Statistical Methodology , 87(2):362-388, 2025. [Cited on p. 2.]

Yifan Cui, Hongming Pu, Xu Shi, Wang Miao, and Eric Tchetgen Tchetgen. Semiparametric proximal causal inference. Journal of the American Statistical Association , 119(546):1348-1359, 2024. [Cited on p. 2.]

Afsaneh Mastouri, Yuchen Zhu, Limor Gultchin, Anna Korba, Ricardo Silva, Matt Kusner, Arthur Gretton, and Krikamol Muandet. Proximal causal learning with kernels: Two-stage estimation and moment restriction. In International Conference on Machine Learning , pages 7512-7523. PMLR, 2021. [Cited on p. 2.]

Liyuan Xu, Heishiro Kanagawa, and Arthur Gretton. Deep proxy causal learning and its application to confounded bandit policy evaluation. In Advances in Neural Information Processing Systems , volume 34, pages 26264-26275, 2021. [Cited on p. 2.]

Ibrahim Alabdulmohsin, Nicole Chiou, Alexander D'Amour, Arthur Gretton, Sanmi Koyejo, Matt J Kusner, Stephen R Pfohl, Olawale Salaudeen, Jessica Schrouff, and Katherine Tsai. Adapting to latent subgroup shifts via concepts and proxies. In International Conference on Artificial Intelligence and Statistics , pages 9637-9661. PMLR, 2023. [Cited on p. 2, 3, 5, and 29.]

Pang Wei Koh, Thao Nguyen, Yew Siang Tang, Stephen Mussmann, Emma Pierson, Been Kim, and Percy Liang. Concept bottleneck models. In International Conference on Machine Learning , pages 5338-5348. PMLR, 2020. [Cited on p. 2.]

Katherine Tsai, Stephen R Pfohl, Olawale Salaudeen, Nicole Chiou, Matt Kusner, Alexander D'Amour, Sanmi Koyejo, and Arthur Gretton. Proxy methods for domain adaptation. In International Conference on Artificial Intelligence and Statistics , pages 3961-3969. PMLR, 2024. [Cited on p. 2, 3, 5, 10, and 29.]

Jonas Peters, Peter Bühlmann, and Nicolai Meinshausen. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society. Series B (Statistical Methodology) , pages 947-1012, 2016. [Cited on p. 3.]

Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893 , 2019.

Mateo Rojas-Carulla, Bernhard Schölkopf, Richard Turner, and Jonas Peters. Invariant models for causal transfer learning. Journal of Machine Learning Research , 19(36):1-34, 2018.

Rune Christiansen, Niklas Pfister, Martin E Jakobsen, Nicola Gnecco, and Jonas Peters. A Causal Framework for Distribution Generalization. IEEE Transactions on Pattern Analysis and Machine Intelligence , 44 (10):6614-6630, 2022.

Cian Eastwood, Shashank Singh, Andrei L Nicolicioiu, Marin Vlastelica Poganˇ ci´ c, Julius von Kügelgen, and Bernhard Schölkopf. Spuriosity didn't kill the classifier: Using invariant predictions to harness spurious features. Advances in Neural Information Processing Systems , 36:18291-18324, 2023. [Cited on p. 3.]

Bénédicte Colnet, Imke Mayer, Guanhua Chen, Awa Dieng, Ruohong Li, Gaël Varoquaux, Jean-Philippe Vert, Julie Josse, and Shu Yang. Causal inference methods for combining randomized trials and observational studies: a review. Statistical science , 39(1):165-191, 2024. [Cited on p. 3.]

Klaus-Rudolf Kladny, Julius von Kügelgen, Bernhard Schölkopf, and Michael Muehlebach. Causal effect estimation from observational and interventional data through matrix weighted linear estimators. In Uncertainty in Artificial Intelligence , pages 1087-1097. PMLR, 2023. [Cited on p. 3.]

Judea Pearl. Causality: Models, Reasoning and Inference . Cambridge University Press, 2nd edition, 2009. [Cited on p. 3 and 22.]

Stephan Bongers, Patrick Forre, Jonas Peters, and Joris M Mooij. Foundations of structural causal models with cycles and latent variables. Annals of Statistics , 49(5):2885-2915, 2021. [Cited on p. 3.]

Steffen L Lauritzen, Alexander P Dawid, Birgitte N Larsen, and Hanns-Georg Leimer. Independence properties of directed markov fields. Networks , 20(5):491--505, 1990. [Cited on p. 4 and 22.]

Judea Pearl. Comment: Graphical models, causality and intervention. Statistical Science , 8(3):266-269, 1993. [Cited on p. 4.]

Peter Spirtes, Clark Glymour, and Richard Scheines. Causation, Prediction, and Search . Springer New York, 1st edition, 1993. [Cited on p. 4.]

Gilbert Strang. Linear Algebra and its Applications . Academic Press, 3rd edition, 1988. [Cited on p. 5.]

Anthony Christopher Davison and David Victor Hinkley. Bootstrap methods and their application . Cambridge university press, 1st edition, 1997. [Cited on p. 9 and 32.]

Adam Woznica, Ben Hamner, Dan Friedman, and SSA_Expedia. Personalize Expedia Hotel Searches ICDM 2013. https://kaggle.com/competitions/expedia-personalized-sort , 2013. Kaggle. [Cited on p. 9.]

