Transferring Causal Effects using Proxies
Authors: Manuel Iglesias-Alonso
Felix Schur
Julius von Kügelgen
Jonas Peters
Venue: NeurIPS 2025 poster ()
Keywords: causal inference
domain shift
unmeasured confounding
proxies
identifiability
causal effect estimation
Primary area: probabilistic_methods
TL;DR: We identify the causal effect in a multi-domain setting with unobserved confounders using proxies, given only measurements of the proxy in the target domain, and provide consistent estimators.

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
Manuel Iglesias-Alonso" ETH Zürich
Felix Schur!
Seminar for Statistics ETH Zürich
Julius von Kügelgen! Seminar for Statistics ETH Zürich
Jonas Peters
Seminar for Statistics ETH Zürich
Abstract
We consider the problem of estimating a causal effect multi-domain setting. The causal effect of interest is confounded by unobserved confounder and change between the different domains. We assume that we have access to proxy of the hidden confounder and that all variables discrete categorical. We propose methodology to estimate the causal effect in the target domain, where we assume observe only the proxy variable. Under these conditions, 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 real-world example studying the causal effect of website rankings on consumer choices.
Introduction
Estimating the causal effect of a treatment or exposure X on an outcome or response Y of interest is core objective across the social and natural sciences [1-4]. However, major obstacle to such causal inference from observational (passively collected) data is the existence of unmeasured confounders U. i... unobserved variables that influence both the treatment and the outcome. For example, suppose that we wish to study the causal effect of new experimental drug X survival Y. Since such drugs tend be administered only severe cases, i.e., to patients 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 measured and correctly adjusted for [6].
Since difficult rule 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 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 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.
Not with ETH anymore, but most of this work was done while MIA was student this university. Correspondence igalonsomanuele gmail.com. Joint supervision.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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E
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U
W
X
X
(a) Source domains
(b) Target domain
Figure 1: Causal effect estimation in unseen domains using proxies. We seek estimate the causal effect of a treatment X on an outcome Y in the presence of an unobserved confounder U. The main learning signal takes the form of proxy measurements W of U. Moreover, we observe data from multiple domains or environments E, which differ through shifts in the distribution of U. (a) In the available source domains, we observe E, X, Y, and W. (b) In the target domain (E = er) for which we aim to estimate the causal effect, only W is observed. We prove that the available data can suffice identify the target interventional distribution Q (==))) pol pdo(X: K:zz,B:err
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 may provide sufficient information 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 estimate the confounder explicitly but instead seek devise more direct estimation methods. In contrast to literature on transportability and data fusion [e.g. 19-21] where identifiability is determined based purely 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, observe a single proxy W of U which is independent of X and Y given U (e.g., resulting from non-differential measurement error; and both U and W are discrete [22-24]. J 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, i.e., the conditional distribution TP( (PWI 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 the outcome.
U
W
inset figure),
Learning from multiple proxies. As shown by Kuroki and Pearl [12]. PUW(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 (for- malised via invertibility of certain matrices of conditional probabilities). Miao et [27] extend this result the more general setting in which W
X
Y
can influence Y (where P(WIU) (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 [3 1]. or deep learning [32].
Proxies for domain adaptation. Alabdulmohsin et al. [33] use proxy methods for single-souree unsupervised domain adaptation, i.c., 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 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 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 multiple, sufficiently diverse source domains are available (see Fig. 1a) and both W and X are observed in the target domain. Prashant 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.

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E
W
E
W
Z
X
X
Figure 2: Examples of causal graphs satistying identifiability conditions. Whereas the paper mostly focuses on the scenario from Fig. our identifiability results hold for broader class causal models including additional covariates Z and unmediated covariate shift via E + X, see Appx. B.
Learning from multiple domains. Causal approaches domain generalization [e.g.. 36 36-40] typically seek to discard spurious features and learn one or more domain-invarisn 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 in the presence of a discrete unobserved confounder U, given 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 Fig. 2. $ 3.1, and Remark in Appx. B for 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 al. seek estimate the conditional mean EE X x] under the target distribution Q our focus is on the interventional distribution Q(Y do(())) Moreover, we only need to observe W in the target domain, as shown Fig. lb; 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 effect of X on Y may differ across domains due to shifts in the distribution of U (even though the joint causal effect of and on invariant [e.g.. 36]). For ease of presentation, we focus on setting in which all variables are discrete. We discuss some extensions to continuous variables in 3.1 and $
[13] is that in our case, the causal
The remainder the paper is organised follows. First, we formally state the considered problem setting ($ 2) and establish sufficient conditions for identifiability ($ 3). We then propose two es- timators 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 (Y))))()) is identified (Thm. 1) if the available source domains are sufficiently diverse (Asm. 1). Furthermore, 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-ade adjyyaddjsttment baselines (Fig. 4) and that the derived confidence intervals have valid coverage and decreas- ing length as sample size increases.
2 Intervention distributions under domain shifts using proxies
Data generating process. Formally, assume that the data generated by structural causal model (SCM) [43, 44] C = (S, xP)) with the following collection S of assignments:
E (NE)),
U = fo (E,Nu), == (x,,
W = fw (U, Nw),
(1)
X = Nx),
3

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The noise variables N := NN, Nw,NN, Ny) follow the joint distribution PN and are jointly independent. The SCM induces the causal graph 9 shown Fig. 1 and a distribution over the random variables (F,,U , W, X, Y), defined as the push-forward of xPx via (1), which we denote by pC or P. As proved, e.g., by Lauritzen et al. [45], this induced distribution Markov with respect to (w.r..) g. (Since our work does not consider counterfactual statements, one could also use causal graphical models and assume that the observed distribution Markov w.r.t. the graph in Fig. ((a).)
Generalization. For sake of accessibility, we focus our presentation on the setting from Fig. 1. However, our identifiability results directly apply to broader class of causal models (that may contain additional covariates Z), whose induced causal graphs G satisfy the d-separation [9] statements
XX,
W Ll 9 (U,)),
1 o Y L 6x XI(
where Gx and Cx 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 more detailed discussion. We believe that our estimators and inference results can also be generalised to these settings analogously.
Domains. We consider kg source domains. The variable E is categorical domain indicator taking values in {ei...., e.g . The labels e; are arbitrary and serve only to distinguish domains. We assume no structure or prior knowledge about the domains. The target domain corresponds to E F= er (not necessarily contained in {eI,.... Ckg }) and is described by the distribution pC:do( which we denote by Q 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 discrete random variable V by supp(V) {ei,... Dky so, e.g., supp(X) {=I,..... Tax J. We assume that pc has full support over (E, U, W. X. Y). i.e. s U, X, X,)) = supp(E) x supp(U) x supp(W) x supp(X) 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(yle, PY = y E = e, X = x) or g(w) = QWW
.......,(a,,, p(ak T) and conditional pmfs as matrices, B) = (p(a))6, 16,) ))j with the jth row given by P(a [) and the 5th column by P(A(b;). For additional discrete random variables C and D. we write P(a = (ofalb,.)...... ((ab:,,g() target domain, we use Q instead of P. The distributions induced by the last four structural assignments of the SCM C' in Eq. (I) are thus described by the matrices (P(U)E), Q(U), P(W PUX and (P W, w ) (x. () respectively.
Matrix notation. We write marginal pmfs as column vectors, P(A) := (p(a1).... .....
and P(a] B, C, d) = (p(a B1 (1,d) (.),,..,,p(,,, [ Br d) d) In the
Data. As illustrated in Fig. 1, we assume that we obtain a sample of fare independent and identi- cally distributed (i.i.d.) source realizations (Fi, Wi, X1, Yi),..,, (Fmmee) W Noggy X Pare? Y Yare) from Po:= PC (E.) and nlgt == n Darc i.idd. target realizations W mac +1...., Wn from Q.
Estimand.
The goal is to estimate
((yldo(z) = Q(Y = y)do(X := x)),
(2)
which, by slight abuse of notation, we call the causal effect from X on Y (in the target domain er). Since the domain is allowed to affect the hidden confounder, the causal effect in the target domain is, in general, different from plyle, do(x)) when e E (I,..., is a source domain.
3 Identifiability
We now show that (2) is identifiable from Po and Qw. More formally, under suitable assumptions, we prove the following statement: if C₁ and C2 are of the form of (1) such that IPC: Iplss and Q6 Q then qft (y)do(=)) gf (y)do(=))) The key idea of the proof (inspired by ideas presented by Miao et al. [27, Section 21) is to consider the covariate adjustment formula [6, 46, 47]
kw Σ
(l)ld(=))
a(ylu, ( x) g(u)).

