Do Finetti: On Causal Effects for Exchangeable Data
Authors: Siyuan Guo
Chi Zhang
Karthika Mohan
Ferenc Huszár
Bernhard Schölkopf
Venue: NeurIPS 2024 oral (2024)
Keywords: Causality; Exchangeability
Primary area: causal_inference
TL;DR: Causal effect in exchangeable data that adhere to the independent causal mechanism principle

Abstract: We study causal effect estimation in a setting where the data are not i.i.d.$\ $(independent and identically distributed). We focus on exchangeable data satisfying an assumption of independent causal mechanisms. Traditional causal effect estimation frameworks, e.g., relying on structural causal models and do-calculus, are typically limited to i.i.d. data and do not extend to more general exchangeable generative processes, which naturally arise in multi-environment data. To address this gap, we develop a generalized framework for exchangeable data and introduce a truncated factorization formula that facilitates both the identification and estimation of causal effects in our setting. To illustrate potential applications, we introduce a causal Pólya urn model and demonstrate how intervention propagates effects in exchangeable data settings. Finally, we develop an algorithm that performs simultaneous causal discovery and effect estimation given multi-environment data.

Do Finetti: on Causal Effects for Exchangeable Data
Siyuan Guo 14
Chi Zhang²
Karthika Mohan
Ferenc Huszár
Bernhard Schölkopf
Maa Planck Institute for Intelligent Systems Toyota Research Institute
Oregon State University University Cambridge
i Equal supervision
Abstract
We study causal effect estimation in setting where the data not i..d. (inde- pendent and identically distributed). We focus on exchangeable data satisfying an assumption of independent causal mechanisms. Traditional causal effect estimation frameworks, e.g., relying on structural causal models and do-calculus, are typically limited i.i.d. data and do not extend more general exchangeable generative processes, which naturally arise in multi-environment data. To address this gap, we develop generalized framework for exchangeable data and introduce truncated factorization formula that facilitates both the identification and estimation of causal effects in our setting. To illustrate potential applications, we introduce a causal Pólya urn model and demonstrate how intervention propagates effects in exchange- able data settings. Finally, we develop an algorithm that performs simultancous causal discovery and effect estimation given muult-ennironment data.
Introduction
Inferring causal effects from observational data is central task for scientific applications from health and epidemiology to the social and behavioral sciences. Scientists aim to understand Nature's 's mecha- nisms to discover feasible intervention targets and estimate intervention effects. The existing causal
effect estimation theory is often formulated for structural causal models (Pearl) 2009), focusing on the study of independent and identically distributed (i.i.d.) data. Indeed, the do-calculus [Pearl] [012] [012] and its extensions and applications (e.g., mediation analysis [Pearl] (022), stochastic interventions [Correa and Bareinboim 2020) developed in the i.i.d. framework has been widely used in practice.
A more recent line of work relaxes the i.i.d. assumption and considers causal inference under the assumption of independent causal mechanisms (ICM) (Schölkopt al. 2012 Pearl 2009) Guo et al. 2023a]. which postulates that distinct causal mechanisms of the true underlying generating process do not inform or influence one another. The Causal de Finetti theorems of Guo al. [2023a] provide mathematical justification of the ICM assumption in exchangeable data-generation processes, thus establishing a foundation for studying causality exchangeable data. Exchangeable data generation processes that adhere to the ICM principle are referred to as ICM generative processes, or ICM (CM-exchangeable data.
Guo et al. [2023a] showed that data from ICM generative processes, or more succinctly ICM- exchangeable data, are more informative than i.i.d. data in that they often permit unique causal structure identification in cases where i.i.d. data does not. In contrast, the present work focuses on causal effect identification and estimation. Standard formalisms in the i.idd. framework, e.g.. using structural causal models and do-operators, do not apply in the exchangeable non-i.i.d. setting. We study the meaning of interventions, feasible intervention targets, and finally causal effect identification and estimation for ICM generative processes. In doing we make three contributions:
CCorespondence to siyuan. guoötuebingen. .mg.ppg.... de
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Lid. process
(A): PLY X a²) P(TT))
X
Y;
Xi
Yi
(b): P(Y = P PLY olpps)dd
B
X
(n): P( - =): 12, P())))))
X
(b): PIV]ddoXX = 1) Xy) - (a.
(b): ICM generative process
(6)
Figure 1: bivariate illustration demonstrates differences in causal effect between i.i.d. processes and CCM--enerative processes. Suppose g X - Y. The hammer represents an intervention on the closest node. (a): Data generated according to & under an i..d. process; (b): Data generated under an ICM -eneerttee process (using plate notation). Block A shows how ((Y)))(X)) differs between the i.i.d. (a) and the exchangeable (b) case. Note that the causal effect under i..d. a special case of that under exchangeable processes with p(1²) 5( who). for some value tho. Corollary [ below justifies that we omit position indices from Block A in ICM-generative processes. Block B shows the difference in intervention effect when conditioned on other observations, for i.i.d. (a) and ICM-gencrative (b) processes. In the i.i.d. case (a), due to (1) I (Y2, X2), conditioning on (X2, Y2) conveys no information the prediction of the interventional effect on Yi. In contrast, for an MiM-gencrative process (b), observing (X2, Y2) does provide additional information about the effect on Y1 when intervening
X1. Graph (c) illustrates the graph surgery performed on ICM(() (cf. Def. 7 in Appendix A). We observe that conditioning on a collider Y2 provides additional information about the effect on Y1 when intervening on X1.
We establish the operational meaning of interventions and the feasible intervention targets in ICM generative processes that differ from the traditional framework in i.i.d. data.
Causal effect in ICM generative processes (Section 3
Our main causal effect identification and estimation theorem (Section 3.1 provides an explicit truncated factorization formula for ICM generative processes that solves the problems of identification and estimation of causal effects. In particular, we introduce a causal Pólya urn model (Section (.2) and show that the post-interventional distribution changes when conditioning on other observations property that does not exist in i.i.d. data.
Causal effect estimation in multi-environment data (Section 4 we connect ICM processes with multi-eeniirommet data, showing that extending the causal framework to exchangcable data does not necessarily mean a decrease of its ability in graphical identifica- tion and effect estimation. Theorem 2 informally stated, shows that with no prior graphical assumptions, data adhering to the ICM principle and generated by an exchangeable process alone is sufficient for both graphical and effect identification. We developed Do-Finetti algorithm for enlti-enviroomeet data, and empirically validate our results in Section 5
Fig. [ illlustrates the main differences in causal effects between i.i.d. and ICM generative processes.
2 Preliminaries
Notation Denote X as random variable with realization e with boldsymboled X set of random variables with realizations x. Data generated by either i.i.d. or ICM generative processes is sequence of random variables X,,, X X....... where X.m := (X ...,............ and d denotes the number of variable indices and n denotes the position index within a sequence. For example, X denotes sequence. Often in this work, N denotes
........
the i-th random variable observed at the n-th position the number of positions observed in sequence and [N] == [1....,N as the set of positive integers less than or equal to N and [-S] denotes the set of positive integers less than equal to N excluding values contained in S. To abbreviate notation, write X.|N] := (Xi...... X For simplicity of illustration, we use bivariate examples with Xn, Yn, as a pair of random variables observed at the n-th position. Putting the bivariate notation in the above multivariate context, X13m corresponds to X, and 2n corresponds to Y, where the first index indicates different variables in the same position. Denote uppercase P for probability distribution and lowercase for probability density functions.