Raluca M Ursu. The power of rankings: Quantifying the effect of rankings on online consumer search and purchase decisions. Marketing Science , 37(4):530-552, 2018. [Cited on p. 9.]

James E Gentle. Matrix Algebra . Springer New York, 3rd edition, 2024. [Cited on p. 22 and 27.]

Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms . The MIT Press, 2017. [Cited on p. 22.]

Anirban DasGupta. Probability for statistics and machine learning: fundamentals and advanced topics . Springer, 1st edition, 2011. [Cited on p. 30.]

Richard H Byrd, Peihuang Lu, Jorge Nocedal, and Ciyou Zhu. A limited memory algorithm for bound constrained optimization. SIAM Journal on scientific computing , 16(5):1190-1208, 1995. [Cited on p. 30.]

NeurIPS Paper Checklist

i.) Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: All the claims made in the abstract and introduction correspond to the contributions made in the paper.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

ii.) Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: The limitations and possible extensions of the methods presented in this paper are mainly discussed in § 3.1 and § 7.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

iii.) Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: Assumptions are indicated for every theoretical result and Appx. A includes the proofs of theoretical results.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

iv.) Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: The code developed for this paper, as well as the dataset used in § 6 are available. Moreover, the algorithm used to generate the synthetic data for the simulations is described in Appx. C.1.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

v.) Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: The code required to reproduce the experiments of this paper is available in a public github repository under the MIT License.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

vi.) Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: The hyperparameters chosen in the simulation experiments are included together with the results in § 5 and the appendices.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

vii.) Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: The results are accompanied by confidence intervals whenever is possible and the methods used to calculate them are derived or explained in the paper.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

viii.) Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: The computation time is not the main scope of this paper, but we present a runtime analysis in Appx. C.5.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

ix.) Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: The paper complies with the NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

x.) Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: The positive impact of causal effect estimation is discussed at the beginning of § 1. We consider that there are no clear negative societal impacts of the research performed.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

xi.) Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [Yes]

Justification: Downloading the dataset used in § 6 requires accepting the terms of use in the Kaggle competition webpage.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

xii.) Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: The dataset corresponding to Expedia hotel searches used in § 6 is properly credited. The terms of use (it can be used for research but not by Expedia competitors) are respected in this paper.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

xiii.) New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: The code for this paper is available in a public repository under the MIT License. The details about these assets are provided in this paper and the README file.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

xiv.) Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The paper does not involve experiments with human subjects or crowdsourcing.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

xv.) Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The paper does not involve experiments with human subjects or crowdsourcing.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

xvi.) Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: The paper does not use LLMs as a core component of the research.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Appendices

Table of Contents

A Proofs

A.1 Proof of Thm. 1

Theorem 1. Under the data generating process described in § 2 (ignoring the paragraph 'Generalization') and assuming Asm. 1, we have for all x ∈ supp( X ) and y ∈ supp( Y ) :

Therefore, if ( E,W,X,Y ) is observed in the source domains and W in the target domain, the causal effect of X on Y in the target domain e T is identifiable.

Proof. The global Markov condition holds in SCMs [e.g., 43, 45], so we have

For every pair of matrices A ∈ R m × n and B ∈ R n × p , the following inequality holds: rank( AB ) ≤ min { rank( A ) , rank( B ) } [52]. Using this inequality in Eq. (15), we get

where the last equality is due to Prop. 6. The matrix P ( W | U ) has dimensions k W × k U , hence

Using Eq. (16) and the inequality k W ≥ k U derived from Asm. 1, we conclude that rank( P ( W | U )) = k W = k U . Therefore, P ( W | U ) is an invertible matrix and this can be used in Eq. (15) to obtain

The substitution of this expression in Eq. (14) yields

The adjustment formula (3) states that q ( y | do( x )) = Q ( y | U, x ) Q ( U ) and the equality Q ( y | U, x ) = P ( y | U, x ) holds because of modularity or invariance [e.g., 53]. Combining Eqs. (3) and (17), we get

.

Using the law of total probability for the last two terms, we get our final result

We then have and therefore

A.2 Proof of Prop. 3

Proposition 3. For all x ∈ supp( X ) and y ∈ supp( Y ) we have 7

q

(

y

|

do(

x

)) = diag

(

P

(

y

|

U, W, x

)

P

(

W

|

U

)

)

Q

(

U

)

.

(5)

Proof. By modularity, we have Q ( y | U, x ) = P ( y | U, x ) . The statement then follows by the covariate adjustment formula (3):

A.3 Proof of Prop. 4

Proposition 4. Suppose that Asm. 1 holds. The estimator ̂ q C,n ( y | do( x )) is consistent, that is,

Proof. We consider a parameter whose components are the joint pmfs that appear in the likelihood function in Eq. (7)

and denote by µ 0 its true value. Moreover, we denote by f ( ̂ θ n ) the vector obtained when calculating these joint pmfs from the optimal vector ̂ θ n . Hence,

The conditional log-likelihood equals

where we define 0 · log 0 = 0 . (It suffices to consider the conditional log-likelihood, as it differs from the unconditional log-likelihood only by an additive term that models the marginal probabilities of the domains.) Instead of ℓ n , we now consider

which satisfies argmax ℓ n = argmax ˜ ℓ n (this is because ℓ n can be optimized by considering the likelihood for each domain separately, so multiplying each domain likelihood by a constant 1 /n ( e l ) or 1 /n tgt does not change the optimal values for the probabilities). The unique value for µ maximizing both ˜ ℓ n and ℓ n , denoted by ̂ µ n , consists of the relative frequencies