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Clearly, U is unobserved, so we cannot exploit this equation directly. Instead, we will rewrite Q(y)U,z) so that it depends on 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 PUW which is guaranteed by the following assumption:
Assumption 1. For all x E supp(X), we have rank(P(W[E)) x)) 2 ky.
Intuitively, Asm. 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 = I. As we show in Prop. 6 in Appx. B. Asm. implies that the proxy W can be transformed in such way that the matrix P(W|E E, x)) has linearly independent rows. We therefore assume this without loss of generality (see Appx. B.1 for more detailed treatment). Finally, this implies that the matrix P PWWE, P(W [W|E,a, E, is invertible [48], so right pseudoinverse P(W|E, x) exists."
Similar assumptions about the rank of conditional probability matrices are common in -osse identifiability results [e.g. 12, 13, 35].
We are now able to state the main identifiability result. Its proof, together with other proof' of this work, can be found in Appx. A.
Theorem 1. Under the data generating process described in $ 2 (ignoring the paragraph "General- ization") ") and assuming Asm. 1, we have for all x E supp(X) and y E supp(Y)):
((yldo(=)) P())E,)) P(W[E,, x Q(W).
Therefore, (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 er is identifiable.
Remark 2. If. within our setting, Asm. does not hold, identifiability is, in general, lost (see Appx. B.2 for a counterexample) Thus, in this sense, Asm. I 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) X and are not discrete but continuous. In this case, P(yl E,x) a vector of evaluations of a conditional probability density function (pdf), for example. Furthermore, we do not assume faithfulness, and Thm. still holds 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 in Appx. B.4 for characterisation of the necessary graphical assumptions.
Incorporating covariates/observer confounders. In practice, we often observe additional covari- ates that we want to include the model. For the case of observed confounders of X and Y, we show that the (target-domain) conditional causal effect ()r)) =) given Z is identified if Z observed in both the source and target domains and subject to an assumption similar to Asm. for the matrix P(W 2). see Asm. 2 and Thm. 8 in Appx. B.4 for details.
(Conditional) average treatment effects. Thm. directly implies that the (target-domain) average treatment effect (ATE), given by [[ == 1)] [[ 0)] under Q. is identifiable, too. Moreover, due to the identifiability of the causal effect conditional on observed confounders Z we can also identify the (larget-domain) conditional average treatment effect (CATE), given by E|Y [X = 1), Z 2] E[Y do = 0), = 2] 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 discrete proxy variable that satisfies Asm. 1, see Asm. 3 and Prop. 10 for details.
4 Estimation and inference
Having established the identifiability of ((v)do(x)), we now construct estimators for this causal effect. Specifically, propose two estimators based on different decompositions of the causal effect,
GGivnn matrix A E mmx s.t. AAT is invertible, its right pseudoinverse is defined A == A (AAAT))-
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denoted by Ic, (v)((()) and 0 (y]do())) respectively. The subscript indicates the sample size used each case. By a slight abuse of notation, we use the same symbols denote the estimators and their values evaluated on specific dataset. Code implementing both estimators, along with the experiments, is available on https:// /tte:://gtttbb poom/mamueligga// proxy- intervention.
4.1 Causal parametrisation
As discussed in $ 2, the structural assignments in Eq. (1) can be represented by the matrices P)U Q(U), P(W[U), P(XJU) and (P(y)U, () xxeupp((X)
For our first approach we explicitly parametrise these matrices, thereby capturing the underlying causal mechanisms. Prop. 3 shows that the estimand (y)d)(x)) can expressed in terms of subset of these (unknown) matrices.
Proposition 3. For all supp(X) and y supp(Y) we have
((vldo(=)) diag ( P(W)U)) Q(U).
(5)
We now proceed by maximum likelihood estimation. Consider a parameter 0 whose components are the entries of P())E), Q(U), P(WIU), P(X and {P(y]U, W, tt) (nesupp(X)) (y By an abuse of notation, denote the entries of 9 by the corresponding probabilities with subscript 8, i.e.
= (po(u)lea))..... (.otull))).....P(((...... lekg). ... qo (ukv ), (wr [u1), ......... pok(ku |uku
(6)
po(2ilu1)) (u1), (x (uke ), pe (y U1, w1, æ1), pe (yky luke wkw Ikx))
The entries are restricted to be non-zero, each column of E) Q(U), P(W and P(X (U) sums to 1, and similarly Σes p[y: [1, w, 1 for all E supp(U), w E supp(W), and supp(X). To ensure these constraints are satisfied without explicitly enforcing them during optimisation,
real-valued parameters (representing logits) and transform them into valid pmfs using the softmax function.
Naturally, the joint distributions of the observed variables Fw.x,y|1 LE in the source domains and Qw in the target domain are also parametrised by 0. Specifically, for all i E (1,...,,y k. E (1,..., kw }. E (1,...,ke kx). and E {1,..., kÆ), have that
kp
Polyi Fs wyle)) =
po (y Wy, x&) po po(w,lu,) po po(uul))))
go (w;) = Σ (wy (u,) (ur),
where both of these equations hold for the true (conditional) pmfs. The (conditional) likelihood over the observed variables is thus given by
by Ax kw k
kw
L(0) = IIIIIIIIII po (Vi, T,, Wylee) Ma w.
II j=1
go (wcy) m((ss)
where n1ys, Si,, Wj, ei) denotes the number of observations for the tuple (yi, Ts. Wj, et) in the source domains and n n(w)) denotes the number of observations of wy in the target domain.
For all supp(XX) and y supp(Y), define the function
92.y Ju.y (0) diag PolylU, Wx (O(IU) Qo(U),
(8)
where the entries of these matrices components of 8. We denote by l the true value of 6, so by Prop. have that (0,) ()))d(())). Therefore, if denote by 0,, maximizer of
'The diag operator applied to square matrix A = (Ars) E R is defined as diag(A) (A) = (A11, A2,2. Ann) _e".

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the likelihood function in Eq. (7), we define the plug-in estimator of the causal effect in the causal paramctrisation for all supp(X) and y sY)p(Y) qc.n (c.n())(((z) Ir, ge,,y (On).
Even if the estimator 0n does not converge, its induced probabilities over the observed variables concentrate around the true values, and the estimator 9c.. (v)(((z)) is consistent.
Proposition 4. Suppose that Asm. holds. The estimator Ic., (v/do(x)) is consistent, that is, Ic (y]()))o((=) ((uld((r)) in probability as Thare: Righ
4.2 Reduced parametrisation
Although ((y)do(z)) identifiable, not all the components of the vector that we optimise in the causal parametrisation can be identified from the observed data - the estimator is based on an overparamettisation of the problem. Our second approach estimates only the components necessary to compute Eq. (4), To do so, for all x E supp(X) and E sYpp(Y),, we consider one long parameter vector Ma.y describing all entries of Q(W,er) q(er). P(W, z, E), Ply. x, E). and P(x, E) (see Appx. B.3 for details). Given an 722y, we can then not only form the matrices Quni (W,eT) eT). Ans. (eT), P 2W, Pu.., (,, E) and P..., (z, E) but also Qn,,,W), Thes (WIE, x), and (y)
which we use to define an estimator based on Eq. (4). The
components of F2 are probabilities, so we can estimate their true value by empirical frequencies". that nw...n = n -E V2 where, for all E {L....,.2) nJ.
may =
I(W; w1, P.g = eT)..... 1(W; = wkw 1, F. = =TT) [(F = eT),
I(W; wi; X; x F. =)), 1 IW; Wkw 1, X; I., Fb ekg);
(9)
1(Y; - y, = x, Es = 1e)....,,
1(Y; =
Xi = x. Ei = Exg),
I(X: x, E = ei)...., 1(;; = x, Es = ekgg)
Then, for example,
1, EI)
P T (W[E,z))
(10)
EE 1(X, = x, E, =e)
For all supp(X] X) and y supp(Y), define the function
h: Ma.y - h(nz.))) PR... E,)E, x) Po.., (W|E,2) x)) Qn (W),
(11)
and define the estimator the causal effect the reduced parametrisation as
(r.((v()((z) = h(fea.n))
(12)
The following proposition proves consistency of @n.n (o)))( and provides confidence intervals. Proposition 5. Suppose that Asm. 1 holds. For all E supp(X) and y E supp(YY), the estimator TR. (y)do(z)) defined in Eq. (12) is consistent estimator of q(y)do(x)) Furthermore,
Vn Gz.
(g.do()))(())) ((vldo(z))) D, N((0
where
2. Vh((fagg.))) Euy Vh(TT..)))
with Exy being the sample covariance matrix of (nh..n..... ........)))) nr
"To simplify notation (for inference), (Fi, Wi, X1,Y).. Yi).... (En, W., X.., Yn) with E, E (ei...., Ckg U fer]: whenever E, = er, the entries of X and Y are missing. This way. n is fixed, while Plre E I(E, o er) and Negt := Σ, ( = er) are random. For all j {1...., kw } ((w,)) = g(wyle (er) can now also be considered conditional probability. 'Strictly speaking. we consider a slightly modified version: on the events where the matrix in Eq. (10) does not have linearly independent rows, we change N...... to ensure that it does. This happens with probability vanishing zero (see proof of Prop. 5) but ensures that Eq. (12) always well-defined.
section we model the sample as an i.i.d. sample
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Causal
Reduced
log( K (W[E,x)))
0.8
Absolute emor
0.4
0.2
0.0
2.5
5.0
7.5
10.0
5.0
7.5
10.0
log( K Pn (W]E,x)))
Figure 3: Absolute estimation error increases for near non-invertible P(W[E,z). For M distinct data generating processes, we draw N= datasets each, consisting of k==2 source domains and n-20 000 realizations. The errors for both estimators increase with the condition number of the matrix P(W|E, E, x) (or its estimated counterpart), measure of non-invertibility.
By Prop. 5, pointwise asymptotic confidence intervals for q(y]do(z)) at level (1 a) are given by
()n(v))((z))
T2.U Vn 21 3. (y.n(y/)((z)))
Z.
21
(13)
n
where 2i denotes the --quantile of a standard normal distribution.
Both Tn.n (o))) and the bounds of the confidence interval can lie outside [0. 1] because the data-generating process adds further constraints on the combination of Ply F,, x)) P(W|E,,) and Q(W) that are not reflected by the estimator. To reduce bias and variance, we thus clip the estimate IR. (z)) and the confidence intervals to [0, 1].
Simulation experiments
5.1 Data generation
The data generating mechanism can described using the matrices P(U)E), QUU), P(W]U), PUX[U) and UP W. different sets of such matrices. For each choice, we generate N i.i.d. data sets with sample size N. Appx. C.1 provides details on how we choose the matrices and the size of the causal effects.
We randomly and independently generate M E N
5.2 Point estimation
We first compare the point estimates obtained from the two proposed estimators. To do so, fix values r E supp(X) and supp(Y) and compute. for several data sets, the absolute estimation errors (y)oz)))) q(y)d)(=)) and ())zz)))
(z))
In our setting, we can assess Asm. by considering the condition number x( x(P(W) E, x)). When ((P((WEE, x)) is large, the almost-violation of Asm. By Eq. (4), we thus expect that the performance of âc.. (v)(()) and TR (z) 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 x(P(W|E, that H( (()) R ((P(WFF, so we can estimate x directly from the data. Whenever R(P(W [E,)) is large, we can flag the corresponding estimates as potentially unreliable.
of P(W]E, are almost linearly dependent, indicating
It also shows
The results suggest that both estimation procedures perform similarly in terms of absolute estimation error, with 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 1000 and n 100 000 supporting the theoretical consistency result. 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, 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 i..d. realizations of the target intervention