The first inequality is due to the definition ̂ µ n := argmax ℓ n . Regarding the second inequality, we have

(where the last equality is due to log L = ℓ n ◦ f , which holds by definition of f ); and since there exists θ 0 such that µ 0 = f ( θ 0 ) , we have

We have that ̂ µ n P - → µ 0 due to the law of large numbers and we use the continuous mapping theorem to get

where n min := min { n ( e 1 ) , . . . , n ( e k E ) , n tgt } is the smallest sample size across all source and target domains. Using the squeeze theorem on Eq. (19),

For all n ∈ N ,

Furthermore, we have the inequality

To prove this inequality, we consider the decomposition

corresponding to the sum of the likelihoods in each domain. For all l ∈ { 1 , . . . , k E , T } , the vector ̂ µ n is the maximizer of ℓ e l n , so we have

Therefore,

where the inequality holds due to Eq. (20). Therefore, considering the Kullback-Leibler divergence, we get the following convergence in probability 8 :

which implies that each of the summands on the left-hand side converges to zero in probability. Then, by Pinsker's inequality and the continuous mapping theorem, we have

Hence,

A.4 Proof of Prop. 5

Proposition 5. Suppose that Asm. 1 holds. For all x ∈ supp( X ) and y ∈ supp( Y ) , the estimator ̂ q R,n ( y | do( x )) defined in Eq. (12) is a consistent estimator of q ( y | do( x )) . Furthermore, where

Finally, we apply the continuous mapping theorem with the function ˜ g x,y : µ 0 ↦-→ ˜ g x,y ( µ 0 ) := q ( y | do( x )) (defined, for example, using Eq. (4), where all the components can be derived from the joint pmfs in µ 0 ) to get the final result:

with ̂ Σ x,y being the sample covariance matrix of ( η 1 x,y , . . . , η n x,y ) .

8 Here ̂ µ e 1 n := ( p ̂ µ n ( y 1 , x 1 , w 1 | e 1 ) , . . . , p ̂ µ n ( y k Y , x k X , w k W | e 1 )) and analogously for the other variables.

Proof. Consider the i.i.d. sample { η i x,y } n i =1 obtained from { ( E i , W i , X i , Y i ) } n i =1 using the definition in Eq. (9). The random vector η i x,y has mean equal to the true parameter η x,y, 0 . We define ̂ η ∗ x,y,n to be the sample mean of η x,y . Here, we use the asterisk to stress the difference to the estimator ̂ η x,y,n , which is equal to ̂ η ∗ x,y,n except for when P ̂ η ∗ x,y,n ( W | E,x ) does not have linearly independent rows.

To prove asymptotic normality, we apply the central limit theorem to { η i x,y } n i =1 . We use again that their mean is equal to η x,y, 0 and their sample mean is the estimator ̂ η ∗ x,y,n . Furthermore, we denote by Σ x,y the covariance matrix of η 1 x,y , obtaining

̸

. Indeed, for all

We now prove that holds, too. Let B n be the event that P ̂ η ∗ x,y,n ( W | E,x ) has linearly independent rows. We first show that P ( B n ) → 1 . Indeed, by Asm. 1 and Prop. 6, the population matrix P ( W | E,x ) has full row rank. Thus, there is a k W × k W submatrix M s.t. det( P ( W | E,x ) M ) = c = 0 . By continuity of the determinant, there is a δ > 0 s.t. 9 ∥ P ( W | E,x ) M -P ̂ η ∗ x,y,n ( W | E,x ) M ∥ F < δ (let us call such events A n,δ ) implies det( P ̂ η ∗ x,y,n ( W | E,x ) M ) = 0 , so P ̂ η ∗ x,y,n ( W | E,x ) has full row rank. By the law of large numbers, P ( A n,δ ) → 1 and thus P ( B n ) → 1 . Defining ˆ a n := √ n ( ̂ η x,y,n -η x,y, 0 ) and ∗ √ ∗ D

n

ˆ

a

:=

n

(

̂

η

x,y,n

-

η

x,y,

0

)

, we can now prove

̸

ˆ

a

n

- → N

(0

,

Σ

x,y

)

b

, we have

which proves Eq. (22). The second equality holds because, on B n , ˆ a ∗ n and ˆ a n are identical and lim n →∞ P (ˆ a ∗ n ≤ b ∩ B C n ) = 0 .

The function h defined in Eq. (11) is continuous, so we apply the delta method to Eq. (22) to get

where the variance σ 2 x,y is given by

Consistency follows by Slutsky's theorem.

To estimate σ 2 x,y , we can replace Σ x,y by the sample covariance matrix ̂ Σ x,y,n of ( η 1 x,y , . . . , η n x,y ) and η x,y, 0 by ̂ η x,y,n , which are both consistent estimators by the law of large numbers. This yields

By the continuous mapping theorem, ̂ σ 2 x,y is a consistent estimator of σ 2 x,y . Therefore, using the continuous mapping theorem again, together with Slutsky's theorem, we have

B Details and extensions of theoretical results

In this appendix, we provide several additional results that highlight how the assumptions and the setting studied in the main paper can be relaxed.

B.1 Relaxing Asm. 1

Under Asm. 1, the proxy W can be transformed in such a way that the matrix P ( W | E,x ) has linearly independent rows.

Proposition 6. Under Asm. 1, for all x ∈ supp( X ) , there exists a W -measurable random variable ˜ W such that

and ˜ W satisfies the independence statements ( X,Y, ˜ W ⊥ ⊥ E | U ) and ( X ⊥ ⊥ ˜ W | U ) , and supp(( E,U, ˜ W,X,Y )) = supp( E ) × supp( U ) × supp( ˜ W ) × supp( X ) × supp( Y ) .

Lemma 7. Let w 1 , . . . , w n be linearly independent vectors in a vector space V , and let

̸

with scalars λ i ∈ R . If λ 1 = -1 , then

are linearly independent.

Proof. Write

̸

are linearly independent.

Proof of Prop. 6. Fix x ∈ supp( X ) . If rank( P ( W | E,x )) = k W the result follows considering ˜ W = W . Assume r := rank( P ( W | E,x )) < k W . We proceed as follows: Choose r + 1 rows of P ( W | E,x ) , denoted by v 1 , . . . , v r +1 (with row indices denoted by a 1 , . . . , a r +1 ), such that v 1 , . . . , v i -1 , v i +1 , . . . , v r +1 are linearly independent and v i is a linear combination of the others, that is, there exist λ 1 , . . . , λ i -1 , λ i +1 , . . . , λ r +1 ∈ R such that v i = ∑ j = i λ j v j . Since the entries of P ( W | E,x ) are non-negative, there exists k ∈ { 1 , . . . , r +1 } \ { i } such that λ k = -1 . We define a new random variable ˜ W := W 1 { W = w a i } + w a k 1 { W = w a i } . By Lemma 7 we have that P ( ˜ W | E,x ) is a ( k W -1) × k E -dimensional matrix and rank( P ( ˜ W | E,x )) = rank( P ( W | E,x )) . It also holds that supp(( E,U, ˜ W,X,Y )) = supp( E ) × supp( U ) × supp( ˜ W ) × supp( X ) × supp( Y ) . Continue this procedure until all rows are independent. By an abuse of notation we call the resulting random variable again ˜ W . For all random variables A,B,C and f measurable A ⊥ ⊥ B | C = ⇒ f ( A ) ⊥ ⊥ B | C . It therefore holds that

Suppose there exist scalars a, b 2 , . . . , b n such that

Substituting the expansion of v + w 1 gives

Collecting coefficients on the independent set { w 1 , . . . , w n } yields the system

Since λ 1 +1 = 0 , the first equation forces a = 0 , which implies b 2 = . . . = b n = 0 . Hence, the only solution is the trivial one, proving that

̸

We have that rank( P ( ˜ W | E,x )) = rank( P ( W | E,x )) ≥ k U and P ( ˜ W | E,x ) has full row rank.

̸

̸

B.2 Asm. 1 and identifiability

We now present a counterexample showing that identifiability is, in general, not possible if Asm. 1 is violated. Concretely, we now construct two different SCMs. Both of them satisfy supp ( X ) := { 0 , 1 } and supp ( Y ) := { 0 , 1 } . We consider x := 0 ∈ supp ( X ) and y := 0 ∈ supp ( Y ) and are interested in inferring q ( y | do( x )) . Both SCMs satisfy supp ( W ) := { w 1 , w 2 , w 3 } , supp ( U ) := { u 1 , u 2 , u 3 } and, for both SCMs, P ( W | U ) and P ( y | U, W, x ) are such that, for all j ∈ { 1 , 2 , 3 } ,

Furthermore, we choose, again for both SCMs, P ( E ) , P ( U | E ) and P ( X | U ) , such that the induced source distribution has full support over ( E,U,W,X,Y ) (this is possible, as sums of products of strictly positive numbers are strictly positive). Denoting by c 1 , c 2 , and c 3 the columns of P ( W | U ) , we have

with c 2 and c 3 being linearly independent, so rank( P ( W | U )) = 2 < k U . Using Eq. (15) and the inequality rank( AB ) ≤ min { rank( A ) , rank( B ) } [52], we get that rank( P ( W | E,x )) < k U , so in both SCMs Asm. 1 is violated.

The two SCMs differ in the target distribution of U : we define

The induced target distributions of W for the two SCMs are given by the law of total probability:

Therefore, both SCMs induce the same target distribution of the variable W . Thus, both SCMs induce the same source distributions over ( E,X,Y,W ) and the same target distribution over W , and thereby agree on the distributions over all observed quantities.

However, if we calculate the causal effect q ( y | do( x )) for each SCM using the covariate adjustment formula (see Eq. (3)), we find that they differ:

̸

In conclusion, the causal effect from X to Y is not identifiable.

Logically, the counterexample implies that if the causal effect from X to Y is identified via Eq. (4) for all SCMs satisfying all assumptions except for possibly Asm. 1 (implying, in particular, that the pseudo-inverse P ( W | E,x ) † exists), then Asm. 1 must hold, too.

B.3 Reduced parametrisation

The parameter vector η x,y should have components corresponding to the following probabilities:

We denote the number of components of this vector by k η := k W +( k W +1) k E , so the corresponding parameter space is contained in H = [0 , 1] k η .

E

U

W

X

Y

Z

(a)

E

U

W

X

Y

Z

(b)

E

Z

U

W

X

Y

(c)

B.4 Additional covariate/observed confounder

We now consider the case in which we have an additional observed confounder Z . Specifically, suppose that the SCM induces one of the graphs depicted in Fig. 6.