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0.3
Absolute error
0.2
0.1
0.0
Oracle
Causal
Reduced
NoAdj
NoAd
WAoj
WAd]+
Method
Figure 4: Comparison of the different estimation procedures. We use the parameters kg = 3. n = 20 000, .M = 10, and = The oracle presents the lowest error due to the use of the data from the intervention distribution. The reduced and causal parametrisation estimators show similar distribution of the error, with their medians close to zero. Their performance is better than that of other baseline methods, whose absolute estimation error distribution is shifted towards larger values. The estimators that use the distribution of (X, Y) in the target domain (with *) obtain better results in terms of the absolute estimation error than their pooled counterparts.
distribution (using data not available to other methods); the no-adjustment estimator, which estimates the distribution Q2 by Ipc (NoAdj) or 2r (NoAdj"; using data not available to other methods); and the W- 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) from the target domain (WAdj": using data not available 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. proves asymptotic normality of (the unclipped version of) (y/dd(/o 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, 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 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 the nominal level but comes the cost of longer confidence intervals.
6 Application: Hotel searches
We consider the Expedia Hotel Searches dataset [50], in which each observation corresponds to 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 the position the hotel the list shown to the user (X) on whether the user clicked on the hotel to obtain more information (Y). We regard it plausible that there an unmeasured confounder U of 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 are comprised of searches in which a given hotel ap- pears. Tab. provides 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 cho- 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

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Table 1: Variables included in the hotel ranking real-world application.
Variable
Description
Values
E
Hotel unique ID
Integers
X
Position of the hotel in the search results
{I...., 40]
Y
Whether the user clicks on the hotel Price range per night USD
0: No click; Click
{00 75). (75, 125), (125, 175), (175, 225), (225, oo)
Go -Y-Y- -1)
Reduced
NoAdj*
0.2
NoAdj
Oracle
WAdj"
WAdj
0.0
Target Domain ID
Figure 5: Reduced parametrisation estimator compared with four baselines and the ground truth for a real dataset. We compare the estimates of the causal effect g(Y = 1|do(X =1) using the reduced estimator and the no-adjustment baselines with the oracle confidence intervals using 25 source and 18 target domains. For both of them, there is overlap between their confidence intervals and the ones from the oracle in all the target domains. The reduced estimator yields estimates closer to the oracle in more target domains and slightly smaller confidence intervals than NoAdj*
the ranking the customer choice, but only considers the part of the dataset which hotels are ranked uniformly at random. This is different from our setting, since we estimate the causal effect q(== = 1/do(X X = =)) in 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 diffferent queries the randomized dataset. This ensures that we can obtain reasonable estimates using the oracle, which we consider the ground truth this experiment. queries for the source and target domains, respectively. Fig. 5 compares the confidence intervals obtained by the reduced estimator with the oracle ones. 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 eиrоr of 0.044 (Reduced), 0.051 (NoAdj), 0.080 (NoAdj*), 0.053 (WAdj) and 0.075 (WAdj*)) and contidence intervals that are on average shorter than the ones from the no-adjustment baseline (median length of 0.14 versus 0.17). In second experiment, causal effect gY = 1 do x)) for different values of x E {1...., 20} in fixed target domain to study the causal effect of the different positions the clicks of the users. Again, the confidence intervals produced by the reduced estimator overlap with all oracle ones, see Appx. for details.
total, we obtain ca. 64 000 and 000 observations from 8400 and 4000
estimate the
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 as an interesting setting, too; here, the ideas discussed by Miao et al. [13] and Tsai et al. [35] could prove helpful.
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Acknowledgments
The project that gave rise these results received the support of fellowship from "la Caixa" Foundation (ID 100010434). The fellowship code is LCF/BQ/RUU2311200008
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20

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Appendices
Table of Contents
A Proofs
22
A.1 Proof of Thm.
22
A.2 Proof of Prop. 3
22
A.3 Proof Prop.
23
A.4 Proof Prop.
24
B Details and extensions of theoretical results
25
B.1 Relaxing Asm. 1
26
B.2 Asm. I and identifiability
27
B.3 Reduced parametrisation
27
B.4 Additional covariatc/observed confounder
28
B.5 Continuous proxy variable.. ................................
29
C Implementation of the experiments and further simulation results
30
C.I Data generation
30
C.2 Optimization.........
30
C.3 Point estimation
30
C.4 Coverage of confidence intervals
32
C.5 Runtime analysis
32
D Comparison of the effect of different ranking positions on user's choices
33
21

---

A Proofs
A.1 Proof of Thm.
Theorem 1.
Under the data generating process described in $ 2 (ignoring the paragraph "General- ization") ") and assuming Asm. 1, we have for all supp(X) and y supp(Y'):
g(yldo(æ=) P(ul a)) P(W x)) Q(W).
(4)
Therefore, (E, W, X) is observed in the source domains and W in the target domain, the causal effect of X on in the target domain er is identifiable.
Proof.
The global Markov condition holds in SCMs [e.g., 43, 45], so have
X,Y,W W 1 LEU
X W U.
We then have
Ply E, P(ylU, E,)) P(WIE, P(WIU) 2()
(14)
(15)
For every pair of matrices A R= and B TR"XP, the following inequality holds: rank(AB) S min (orn(rnn)))))) A), rank(B)) [52]. Using this inequality in Eq. (15), we get
2 rank rank(P(W [, [E)))) kw, where the last equality is due to Prop. 6. The matrix P(W]U) has dimensions kw x ku, hence min(kw. ku 2 (ank(P(W(U)) 2 kw Using Eq. (16) and the inequality kw rank ((W = kw = ky. Therefore, P(W]U) an invertible matrix and this can be used in Eq. (15) to obtain
rank(P(W))
(16)
ky derived from Asm. 1, conclude that
P(UIE x) = P(W)))) P(W(E,,)
The substitution of this expression Eq. (14) yields
P(ylF,x) x) = -()(U, P(W]U)- F(W)F,
and therefore
P() P... P(W(U).
(17)
The adjustment formula (3) states that (())d(=)) (, Q(U) and the equality Q(y x)
Pyy[U, x) holds because of modularity or invariance [c.g.. 53]. Combining Eqs. (3) and (17), we get
((yldo())) = P(ylU,,) Q(T) = P(y) F, P(W E, Exx P(W)U)Q(U).
Using the law of total probability for the last two terms, we get our final result
((yldo(z)) = P,) P(W] x Q(W).
(18)
A.2 Proof of Prop.
Proposition 3. For all supp(X) and supp(Y) we have?
((vld((z)) = diag P(u|U, w. P(WIV)) Q(U).
(5)
Proof. By modularity, we have Q(y[U,x) U.) The statement then follows by the covariate adjustment formula (3):
ky
(luld((z)) = P(ylU, Q(U) = Σ p(ylu,, 2)) g(ur)
ke
kow
plylu, Wj, x) p(usplls (ur) a(u,) = diag P(ylU, w, P(W())) Q(U).
L
'The diag operator applied to square matrix A = (Acs)isssSn E Raxn is defined as diag(A) = (A11, A2,2. Ann) me..
22