We replace Asm. 1 with the following assumption.

Assumption 2. The proxy variable W is such that k W ≥ k U and, for all x ∈ supp( X ) and z ∈ supp ( Z ) , the matrix P ( W | E,z,x ) has linearly independent rows.

Theorem 8. Assume an SCM whose induced graph is one of those shown in Figure 6. Let E,U,W,X,Y,Z denote the random variables generated by this SCM, and suppose their joint support factorizes as supp( E,U,W,X,Y,Z ) = supp( E ) × supp( U ) × supp( W ) × supp( X ) × supp( Y ) × supp( Z ) . Suppose that Asm. 2 holds. We then have for all x ∈ supp( X ) , z ∈ supp( Z ) , and y ∈ supp( Y ) :

and

Therefore, if ( E,W,X,Y,Z ) is observed in the source domains and ( W,Z ) in the target domain, the conditional causal effect of X on Y given Z = z and the total causal effect of X on Y in the target domain e T are identifiable.

Proof. Since the global Markov condition holds for the induced graph and distribution of SCMs, we have for the graphs in Fig. 6 that Y ⊥ ⊥ E | ( U, X, Z ) and W ⊥ ⊥ ( E,X ) | ( U, Z ) and thus for all x ∈ supp( X ) , y ∈ supp( Y ) , z ∈ supp( Z ) :

Under Asm. 2, we follow the same steps as in the proof in Appx. A.1 to get

The target conditional interventional distribution is given by

where the first line follows from the law of total probability; the second line follows the fact that U ⊥ ⊥ X | Z in the post-intervention graph G X (formally, do-calculus rule 3); the third line follows

from parent adjustment applied to x (formally, do-calculus rule 2); and the last line follows from the assumed modularity or invariance. Substituting the expression from Eq. (24) then yields

Hence, if Asm. 2 holds and ( W,Z ) is observed in the target domain, the conditional interventional distribution q ( y | do( x ) , z ) is identified.

Furthermore,

where the first line follows from the law of total probability; the second from Z ⊥ ⊥ X in the postintervention graph G X ; and the last by substitution of (26). Therefore, if ( W,X,Y,Z ) is observed in the source domains and ( W,Z ) in the target domain, the causal effect q ( y | do( x )) is identifiable.

Remark 9 (Relaxing the latent shift assumption via E → X ) . The existence of an edge from E to X in the graphs in Fig. 6 means that the latent shift assumption required by Alabdulmohsin et al. [33] and Tsai et al. [35] (which we also rely on in the main paper) can be relaxed in our setting. Specifically, we can also allow for (direct or unmediated) covariate shift, i.e.,

This applies not only to Thm. 8 but also (in the form X ̸⊥ ⊥ E | U ) to Thm. 1, i.e., the identifiability result without observed confounders Z presented in the main paper.

The reason for this possible relaxation is that Thm. 1 only makes use of

while Thm. 8 only requires the analogous statements given Z ,

where G X and G X denote the graphs obtained by respectively removing the edges coming into X and going out of X from G . All these relations still hold when including E → X .

B.5 Continuous proxy variable

We can extend our results to continuous proxy variables W with similar ideas as the ones used in Appx. B.1. Let m ∈ N and V 1 , . . . , V m ⊆ supp( W ) be pairwise disjoint measurable sets such that supp( W ) = ⋃ i ≤ m V i . Define ˜ W V := ∑ i ≤ m i 1 { W ∈ V i } .

Assumption 3. Assume that, for all x ∈ supp( X ) , there exist m ∈ N and V 1 , . . . , V m ⊆ supp( W ) pairwise disjoint measurable sets such that supp( W ) = ⋃ i ≤ m V i and rank( P ( ˜ W V | E,x )) ≥ k U .

Proposition 10. Under Asm. 3, for all x ∈ supp( X ) , we have for the W -measurable random variable ˜ W V that, for all y ∈ supp( Y ) ,

Therefore, if ( E,W,X,Y ) is observed in the source domains and W in the target domain, the causal effect of X on Y in the target domain e T is identifiable.

Proof. The result follows directly from Thm. 1 with W = ˜ W V and the observation that

Causal

Reduced

2

4

6

2

4

6

0.0

0.2

0.4

0.6

0.8

log(

k

(

P n

(W|E,x)))

Absolute error

log(

k

(

P(W|E,x)))

2.5

5.0

7.5

10.0

C Implementation of the experiments and further simulation results

C.1 Data generation

The algorithm to generate the data for the simulation studies is based on the SCM introduced in Eq. (1). The algorithm has the following steps:

(i) Generate P ( U | E ) , Q ( U ) , P ( W | U ) , P ( X | U ) and { P ( y | U, W, x ) } x ∈ supp( X ) ,y ∈ supp( Y ) at random. The sum of each column of P ( U | E ) , Q ( U ) , P ( W | U ) and P ( X | U ) has to be equal to 1; we generate each column of length k from a Dirichlet distribution with parameter α = (1 , . . . , 1) ∈ R k , so the values are uniformly distributed in the ( k -1) -dimensional simplex [54]. Similarly, for all r ∈ { 1 , . . . , k U } , j ∈ { 1 , . . . , k W } and s ∈ { 1 , . . . , k X } , ∑ n i =1 p ( y i | u r , w j , x s ) = 1 ; we generate the values p ( y i | u r , w j , x s ) from a Dirichlet distribution with parameter (1 , . . . , 1) ∈ R k Y . The variable E is generated from a discrete uniform distribution in { e 1 , . . . , e k E , e T } , where e 1 , . . . , e k E are considered the source domains and e T is the target domain.