---

A.3 Proof of Prop. 4
Proposition Suppose that Asm. holds. The estimator Ic.. (c.n(o))((z)) is consistent, that
(c.(()))(z)) ((y)do(x)) probability as Tere: Ptgt > co.
Proof. We consider a parameter whose components are the joint pmfs that appear in the likelihood function in Eq. (7)
# (yr 21, wile wilee)...... ..... Py (yky 2kx Wkw (eke) 9u (w1), ..., .4 (Wkwe) ))
and denote by /o its value. Moreover, denote by f f(0x) the vector obtained when calculating these joint pmfs from the optimal vector On. Hence,
f(...) (Pa, (yi, Iı, wlle)), ................................ Pä. (Jky Tax: Nkw lexx) (. qz (Wi).... .8, (Wkw )).
The conditional log-likelihood equals
kx
kw ky
kw
(1)
1=1 ΣΣΣΣ
n(v; Fas wy, ei) log(P (Ms, wy wylez)
Σ n(wy) log(a (wrg):
where we define 0- log 0 = O. (It suffices to consider the conditional log-likelihood as differs from the unconditional log- ligeeloodd only by additive term that models the marginal probabilities of the domains.) Instead Ln, we now consider
kv kx kw kg
lw
(lli Fs. et) n(ei)
n(w;) Nsgt
(x))) = Σ
log log(p,(Us,,,, wylei)
log (w)(),(w;)),
which satisfies argmax argmaxx l. (this is because &, be optimized by considering the likelihood for each domain separately, so multiplying each domain likelihood by constant 1/n(ei) or 1/ngg does change the optimal values for the probabilities). The unique value for μ maximizing both bn and e,, denoted by P.. consists of the relative frequencies
Pü. (Vis Fsz wyles) = (1 Tar Wgy
n(ei)
n3, (w;) = n(w,)
Nlgt
For all n E N,
e. (T. 2 (Q()) 2 en (140)
(19)
The first inequality is due to the definition fin == argmax Lu.. Regarding the second inequality, have
0, argmax argmax log argmax t, o.
(where the last equality is due to log L en f. which holds by definition of f'; and since there exists fo such that po f(o), we have
e. (..))) 2 e, (f(oo) = en (HO).
We have that G. Eg pn due to the law of large numbers and we use the continuous mapping theorem to get
l (n())) (n()) P, 0,
Rmin
where Pemin = min(n(e1),...... (( Digt] is the smallest sample size across source and target domains. Using the squeeze theorem Eq. (19),
(P( e, (fw)) F, 0.
Pemin
23

---

Furthermore, we have the inequality
tn (Pn) (n((((n))
2 -(
Tmmin To prove this incquality, we consider the dccomposition
kg
(1)
Σ
t (1) + t0 (1)
corresponding to the sum of the likelihoods each domain. For all {l,..., KE, T}, the vector pi is the maximizer of t so have
e (Pn) e (f(O,)) 2 0.
(20)
Therefore,
(Pr)) (f((
hx kw kg
(VVi 2s Wj,el ei) n((e)
Pa. (yl, Js: wyye ee))
kw
n(ww)) log
Ye (w,)
log
Pě (Yi, Taa wgy (0yles)
Megt
a0 (w,)
E. (Tm) -2: -e +
1 (Mn) en (f(O))) (
n(e)
7ttt
kR
<
Σ
C7 (f(Or))) + (eT (Pt)) exr (f(0))) a (pr))
Qmiin
(Pn) - (n(
Timio
where the inequality holds due to Eq. (20). Therefore, considering the Kullback-Leibler divergence, we get the following convergence in probability
kg
DKL (p (()))(()))) DKL (1)1)(()))T)) - -
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
k
11 - -000000)
2DK (p (1(1))(0)))) + 22DL (piT11 (l Ps
Hence,
f(...) E No-
Finally, we apply the continuous mapping theorem with the function Lz.v : Mu + I2.y(1a) = q(y]do(z)) (defined, for example, using Eq. (4), where all the components can be derived from the joint pmfs in μo) to get the final result:
(v)o(o()(()) P dlyldo(z)).
A.4 Proof of Prop.
Proposition 5. Suppose that Asm. 1 holds. For all supp(X) and y E supp(Y', the estimator TR ()) defined in Eq. (12) is consistent estimator of q(v)do(z)). Furthermore,
Vn
(n a(v)do(2))) 2, No(,
where
2² Vh(feenn) EEY Vh(îzag.)
with Ea.y being the sample covariance matrix of (m..g.... nr
Here D (pa., (y1, Iı, wi lei).........)........). .... Pp. (yky, Økx wkw ee1) and analogously for the other variables.
24

---

Proof. Consider the i..d. sample (vz.g) ri= obtained from {E,, W, X, YO using the definition in Eq. (9). The random vector yay has mean equal to the true parameter g. We define ye...n to be the sample mean of My.... Here, we use the asterisk to stress the difference to the estimator Te.y. which is equal to Te.y. except for when Pr:.... [ x) does not have linearly independent rows.
To prove asymptotic normality, we apply the central limit theorem to ( We use again that their mean is equal to 77.y.0 and their sample mean is the estimator Mz.... Furthermore, we denote by ,x,y the covariance matrix of va.g" obtaining
vn Tann.n - (..y10)
2, (0,
(21)
We now prove that
Vn (T.. 72, 23 (0
(22)
holds, too. Let B, be the event that Pir... (WI,,)) 2:) has linearly independent rows. We first show that IB,,,, + 1. Indeed, by Asm. 1 and Prop. 6. the population matrix 2()) has full row + 0. By continuity of events An,5) implies (Pi 2 + 0, so Pit s.n (W E, 2). I|
rank. Thus. there is a kw x kw submatrix M s.t. det P(W E, x) x)
(let us call such EW|E, x) has full row rank. By the and thus P(B,) + 1. Defining ân := Vn na and a, = Vn(f2. N,,y,0), we can now prove ån 2, N00, En,y). Indeed, for all b, we have lim P(ân < b) lim (P(@n, < bn B.) + P(ū, < bn B2))
the determinant, there is 5 > s..." [P(W] 2) M Po
law of large numbers, (AA,,))
lim (P(6 < bn B,) P(ā, S bn B + P(O (b)
n-+30
lim (P(0) < b) + (â. S (BBB))
n=+00
lim P(a,, < b), ==
which proves Eq. (22). The second equality holds because, Bas 0z and â, are identical and lim P( < b n B = 0.
The function h defined in Eq. (11) is continuous, apply the delta method to Eq. (22) to get Vnl (Tna(y)do(x) g(vldo(x))) 2, NNO,0022 22
where the variance o2, is given by
2 Vh(ne.v.)) T Ez.y Vh(na.gu).
Consistency follows by Slutsky's 's theorem.
To estimate o.. we can replace V2. by the sample covariance matrix Eman of (T n2,9) and Nz.y, by Nz.y,n which are both consistent estimators by the law of large numbers. This yields H
Vh (fz.g..)) En,yy. Vh (fz.g..).
By the continuous mapping theorem, 02 is a consistent estimator of 02 22 Therefore, using the continuous mapping theorem again, together with Slutsky's theorem, we have
(vldoo))()) - ((vld((z)) P, (00,)).. Oz,
nt
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 be relaxed.
oIl || denotes the Frobenius norm.
25

---

B.1
Relaxing Asm. 1
Under Asm. 1, the proxy W can be transformed in such way that the matrix E, has linearly independent rows.
Proposition Under Asm. for all E supp(X), there exists a W-measurable random variable W such that
kw rank(P(W]E, E, x)) = rank(P(W|EE.)
and W satisfies the independence statements (X, (X,YWW TL EIU) and (X JL WU),, and supp((E, U, W, X,,X, sEp supp(U) x sup supp(X) sYpp(Y).
Lemma 7. Let w1,.... Wn be linearly independent vectors in a vector space V, and let
Ws
with scalars X E RR. 1f -1, then
+ wT W2, Wn
are linearly independent.
Proof. Write
n
+ 01 (A1 1) w Σ X wi-
Suppose there exist scalars a, bz,..., bn such that
(v + w1) + Σ b, we = 0.
a
Substituting the expansion of v + w1 gives
n by w. 0.
af [(11 + 1)w1 X2W2 +
Wn
Σ
Collecting coefficients on the independent set [Wi,.... wn) yields the system
(A + +1 0, aa + b, = O, (2......
Since X + 1 &f O, the first equation forces = 0, which implies by _... b, = 0. Hence, the only solution the trivial one, proving that
+ W1 W2. Wn
are linearly independent.
Proof of Prop. 6. Fix a E supp(X). If Ea) kw the result follows considering W W. Assume = rank(P(W]EE,
< kw. We proceed as follows: Choose + rows of a(W),, denoted by V1...., Ur41 (with row indices denoted by o1...., such that 11- 11...., Ui+...... Ur+1 are linearly independent and e is a linear combination of the others, R such that o; = Σjær dyu Since the entries of P(W|E,2) x) are non-negative, there exists (1. r + 1} \, {i} such that d + -1 define a new random variable W == WI(W * (W,)) + Wa, I(W = ) By Lemma 7 we have that P(W)E, æ) is (kw - 1) x -gddimensional matrix and rank(P(W E x)) rank (Ank(P x)). It also holds that SE U, W, xx X)Y supp(E) x sUp)(U') x supp(W)) x supp(X) x supp(Y). Continue this procedure until all rows are independent. By abuse of notation we call the resulting random variable again W. For all random variables A, B, C and f measurable A
that is, there exist Ai..... A Art1....,
f(A) IL B C.. It therefore holds that
X,Y,W W IL LEU
X J W/U.
We have that rank(P((((E rank (PWW) 2 ky and P[W|E, has full row rank.
26

---

B.2 Asm. 1 and identifiability
We now present a counterexample showing that identifiability is, general, not possible Asm. is violated. Concretely, we now construct two different SCMs. Both of them satisfy supp(XX) : {0. } and supp(Y) := {0, 1}. We consider x 0 E supp(XX) and y = 0 E supp(Y) and are interested in inferring ( do x)) Both SCMs satisfy supp(W") == {wr, w2, w3 supp(U) = {u1, 12 Uz) and, for both SCMs, P(WIU) and P[ W,x) are such that, for all j (1,2,3).
00.23
0.3
0.2
P(WIU)-
0.46
0.6
0.4
and
P(ylU, w,, P(y
- (0.5 0.2 0.3).
0.31
0.1
0.4/
Furthermore, we choose, again for both SCMs, P(E), E) and P(XJU), 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 o, o2, and c3 the columns of P(W\U). we have
ei
10 02
10
c3
(23)
with 02 and 03 being linearly independent, so rank rank( (P((U)) 2 < ky. Using Eq. (15) and the inequality bank(AB) I (rinkaan(A)) rank(B3) [52], we get that rank(P(W E, x)) < ky, so in both SCMs Asm. is violated.
The two SCMs differ in the target distribution of U: we define
P1 (11)
0.6
P2 (P2(u1)
0.5
(U)
P1 U2
0.3
and P2(U)
P2l u2)
0.33 P2(us)
(Ug)
0.1
00.17
The induced target distributions of W for the two SCMs given by the law of total probability:
(0.248)
(W) ((W = 0.6c1 + 0.302 + 0.1es =
0.496
(0.256/
0.5ca 0.33e² + 0.17c3 P(WIU)P2(U) P2(W).
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 y(y) (())))()) for each SCM using the covariate adjustment formula (see Eq. (3)), we find that they differ:
3
a1 (y))))()))
Σ plulu,, ()p1 = 0.39
=
* (2 (o())) = Σ p(ul url x)p2(u,) = 0.367.
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, z)) exists), then Asm. must hold, too.
B.3 Reduced parametrisation
The parameter vector Vay,y should have components corresponding to the following probabilities:
(nn. (Wi, GT), ... Ins.y (Wkw -1 CT), 4nr.p (CT), Pmr.o (W E1 EI)....,
Pr.., (Wky 1, W. Okg). ), P (v. x, e1), Pre.y (y, x. Ckg Pr (Ix, E1); Pui.. (x,, ekg))
We denote the number of components of this vector by k. == kw (kw 1 1),E, so the corresponding parameter space is contained in H [0, 1
27

---

E
W
E
W
E
W
X
X
(a)
(b)
(c)
Figure 6: Causal effect estimation in unseen domains with both observed and unobserved confounders. Shown three example graphs that allow for identification of the conditional and unconditional causal effect in the presence of an additional observed confounder, see Thm. 8 for details. The black edges shared across all graphs; the green edges differ.
B.4 Additional covariate/observel confounder
We now consider the case which we have an additional observed confounder Specifically, suppose that the SCM induces one of the graphs depicted in Fig. 6.
We replace Asm. 1 with the following assumption.
Assumption The proxy vuriable is such that kw 2 ky and, for all a E supp(X) and E supp(Z), Z), the matrix P(W|E, 2, has linearly independent rows.
Theorem 8. Assume an SCM whose induced graph is of those shown in Figure 6. Let E, U, W, W,X.Y. X.Y, Z denole the random variables generated by this SCM, and suppose their joint support factorizes as supp(E, U, W, X,Y, Z) supp(E) x supp(U) supp(W) supp(X) supp(Y) x supp(Z). Suppose that Asm. 2 holds. We then have for all x E supp(X), 2 E supp(Z), and y E sYpp(Y):
(v)ld((r), x. (,W
and
((vldo()))
Σ 2, 2)
2Esupp(2)
Therefore, if (E, W, X,Y is observed in the source domains and (W, Z) in the target domain, the conditional causal effect of X on Y given = z and the total causal effect of X on in the target domain ex 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 JL EU(U, X, Z) and W (U, Z) and thus for all E supp(X), supp(Y), supp( (Z):
Ply E, x) x P(U1E,
P(W E, x, xx P(W 2).
Under Asm. 2, follow the same steps in the proof in Appx. A. get
Ply U, x, Ply 2) P( E, 2) P(W(U,,).
(24)
The target conditional interventional distribution is given by ((yldo(z), 2)
QQ 2Q(
Q U, dot) 2(
2(y(U,,,))Q(U1)
where the first line follows from the law of total probability; the second line follows the fact that
U IL Z ZZizz in the post-intervention graph Gx (formally, do-calculus rule 3); the third line follows
28

---

from parent adjustment applied to (formally, do-calculus rule 2); and the last line follows from the assumed modularity or invariance. Substituting the expression from Eq. (24) then yields
((y)do(z), =) P(yl E, x, P E, 2) P(WIU,))Q(UH)
(25)
P(yl E, P(W E,x,,) Q(W]=).
(26)
Hence, Asm. holds and (W, 2) is observed the target domain, the conditional interventional distribution q(vl do(z), 2) is identified.
Furthermore,
[lyldo(z))
E
d())do(z),2)
22
Σ
(l))do(z),2)((2)
=Esupp(2) 2)
Σ
x) Q(W)))))))())
=Esup((2)
where the first line follows from the law of total probability: the second from Z JL. X in the post- intervention graph 9x: and the last by substitution of (26). Therefore, if (W, Y, Z) is observed in the source domains and (W, Z) the target domain, the causal effect qyy do(x)) ) is identifiable. O
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 also rely on in the main paper) be relaxed in our setting. Specifically, we can also allow for (direct or unmediated) covariate shift, ie.,
X A EU, Z).
This applies not only to Thm. 8 but also (in the form X LL E 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. only makes use of
(Y 1 (U, W 1 (E, X)J}
in
g,
ULX Y while Thm. only requires the analogous statements given Z.
X
in
Gx.
in Gx
(Y EJ, X, Z), W (E,X) (U, Z)) U IL X XI Z) in 9x,
9,
in 9x.
where 9x and 9x denote the graphs obtained by respectively removing the edges coming into X and going X from G. All these relations still hold when including E + X.
B.5 Continuous proxy variable
We can extend our results continuous proxy variables W with similar ideas as the ones used in Appx. B.1. Let m E N and Vi...., Vm C supp(W)) be pairwise disjoint measurable sets such that supp(W)) = Uism V,. Define Wy = Σis is il E V,).
Assumption 3. Assume that, for all x E supp(X), there exist m N and Vi...., , Vm S sup pairwise disjoint measurable sets such that supp(W) Uigm Vi and rank((((ylE,.)))
ky.
Proposition 10. Under Asm. 3, for all E supp(X), we have for the W W-measurable random variable Wy that, for all E suup(YY),
q(yldo(z)) = P(y)E,z) P(WVI), Q(Vv).
Therefore. (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 er is identifiable.
Proof. The result follows directly from Thm. with W Wy and the observation that
X, Wy 1L EIU
X ll Wy
29