(ii) Generate an i.i.d. sample with sample size n of ( E,U,W,X,Y ) from the induced distribution of the SCM defined by the matrices from step (i) following the assignments in Eq. (1).

(iii) Repeat step (ii) N times.

(iv) Repeat steps (i)-(iii) M times.

This algorithm generates MN samples (each one corresponding to in total n data points, some of which belong to the target domain and the rest to the source domains). The matrices from step (i) are used to calculate the true value q ( y | do( x )) , which is used to measure the absolute estimation error. Each sample obtained in step (ii) is used to calculate an estimate of the causal effect. The causal effects q ( y | do( x )) generated by this algorithm have a sample mean around 0.5.

C.2 Optimization

The implementation of the causal estimator presented in § 4.1 requires the optimization of the likelihood function in Eq. (7). The components of θ are optimized first over R and later transformed into valid pmfs using the softmax function. The optimization is done using the default optim function in R with the L-BFGS-B algorithm [55]. The initial value for each component of θ is generated from a continuous uniform distribution in [0 , 1] and we consider a maximum of 50 000 iterations.

C.3 Point estimation

The plot presented in Fig. 3 uses a sample size of n = 20000 . We repeat the same procedure but using the sample size n = 1000 in Fig. 7 and n = 100000 in Fig. 8. These three figures suggest that increasing the sample size indeed leads to a reduction of the absolute estimation error for both methods, which gradually approaches zero.

The left panel of Fig. 9 shows a pairwise comparison between the causal and reduced parametrisations and is based on the same samples as Fig. 3. Most of the points are located close to the identity line,

Causal

Reduced

2

4

6

8

2

4

6

8

0.0

0.2

0.4

0.6

0.8

log(

k

(

P n

(W|E,x)))

Absolute error

log(

k

(

P(W|E,x)))

2.5

5.0

7.5

10.0

1e-04

1e-03

1e-02

1e-01

1e+00

1e-04

1e-03

1e-02

1e-01

1e+00

Absolute error causal

Absolute error reduced

1e-04

1e-03

1e-02

1e-01

1e+00

1e-04

1e-03

1e-02

1e-01

1e+00

Absolute error causal

Absolute error reduced

so the absolute error is similar for both methods. The right panel shows the results when adding another source domain and shows smaller differences between the estimates of both parametrisations. Furthermore, we increase the sample size when adding an extra source domain to keep the average sample size per domain approximately constant in both settings.

Regarding the baseline comparison in Fig. 4, we only consider samples from induced distributions corresponding to causal models where | q ( y | do( x )) -q ( y | x ) | > 0 . 1 to avoid those cases in which the causal effect of X on Y was barely confounded. We compare the two estimators introduced in this work with the following five methods.

Oracle. We generate the data directly from the intervention distribution Q do( X := x ) Y . For an i.i.d. sample Y 1 , . . . , Y n from this marginal intervention distribution of Y in the target domain, we estimate causal effect by

Since the oracle estimator has access to the intervention distribution, it should not be considered a competing method but rather a benchmark (or, as in § 6, an estimator for the ground truth). In the simulation experiments, the differences between the oracle estimates and the true value q ( y | do( x )) are due to the sample size being finite.

No adjustment estimator. This estimator ignores the hidden confounding and estimates the causal effect by estimating the conditional distribution Y | X . We consider two variants: the first one (NoAdj) uses an i.i.d. sample ( X 1 , Y 1 ) , . . . , ( X n , Y n ) from the marginal distribution P C ( X,Y ) pooled across all

Level

0.9

0.95

0.99

Bootstrap

Reduced

1e+03

1e+04

1e+05

1e+03

1e+04

1e+05

0.80

0.85

0.90

0.95

1.00

Sample size

Coverage

Bootstrap

Reduced

1e+03

1e+04

1e+05

1e+03

1e+04

1e+05

0.0

0.2

0.4

0.6

Sample size

Median CI length

source domains, while the second one (NoAdj*) uses an i.i.d. sample ( X 1 , Y 1 ) , . . . , ( X n , Y n ) from the marginal distribution Q C ( X,Y ) in the target domain; this, again, is information that is not available in our setting, so (NoAdj*) should not be considered a competing method, either. Both NoAdj and NoAdj* estimate the causal effect with the following expression:

W -adjustment estimator. This estimator uses the adjustment formula (3) with the proxy variable W instead of U . { W } is not a valid adjustment set and, in general, this estimator is not a consistent estimator for the causal effect of X on Y . However, if the proxy W is 'similar' to the confounder U (for example, if they are equal except for a small noise perturbation), the estimate could be close to the true value. We again consider two variants: one which uses an i.i.d. sample ( W 1 , X 1 , Y 1 ) , . . . , ( W n , X n , Y n ) from P C ( W,X,Y ) pooled across the source domains (WAdj) and the other from Q C ( W,X,Y ) , that is, from the target domain (WAdj*); as before, the latter estimator should not be considered a competing method as it uses information that is not available in our setting. In both cases, the estimator is defined as

C.4 Coverage of confidence intervals

We compare the proposed confidence intervals with bootstrap confidence intervals based on the normal approximation method [49]. Here, the variance is estimated using the sample variance ̂ σ 2 B of the bootstrap estimates for q ( y | do( x )) . In that case, the bootstrap confidence interval is given by

Fig. 10 presents the results comparing the bootstrap confidence intervals and the ones obtained through the asymptotic normality of the reduced estimator (see Eq. (13)). The comparison is done based on two properties of these intervals: coverage and median length. With regards to coverage, the empirical values obtained in the simulations are close to the nominal value, being slightly more accurate for the bootstrap method. However, the length of the bootstrap confidence intervals is larger.