---

Causal
Reduced
logi K ((W]E,X)))
Absolute error
10.0
7.5
5.0
2.5
log( (P, (W]E x)))
Figure 7: Comparison of the absolute error for both parametrisations when having a small
sample size. Absolute error for setting with kg = 2, n 1000, M 25 and N = The error is larger in both cases than Fig. 3, even for the low values of the condition number ((P(WIB,x))
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:
Generate B(U Q(U), P(WIU), P(X(U) and [P(y|U, W, (x)ppp(())) dom. The sum of each column of E). Q(U), P(W U) and P(X has to be equal to I; we generate each column of length & from Dirichlet distribution with parameter (1....; E R the values uniformly distributed in the (h 1)-dimensional fI....,ww kw and s E [1..... kx ((,, generate the values p(N (4y wx) from Dirichlet distribu-
ran-
simplex [54]. Similarly, for all r E (1....,ke ky }.
E:= P(y: Wy. F.) = tion with parameter (1,...,1) E R The variable E. is generated from discrete uniform distribution (ei...., Ckg. er). where e1...., Cbg are considered the source domains and er is the target domain.
(ii) Generate an i.i.d. sample with sample size of (F,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)-(ii) M times.
This algorithm generates N samples (each one corresponding to in total n data points, some of which belong the target domain and the rest the source domains). The matrices from step (i) used to calculate the true value q(y]do()) which is used to measure the absolute estimation error. Each sample obtained in step (ii) is used calculate an estimate of the causal effect. The causal effects do(x)) generated by this algorithm have 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 0 are optimized first over R and later transformed into valid pmfs using the softmax function. The optimization done using the default optim function in R with the L-BFGS-B algorithm [55]. The initial value for each component of is generated from a continuous uniform distribution in [0, 1] and we consider maximum of 50 000 iterations.
C.3 Point estimation
The plot presented in Fig. 3 uses a sample size of n = 20 000. We repeat the same procedure but using the sample size 1000 in Fig. 7 and 100 000 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. shows a pairwise comparison between the causal and reduced parametrisations and is based the same samples as Fig. 3. Most of the points are located close to the identity line,
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Causal
Reduced
logí K PIWEE E.WEE. XII)
Absolute error
0.8
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7.5
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log( ( (Pn (W))
Figure 8: Comparison of the absolute error for both parametrisations when having large sample size. Absolute error for setting with Kg = 2, 100000, M = 25 and N = 5. The are more points close to zero than Figs. and 7, although there are still some estimates with large absolute error.
1e400
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Absolute emor reduxed
1e-01
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Absolute error causal
Absolute enor causal
Figure 9: Pairwise comparison of the absolute error for the causal and reduced parametrisations. We compare the graphs with kg 2 (left, 20 000) and kg = 3 (right, line has a slope equal to 1. The results provided by both estimators are better for the causal estimator in the first case, while they become estimator provides slightly more accurate estimates, with an average absolute estimation error of 0.0255 for the causal estimator and 0.0252 for the reduced estimator.
25 000). The dashed
similar when including one extra domain, where the reduced
so the absolute error 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 keep the average sample size per domain approximately constant in both settings.
Regarding the baseline comparison in Fig. 4, only consider samples from induced distributions corresponding to causal models where (g(y (x))) ((y[=)] > 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.
We generate the data directly from the intervention distribution Qy (do((===) For i.i.d. sample YI.... Y, from this marginal intervention distribution of Y in the target domain, we estimate causal effect by
Oracle.
Joracle, n (v/do(
Σ 10 =))
Since the oracle estimator has access to the intervention distribution, it should not considered a competing method but rather a benchmark (or, in & 6, estimator for the ground truth). In the simulation experiments, the differences between the oracle estimates and the true value (z)) 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 YJX. We consider two variants: the first one (NoAdj) uses an i..d. sample (X1, ....... (Xn, Yn) from the marginal distribution PC (x,Y) pooled across all
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Level
0.55
Bootstrap
Bootstrp
1.00
0.95
Medan CI lergh
Coverage
0.4
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Sample size
Sample size
Figure 10: Comparison of the coverage and median interval length of asymptotic and bootstrap confidence intervals as function of the sample size. The coverage is close to the nominal level in both cases but it is slightly better for the bootstrap method. The length of the bootstrap confidence intervals larger than the ones obtained from the asymptotic normality of the estimator, especially for small values of the sample size.
source domains, while the second one (NoAdj*) uses an i.i.d. sample (X1, (1), Mi)...., (Xn, Y,,) from the marginal distribution (xx,y) in the target domain; this, again, is information that is not available in our setting, (NoAdj*) should not be considered competing method, either. Both NoAdj and NoAdj* estimate the causal effect with the following expression:
x = E 1(Y, = y. xx
(27)
Σ 1(,
W -ajjmmmteent estimator. This estimator uses the adjustment formula (3) with the proxy vari- able W instead of U. (W} is not a valid adjustment set and, in general, this estimator 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 the true value. We again consider two variants: one which uses an i.i.d. sample X1, Yi).... (Wa, Xx, Yn) from F (W.X,Y) pooled across the source domains (WAdj) and the other from (. 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
1(Y
= - = W = wyy) W wj)
(E 1(W, (W:=w))) w3) )
]
@w (v/do(=)) =
Σ j=1
1=
T i= 1(1
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 o of the bootstrap estimates for (z)) In that case, the bootstrap confidence interval is given by
(y)oo())) G 21 : qR.n (v)))())) + B 21
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 two properties of these intervals: coverage and median length. With regards to coverage, the empirical values obtained 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
We compare the different estimation methods and baselines presented in Fig. in terms of computation time. For this purpose, we measure the total time needed to compute estimate 50 times (to obtain
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240
orade
causal
noad]
nead
wadj
wadj"
10+03
Sample size
Figure 11: Comparison of the computation time for the proposed methods and baselines. The proposed estimators present a higher computation time than the baselines. However, the reduced esti- mator is only marginally slower than the W-ad -ajjuustment estimators but presents better performance in terms of absolute error in Fig. 4.
0.5
G (X=
Reduced
NoAdj
0.2
Avg click praportion
10
15
Position
Figure 12: Estimation of the causal effect of different ranking positions on the probability of clicking on a fixed hotel considered as target domain. The horizontal dashed line represents the average proportion of clicks that hotel across all positions. The confidence intervals for the positions and for the reduced and oracle estimators are completely above that line, showing that these positions have significant effect on the probability of being clicked.
more accurate time measurements) with each estimator. This value depends on the implementation of each method and the equipment used for the measurement, we focus the comparison between the methods and not on the specific time values. Fig. 11 presents computation time for each method function of the sample size. The proposed estimators have 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.
Comparison of the effect different ranking positions on user's choices
We estimate q(Y 1]do(XX x)) for different values of the position variable æ {L...., 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 in the experiment in Fig. under the atomic intervention of fixing its position to æ. We again compare the reduced estimator with the oracle estimator. Fig. 12 shows the results. As before, there is overlap between the confidence intervals for the reduced estimator and the oracle in all cases, except for those positions in which the position X = was not observed the randomized dataset (positions 5 and 11), so not possible to calculate oracle estimate. All the confidence intervals are 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.
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The horizontal line indicates the average proportion of clicks on that hotel across all positions when they are randomized; this allows to see for which positions the probability of clicking on the hotel under the intervention do X := x) is larger or smaller than the average value. 'hee confidence intervals constructed using the reduced parametrisation and the oracle corresponding to the positions X = 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 I or 3 through an intervention is significantly higher (at level 0.95) than the average proportion across all the positions.
34