C.5 Runtime analysis

Wecompare the different estimation methods and baselines presented in Fig. 4 in terms of computation time. For this purpose, we measure the total time needed to compute an estimate 50 times (to obtain

0

1

5

20

80

240

1e+02

1e+03

1e+04

1e+05

Sample size

Time (s)

oracle

causal

reduced

noadj

noadj*

wadj

wadj*

0.0

0.1

0.2

0.3

0.4

0.5

0

5

10

15

20

Position x

qn

(Y=1|do(X=x))

Reduced

NoAdj*

Oracle

Avg click proportion

more accurate time measurements) with each estimator. This value depends on the implementation of each method and the equipment used for the measurement, so we focus on the comparison between the methods and not on the specific time values. Fig. 11 presents the computation time for each method as a function of the sample size. The proposed estimators have a computation time larger than the baselines, but this difference is marginal for the reduced estimator, staying below one second for this runtime analysis. The time is larger for the causal estimator due to the optimization procedure needed to calculate the estimate.

D Comparison of the effect of different ranking positions on user's choices

We estimate q ( Y = 1 | do( X = x )) for different values of the position variable x ∈ { 1 , . . . , 20 } in a unique fixed target domain, that is, the probability of clicking in a fixed hotel (we consider the hotel with target domain ID equal to 1 in the experiment in Fig. 5) under the atomic intervention of fixing its position to x . We again compare the reduced estimator with the oracle estimator. Fig. 12 shows the results. As before, there is an overlap between the confidence intervals for the reduced estimator and the oracle in all cases, except for those positions x in which the position X = x was not observed in the randomized dataset (positions 5 and 11), so it is not possible to calculate the oracle estimate. All the confidence intervals are at level 0.95 and the ones corresponding to the oracle and no adjustment estimators are calculated from the exact test for binomial data using the default binom.test function in R.

The horizontal line indicates the average proportion of clicks on that hotel across all positions when they are randomized; this allows us to see for which positions x the probability of clicking on the hotel under the intervention do( X := x ) is larger or smaller than the average value. The confidence intervals constructed using the reduced parametrisation and the oracle corresponding to the positions X = 1 and X = 3 are completely above the line representing the average proportion of clicks on the hotel across all positions. Therefore, we conclude that the probability of clicking on that specific hotel when its position is fixed to 1 or 3 through an intervention is significantly higher (at level 0.95) than the average proportion across all the positions.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of estimating causal effects in a multi-domain setting. The authors leverage proxy variables to recover causal effects and propose an estimation method along with its theoretical properties. Finally, they present empirical evaluations and an application to demonstrate the practical utility of their approach.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n\nThis paper is mathematically rigorous.\n\n**Waekness**\n\nMy concern is that the applicability of your result in practical settings may be substantially limited. (I will raise score if my concern is wrong.)\n\nQUESTIONS:\nMy concern is that the applicability of your result in practical settings may be substantially limited.\n\n1. Your identification result states that the causal effect can be recovered from (W, X, Y) in the source domain and W in the target domain.\nAre there any typos? I believe it should be (E, W, X, Y).\n\n2. Your identification result appears to contradict my intuition. The causal effect depends on U, W, and N_Y, where both U and W follow different distributions in the source and target domains. It is therefore not intuitive that the causal effect could be recovered using only W.\n\n3. The source of my concern lies in your specification of E and U. Your setting assumes that all factors responsible for shifts in the distribution of U across domains are fully observed through E. This is a very strong assumption, and if it is indeed being made, it should be stated explicitly and clearly in the text.\nIf N_U differs between the source and target domains, then your identification result does not hold. Is that correct? If that is the case, then the applicability of your result in practical settings may be substantially limited.\n\n4. In relation to the concerns mentioned above, I also have reservations regarding the applicability of your results in practical settings.\nIn your application, E represents the hotel ID. Can all factors responsible for shifts in the distribution of U truly be captured by E alone? I do not think so.\n\nConsider the following setting:\n\nE: Hotel ID\n\nN_{U1}: Customer’s age\n\nN_{U2}: Customer’s gender\n\nN_{U3}: Customer’s occupation\n\nN_{U4}: Hotel location\n\nN_{U5}: Quality of hotel service\n\nThe variables N_{U1} through N_{U5} may vary across different hotels. Consequently, the validity of your results appears questionable under this setting.\nTo ensure the validity of your results, it is necessary to observe all factors that account for shifts in the distribution of U, such as N_{U1} through N_{U5}.\n\n5. Is it necessary to specify e_T for estimation? In the application, E represents hotel IDs, and the results change when the target hotel ID e_T is replaced. This behavior is quite strange.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies a setting where data has been collected from multiple domains and we are interested in estimating the interventional distribution in a specific target domain. In this setting, unmeasured confounding is allowed but to our help we have an outcome-related proxy. The authors propose two novel estimators which require information about the treatment, outcome and proxy in the source domains and only the proxy in the target domain. They prove the estimators are consistent, and asymptotic normality for one of them. In a set of experiments, they show how the proposed estimators are able to reduce the bias from the unmeasured confounding relative to a set of baseline estimators which ignores the confounding or adjusts for the proxy as if it was the confounder.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n- The paper is very well-written overall and has an easy structure to follow.\n- I liked the prior works section in the beginning and it helped me understand the context of this paper.\n- The empirical results seem strong, in particular the real data experiment.\n\n# Weaknesses\n- It is not clear to me why WAdj and WAdj* are missing from the real data experiment, I would apriori have thought that it would make more sense to compare to than the NoAdj and NoAjd* since it showed a lower absolute error in the other simulations.\n- While the paper generally does a nice job mentioning relevant related works, I feel like it is overlooking important work on e.g. data fusion in causal inference. I understand that the domain adaption papers are closely related, but there also exists a rich literature in causality on the multi-domain setting as well. It would be great if the authors also could relate their setting and method to these papers; I've added some references as examples.\n\n#### References\nBareinboim, Elias, and Judea Pearl. \"Causal inference and the data-fusion problem.\" Proceedings of the National Academy of Sciences 113.27 (2016): 7345-7352.\nDahabreh, Issa J., and Miguel A. Hernán. \"Extending inferences from a randomized trial to a target population.\" European journal of epidemiology 34 (2019): 719-722.\nColnet, Bénédicte, et al. \"Causal inference methods for combining randomized trials and observational studies: a review.\" Statistical science 39.1 (2024): 165-191.\n\nQUESTIONS:\n1. Why is not WAdj and WAdj* not included in the real data experiment?\n2. In the real data experiment, for some cases the reduced estimators disagrees with the oracle and the NoAdj is actually closer to the oracle, for instance target domain id 3 or 11. Do you have an hypothesis for why this happens in this particular cases? I wonder if any of the assumptions you make are particularly violated here.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-25"}

comments: SUMMARY:
The paper proposes an approach on identifying and estimating the causal effect of $X$ on $Y$ in the target domain, by using the proxy variable $W$ of latent confounder $U$ affecting $X$ and $Y$ in multiple-domain settings. Also, the work demonstrates its application through experiments on real-world and synthetic datasets.

STRENGTHS AND WEAKNESSES:
* Strengths
1. Identifying $q(y|do(x))$ with $W$ does not require a fully specified causal graph.
2. The work provides (economical) estimator of $q(y|do(x))$ and show that those are consistent (Propositions 2 ~ 4). Asymptotic confidence intervals for $q(y|do(x))$ is also given (Eq. (13)).
3. Experiments performed over real-world and synthetic datasets show effectiveness of the proposed estimators.

* Weaknesses
1. The paper does not seem to provide a method on finding any valid $W$ if one exists (in polynomial time), or a method on enumerating all valid $W$ (although it may be a different research problem).

QUESTIONS:
* Suggestions

It could be useful to have a general runtime analysis, in a form of $O()$ analysis (if applicable), or across different controlled variables (e.g., $n$) and compare the proposed estimators against other baseline estimators, rather than on one specific instance as shown in Footnote 5, page 7.

* Questions

Is the identification formula shown in Eq. (4) of Theorem 1 complete? In other words, is it the case that $W$ is the proxy variable (as defined in Assumption 1) if and only if $q(y|do(x))$ is identified with Eq. (4)?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider the problem of causal effect identification of variable $X$ on $Y$ in the target domain under the presence of an unobserved confounder and proxy variable $W$ from multiple-domain observations (source domain). In their setting, the SCM is the same across the domains except for the equation for the unobserved variable $U$ that depends on the environment. The main estimand in this work is $P(y|do(x))$ in the target domain, that is  by knowing it one can identify Average Treatment Effect or Conditional Average Treatment Effect.\n\nUnder the specific assumption on the rank of matrix $P(W|E, x)$ and domains of variables $U$, $W$, $E$, authors propose an identification result for  $P(y|do(x))$. Further, they proposed two estimation procedures based on different decompositions of $P(y|do(x))$. Finally, the authors propose the theoretical results that the estimators are consistent.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nThe authors consider an important problem of causal effect identification of $X$ on $Y$ under the presence of a latent variable $U$ and one proxy variable $W$. To enable identification results, authors assume access to the observational data from multiple environments. Access to the observational data from multiple environments substitutes the need for knowledge of the distribution of $P(W|U)$ as was done in Kuroki and Pearl [12]. Further authors propose two estimation procedures based on different decompositions of $P(y|do(x))$ and proved that the proposed estimation algorithms are consistent.\n\nAlso, the paper is well-written.\n\nWeaknesses:\n- To enable identifiability of $P(y|do(x))$ in the target domain, the authors require that the number of domains $k_E$ is bigger than the number of realizations $k_W, k_U$ of variables $W$ and $U$, respectively and additionally $k_W$ should be equal to $k_U$. This assumption is unrealistic, and almost never happens in practice. Therefore, I am not sure how significant the result is.\n- How does the estimator that is presented in Section 4.1 relate to Theorem 1? \n- Is the identifiability of $P(y|do(x))$ in Section 4.1 also requires the assumption 1?\n- The assumption that only the equation of unobserved variables changes across the environments is also seemed to be too strong and rarely observed in practice.\n\nQUESTIONS:\nSee Strengths And Weaknesses.\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-05"}