Reviews:
[{"review_id": "AiA9AHdpFI", "reviewer": "Reviewer_hDXV", "summary": "The 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.", "questions": "See Strengths And Weaknesses.", "limitations": "NA", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\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.", "was_revised": true, "final_justification": "Most of my concerns raised by the review were positively addressed by the authors and were resolved to some extent. Therefore I am raising my score just to the 4.", "rebuttal": "## General response to all reviewers and the AC\nWe are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (`hDXV`), “well-written” (`hDXV`, `nez5`), “mathematically rigorous” (`qoW1`), and found the empirical results “strong” (`nez5`) and “effective” (`aTiY`). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (`hDXV`, `aTiY`).\n\nA concern of reviewers `hDXV` and `qoW1` is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer `nez5` for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers `nez5` and `aTiY`, respectively.\n\nOur responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.\n\n## Individual response to reviewer\nThank you for your time in reviewing our work. We respond to your questions and comments below. \n\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_U, k_W 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\nThank you for this question. It addresses an important part of our work. We agree that Assumption 1, which implies $k_W = k_U$, is restrictive. However, this is a simplified version of our assumption, presented in the main text for clarity. We provide a **more general and weaker assumption in Appx. B** (we mention this in Sec. 3.1). To avoid confusion, we decided that for the revised version of our manuscript, we will only state the general result of Appx. B directly in the main part of the paper. \n\n**Clarification of the Necessary Assumption.** Our results only require the rank condition $rank(P(W|E,x)) \\geq k_U$ to hold (see Appx. B). This condition is less restrictive than the one highlighted by the reviewer and only implies that $k_E,  k_W \\geq k_U$. Importantly, this rank condition is also a **necessary** condition for identifiability. If $rank(P(W|E,x)) < k_U$, it is, in general, not possible to identify the causal effect. To illustrate, consider the case where $E$ and $W$ are only correlated with the first $k_U - 1$ values the confounder $U$ can take.  In this scenario, the last value, $u_{k_U}$, can act as a hidden confounder with no proxy, making $P(Y | \\text{do}(X))$ non-identifiable. More precisely, assume we have an SCM with hidden variable $\\tilde{U}$ such that $rank(P(W|E,x)) = k_U-1$. Now modify this hidden confounder such that $U \\coloneqq \\tilde{U} 1(N=1) + u_{k_U} 1(N \\neq 1)$, where $N$ is independent of $E$ and $W$, to get a new SCM. If in the new SCM $X$ and $Y$ are correlated through $U = u_{k_U}$, then this confounding cannot be detected by $E$ or $W$ as the joint distribution of $(E, W)$ does not change between the two SCMs. More generally, if the rank condition is violated, there will always be a linear combination of the confounder's values whose influence on $Y$ and $X$ cannot be detected through $E$ or $W$. We will formalize this intuitive argument in the revised manuscript.\n\n**Significance.** Achieving identifiability in the presence of unobserved confounders is a fundamentally challenging problem that always requires strong assumptions. We contend that our more general assumptions in Appendix B, particularly the generalized rank condition, is no stronger than those made in comparable works that achieve identifiability. In the revised version of our manuscript, we will expand our discussion to better contextualize our assumptions within the broader literature.\n\n> How does the estimator that is presented in Section 4.1 relate to Theorem 1?\n\nThm. 1 is a theoretical result establishing that the causal effect is identifiable under our assumptions. While the theorem's statement directly suggests a \"plug-in\" estimator (i.e., by estimating the matrices on the right-hand side), such an approach is not necessarily the only possible estimator. We therefore also propose an estimator based on Maximum Likelihood of the full causal structure, building off Prop. 2. The connection to Thm. 1 is that our proof of consistency for this estimator (Prop. 3) requires the identifiability result from Thm. 1. In short, the theorem guarantees a unique target exists, which our proposed estimator is proven to converge to.\n\n> Is the identifiability of P(y|do(x)) in Section 4.1 also requires the assumption 1?\n\nAs discussed in our answer to the previous question, the identifiability result presented in Sec. 3 is required for the consistency result of the estimators in Sec. 4.1. For the revised manuscript, we will make this connection clearer.\n\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\nWe thank the reviewer for this comment, as it highlights a point we need to make clearer in the paper.\nOur method is **not** limited to the setting where only the unobserved variables change. This was a simplifying assumption used for the illustrative example in the main text, chosen for pedagogical clarity and its connection to previous work [30].\nIn fact, our theoretical guarantees hold for a much broader class of SCMs. In our manuscript we provide a detailed discussion and examples of these more general cases---where, for example, the mechanisms for observed variables like $X$ also shift across environments---in Appendix C.2 (Figure 5). The key requirement for our method is not the immutability of specific equations, but rather the structural properties that allow for identification."}, {"review_id": "bSdZH1SSFJ", "reviewer": "Reviewer_aTiY", "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.", "questions": "* Suggestions\n\nIt 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.\n\n* Questions\n\nIs 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", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "* Strengths\n1. Identifying $q(y|do(x))$ with $W$ does not require a fully specified causal graph.\n2. 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)).\n3. Experiments performed over real-world and synthetic datasets show effectiveness of the proposed estimators.\n\n* Weaknesses\n1. 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).", "was_revised": true, "final_justification": "I am satisfied with the authors addressing my main concerns.", "rebuttal": "## General response to all reviewers and the AC\nWe are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (`hDXV`), “well-written” (`hDXV`, `nez5`), “mathematically rigorous” (`qoW1`), and found the empirical results “strong” (`nez5`) and “effective” (`aTiY`). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (`hDXV`, `aTiY`).\n\nA concern of reviewers `hDXV` and `qoW1` is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer `nez5` for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers `nez5` and `aTiY`, respectively.\n\nOur responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.\n\n## Individual response to reviewer\nThank you very much for your time. We answer your individual questions below.\n> 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.\n\nThanks for the remark. We will address what we consider as two separate aspects: choosing among known proxies and finding new ones.\n\nRegarding how to select the best proxy from a known set of valid proxies, our framework does not require a selection. Instead, these proxies can be combined into a single, more informative proxy (i.e., consider the combined proxy that takes values in the Cartesian product of the sets of values the individual proxies can take). If any of the proxies satisfy Asm.  2, then the combined proxy will satisfy it as well. Therefore, the problem of selection is sidestepped by combination.\n\nRegarding how to efficiently find or enumerate all valid proxies $W$ in a *known* large, complex causal graph, we agree with the reviewer that this is an interesting and distinct research problem. The focus of our paper is on establishing identifiability and estimation given a valid proxy. Developing a polynomial-time search algorithm to check the graphical conditions listed in Remark 8 in Appx. B is a valuable direction for future work but falls outside the scope of our current contribution.\nIf the causal graph is completely unknown, it does not appear to be testable whether a valid proxy exists. Instead, domain expertise is needed to argue for the validity of a given proxy (similar to justifying instrument validity in IV models).\n\n> 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.\n\nThank you for the suggestion. Since our main focus was not on computational cost of the proposed methods, we only included a specific instance (Footnote 5) to show that it is not really computationally expensive. The main factor that increases the runtime is the dimension (number of categories) of the categorical random variables. However, we will look into this further and will try to augment the manuscript with a O() analysis or more comprehensive runtime numbers.\n\n\n\n> 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)?\n\nYes, we believe the identification formula in Equation (4) to be complete. Thm. 1 shows that $q(y|do(x))$ is identified under Assumption 1. Conversely, if $q(y|do(x))$ is identified with Eq. (4) for any form of the confounder, then the right pseudoinverse of $P(W|E,x)$ exists and, therefore, this matrix has linearly independent rows. Furthermore, in this case, it can be shown that the inequality $k_W \\geq k_U$ is obtained as follows from Equation (4):\n\n$\\forall Q(U), P(y|U,x): \\quad P(y|U,x) Q(U) = P(y|E,x) P(W|E,x)^{\\dagger} P(W|U) Q(U) \\Rightarrow$\n\n$\\forall  P(y|U,x): \\quad P(y|U,x) = P(y|E,x) P(W|E,x)^{\\dagger} P(W|U) \\Rightarrow $\n\n$\\forall  P(y|U,x): \\quad P(y|U,x) = P(y|U,x) P(U|E,x) P(W|E,x)^{\\dagger} P(W|U) \\Rightarrow $\n\n$I_{k_U}= P(U|E,x) P(W|E,x)^{\\dagger} P(W|U) \\Rightarrow $\n\n$k_U = rank(I_{k_U}) \\leq \\min( rank(P(U|E,x)), rank(P(W|E,x)), rank(P(W|U)) ) \\leq k_W $"}, {"review_id": "IXbGB5vhgL", "reviewer": "Reviewer_nez5", "summary": "The 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.", "questions": "1. 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.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# 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.", "was_revised": true, "final_justification": "The authors satisfactorily addressed my concerns. I stand by the same score as before, which was 5. Happy to recommend this paper for an accept to the conference.", "rebuttal": "## General response to all reviewers and the AC\nWe are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (`hDXV`), “well-written” (`hDXV`, `nez5`), “mathematically rigorous” (`qoW1`), and found the empirical results “strong” (`nez5`) and “effective” (`aTiY`). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (`hDXV`, `aTiY`).\n\nA concern of reviewers `hDXV` and `qoW1` is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer `nez5` for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers `nez5` and `aTiY`, respectively.\n\nOur responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.\n\n## Individual response to reviewer\nThank you for your time reviewing our work and your suggestions to extend our section about related literature. We address each point individually below.\n\n> It is not clear to me why WAdj and WAdj* are missing from the real data experiment.\n\nThe reason for excluding WAdj and WAdj* from the real data experiment was the lack of a method for calculating confidence intervals for these estimators, which would reduce the relevance of the comparison presented in Figures 4 and 10. Wald or exact binomial test confidence intervals cannot be used for WAdj and WAdj*, as opposed to the Oracle and NoAdj* estimators, and there is not a simple derivation as in the case of the reduced estimator. The main alternative would be to consider bootstrap confidence intervals. \nTo compare the point estimates of WAdj and Wadj* with the rest of the methods in the real data experiment, we have computed the mean absolute error (MAE) with respect to the oracle during this rebuttal period obtaining the following values: 0.044 (Reduced), 0.051 (NoAdj), 0.080 (NoAdj*), 0.053 (WAdj) and 0.075 (WAdj*). This suggests that the proposed reduced estimator performs better in terms of absolute error than all the considered baselines in this real data example. We will include these results in Section 6 of the main paper. \n\n> I feel like it is overlooking important work on e.g. data fusion in causal inference.\n\nThank you very much for the pointers. We agree that prior works on data fusion and transportability, combining observational and experimental studies, and causal approaches to multi-domain data are related and will include additional paragraphs in Sec. 1.1. To briefly comment on some commonalities and differences:\nThe work of Pearl, Bareinboim et al. on data fusion considers a similar problem. However, in that line of work, transportability is typically determined purely from the extended causal graph (a.k.a. selection diagram) and (extensions of) do-calculus. In contrast, proximal causal inference is less fully nonparametric and typically relies on additional constraints such as Asm. 1 (e.g., requiring invertibility of certain matrices, thus formalising sufficiently informative proxies). In our setting, identifiability thus cannot be concluded solely from the causal graph, but leverages additional assumptions of a different nature than those explored in the transportability literature. \nWork on combining experimental and observational data typically aims to overcome the statistical challenge of small sample size in the experimental study, e.g., by devising biased but lower variance estimators that incorporate observational data. In contrast, in our setting, we do not have any observations of $X$ or $Y$ in the target domain (taking the role of the experimental data), but only observational data from different domains. \nCausal methods for multi-domain data often seek to learn an invariant predictor by discarding spurious features. Like the most closely related work by Tsai et al., they typically target the conditional mean $E_Q[Y|x]$, rather than the interventional distribution $q(y|do(x))$. \n\n> 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\nThe NoAdj and NoAdj* estimators do not consider the existence of an unmeasured confounder $U$. Therefore, in those domains in which the causal effect of $X$ on $Y$ is barely confounded, these estimators can obtain good estimates in terms of absolute error. Even though the change of domain does not directly affect the equations for $X$ and $Y$ in the SCM, the strength of the confounding can change due to the distribution shifts of $U$ when changing the domain. Furthermore, the NoAdj* estimator, which is the one that performs really well in terms of absolute error for the target domains ids 3 and 11 uses an i.i.d. sample $(X_1,Y_1),\\dots,(X_n,Y_n)$ from the target domain, and this information is not available in our setting. Therefore, NoAdj* has access to some data that our proposed estimators do not (see Appendix D.3 for additional details)."}, {"review_id": "xOnAM7psAJ", "reviewer": "Reviewer_qoW1", "summary": "This 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.", "questions": "My 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.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**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.)", "was_revised": true, "final_justification": "My main concern has been resolved.", "rebuttal": "## General response to all reviewers and the AC\nWe are grateful to all reviewers for their time and the useful feedback and suggestions. We are pleased that reviewers consider our work to tackle ”an important problem” (`hDXV`), “well-written” (`hDXV`, `nez5`), “mathematically rigorous” (`qoW1`), and found the empirical results “strong” (`nez5`) and “effective” (`aTiY`). Two reviewers also highlighted the proposal of new, provably consistent estimators as a key strength (`hDXV`, `aTiY`).\n\nA concern of reviewers `hDXV` and `qoW1` is the applicability of the proposed methods. For the former, we refer to Appx. B, where we provide relaxations for some of the conditions over the number of categories of $W$ and $U$. For the latter, we clarify in our response that our model can accommodate arbitrary shifts in the distribution of $U$, even without changes in $N_U$, via the domain indicator $E$. We thank reviewer `nez5` for pointers to related work on data fusion and transportability, which we will incorporate in Section 1.1. We will also include an evaluation of MAE for the WAdj and WAdj* baselines (for which no CIs are available), as well as a runtime analysis for the proposed methods to the baselines, following suggestions by reviewers `nez5` and `aTiY`, respectively.\n\nOur responses to individual reviewers provide further details. We believe to have thus addressed all reviewers’ concerns, but remain available for further discussion and clarification.\n\n## Individual response to reviewer\nThank you for your time in reviewing our work and for your questions and remarks which have helped us to clarify some key aspects of our setting. \n\n> 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. Are there any typos? I believe it should be (E, W, X, Y).\n\nThank you for pointing out this possibly confusing formulation. What we mean here is that ``the causal effect can be recovered from $\\{P(W,X,Y|E=e_1), …, P(W,X,Y|E=e_{k_E})\\}$ and $Q(W)=P(W|do(E:=e_T))$’’, i.e., we do indeed require the domain labels. (The main point we wanted to convey here is that observations of $X$ and $Y$ are not needed in the target domain.) We will adjust the formulation accordingly to clarify.\n> 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\nIt may perhaps be unintuitive at first, but this is precisely the main insight of proximal causal inference: a causal effect can be identified, even in the presence of unobserved confounding $U$, if sufficiently informative proxies $W$ are observed. This insight is not specific to our work, but also central to the works described in Section 1.1, many of which are by now well-established in the causal inference literature. \nWhile it is true that the joint and marginal distributions of $U$ and $W$ may change across domains, a crucial aspect of our setting is that the way in which $W$ depends on $U$---captured by the conditional distribution $P(W|U)$ (or $P(W|U,Z)$ for the more general setting discussed in Appx. B)---remains invariant across domains. This invariance (which follows from the modularity of causal models) is leveraged to transform the adjustment formula from Eq. (3) such that it only involves observable quantities, see also l.130-132. \nImagine that $W$ is a perfect copy of $U$, i.e., $W=U$ and $P(W|U)$ is the identity. In this case, observing $W$ in the target domain suffices, because the terms in Eq. (3) become $P(y|W,x)$ and $Q(W)$. Our identification result shows that this argument extends to noisy or imperfect copies of $U$, provided that they are sufficiently informative as formalised by Asm. 1. In particular, Asm. 1 rules out cases such as that of a completely uninformative proxy $W$ that is independent of $U$, for which identification would become impossible.\nIntuitively, due to the invariance of other conditionals, observations of an informative proxy $W$ in the target domain can suffice to correct for changes in the distribution of $U$. \n\n> 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. If 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\nWe believe there may be a misunderstanding regarding our use of structural equations for modelling how and which distributions change across domains. We will attempt to clarify this key aspect of our setup.\n\n**SCMs and induced distributions.** The noises $N_i$ in Eq. (1) are exogenous (not being modelled) and only serve to introduce stochasticity into the otherwise deterministic model. The variables of interest are $E,U,W,X,Y$. Crucially, the functions $f_i$ and noise distributions $P_{N_i}$ are completely unconstrained. Hence, the SCM can induce any $P(E)$, $P(U|E)$, $P(W|U)$, $P(X|U)$, and $P(Y|U,W,X)$, see, e.g., Sec. 3.4 of Peters et al. [42] for details. In particular, the distributions of $U$ can change in arbitrary ways across domains, i.e., $P(U|E=e_1)$, …, $P(U|E=e_{k_E})$, $Q(U)$ are (a priori) unconstrained. While we use SCMs to formalize the data-generating process, the paper could also be written purely in terms of conditional pmfs (see l. 100-101).\n\n**The role of the environment variable $E$.** The variable $E$ is *not* assumed to explicitly capture “factors responsible for shifts in the distribution of U”. In line with prior literature on causal transportability and domain adaptation, $E$ is simply a **domain indicator** (e.g., $E \\in \\{1, 2, 3, 4\\}$ for experiments run in different labs) that can be thought to switch between different distributions for $U$. We only assume that these distributions are stable within any given domain (i.e., for a fixed value of $E$), which is a standard assumption for many well-established methods. As explained above, the distributions of $U$ can, in general, differ arbitrarily across domains, even without $N_U$ changing; they are only implicitly constrained through Asm. 1.\n\n**Identifiability with Shifting Confounder Distributions.** To answer your direct question: ``If $N_U$ differs between the source and target domains, then your identification result does not hold. Is that correct?'' **No, this is incorrect.** Our identification result is agnostic to why the distribution of $U$ changes. The key requirement is that the rank condition, $\\text{rank}(P(W|E,x)) \\geq k_U$, is met in the source domains. The mechanism that generates $U$ can change arbitrarily. As explained above, the standard assumption that $N_U$ does not change across domains **comes without loss of generality**, but changes to $N_U$ could equally be accommodated without causing issues. \nWe will revise the manuscript to clarify these crucial points and make the role of $E$ as a simple indicator variable and the relation between Eq. (1) and changes to $P(U)$ more explicit. \nIn light of these explanations, we hope that you will agree that our modelling of the relationship between $E$ and $U$ *does not* limit the applicability of our results.\n\n> In relation to the concerns mentioned above, I also have reservations regarding the applicability of your results in practical settings. In your application, E represents the hotel ID. Can all factors responsible for shifts in the distribution of U truly be captured by E alone? \n\nWe hope that our previous response also clarifies this issue.\n[To briefly reiterate the main point: Even without changes to the distribution of $N_U$, arbitrary changes in $P(U)$ can be modelled via $U:=f_U(E,N_U)$ via a domain indicator $E$. One way to see this is inverse cumulative distribution function (CDF) sampling. Consider two domains with arbitrary distributions $P_1(U)$ and $P_2(U)$. Let $N_U$ be uniform and define $f_U(e_1, n_U):=F_1^{-1}(n_U)$ and $f_U(e_2, n_U):=F_2^{-1}(n_U)$ where $F_1$ and $F_2$ are the CDFs of $P_1$ and $P_2$. This construction induces $P_1$ in $e_1$ and $P_2$ in $e_2$. The key is the flexibility of the function $f_U$, so that different values of $E$ can index different functions of the second (stochastic) argument.] \n\n> 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\n$E$ is simply a categorical identifier for distinct domains. In the application, each hotel ID is an arbitrary label. The model does not use the value of the ID itself in any calculation.\nThe key point is that changing the target ID from, say, Hotel A to Hotel B, means we are asking a different question: \"What is the causal effect for Hotel A?\" versus \"What is the causal effect for Hotel B?\". These environments have different characteristics—specifically, different distributions of the proxy variable $W$ (price). That is, changing the target hotel ID also changes $Q(W)$. Our method leverages this information from the proxy in the target domain to transport the causal effect correctly. Therefore, the estimate *should* change. If the results did not change when switching the target from a luxury hotel to a budget hotel, it would imply the model is failing to account for the target-specific context. The observed behavior thus confirms that the model is working as intended.\n\n**We hope that our responses have clarified some misunderstandings and resolved your concerns. In light of this, we kindly ask that you reconsider your evaluation.**"}]

Author rebuttal:

Meta-review: This paper addresses the problem of estimating causal effects in a multi-domain setting with unobserved confounders, proposing a methodology to identify and estimate the target domain's causal effect using proxies of the hidden confounder, with theoretical guarantees of identifiability, consistent estimators, and supporting empirical results.

All reviewer's main concerns were thoroughly addressed: authors clarified assumptions to be more general than initially perceived, strengthened theoretical connections and supplemented experimental details. The responses effectively resolved doubts about applicability, theoretical rigor, and experimental validity. Considering the paper, reviews and discussion overall, I recommend this paper for acceptance.
Decision: Accept (poster)