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2.1 The Causal Framework in i..dd Data
Structural Causal Model [Peters al. 2017 Pearl 2009
Haavelmo al. 1995 Wright 1921
A structural causal model (SCM) M == (F. consists of set of structural assignments F := {ff..... fu} such that X, == f, (PA,. Uy}.j = I....,dd d, where PA, - {XI...... ( and are often called as parents of Xj- The joint distribution PJ over the noise or exogenous variables U assumed to be jointly independent. When such a condition is satisfied, the SCM is Markovian model that assumes the absence of unmeasured confounders. A graph G of an SCM is obtained by creating one vertex X, and drawing directed edges from each parent in PA to X,. We assume 9 is acyclic. Our lack of knowledge in noise variables Pt and together with the structural assignments induces a joint distribution over the observable variables , Such joint distribution satisfies the Markov factorization property with respect to G:
d
P(X Xa)
II P(X, PA.):
Given two disjoint sets of variables X and Y, the causal effect of X on Y, denoted P(Y do(X)): is defined with respect to modifications of existing SCM (also known as graph surgery): for each realization x of X, P(y]do(x) gives the probability of = induced by deleting from the SCM all structural assignments corresponding to variables in X and substituting X in the remaining equations. In Markovian models, given a graph, causal effect is identifiable via Eq. 2
= x))
II P(X,(PA,) XX,EX
(2)
X-x'
where xx=x enforces X1, XI..... Xa to be consistent with realizations of X else Eq. takes value O. This principled approach is known as g-computation formula [Robins] 1986). truncated factorization
[Pearl 2009] or manipulation theorem (Spirtes et al., 1993]. Appendix A details the standard graphical terminology.
Independent Causal Mechanism (ICM) postulates how Markov factors (hereon referred to as causal mechanisms) in Eq. (I) should relate to each other. Schölkopf et al. (012 and Peters et al. 2017] express the insights as follows:
Causal mechanisms are independent of each other in the sense that a change in one mechanism P(X, PA.) does not inform or influence any of the other mechanisms P(X, PA,). for i + j.
This notion of invariant, independent, and autonomous mechanisms has appeared in many forms throughout the history of causality research: from early work led by Haavelmo [1944) and Aldrich [1989] to Pearl [2009]. Studying properties of independent causal mechanisms rigorously demands a statistical understanding of what such independence means between distributions. Guo et al. [2023 provides a statistical formalization of ICM in exchangeable data, thus providing necessary tools to study causal framework in exchangeable data. The next section introduces relevant background.
2.2 The Causal Framework in Exchangeable Data
Definition (Exchangeable Sequence) An exchangeable sequence of random vuriables is a finite or infinite sequence X1, X2, Xs.... such that for any finite permutation of the position indices {1...., N}, the joint distribution of the permuted sequences remains unchanged to that of original:
(X.(I))...,Xx(N)) P(X
Note an independent and identically distributed (i.idd)) sequence of random variables is an exchange- able sequence, i.c., P(Xi,... ......x)) Xx) -T P(X;), but not all exchangeable sequences are i.i.d. Examples of exchangeable non-i.idd sequences include but are not limited to Pólya urn model (Hoppe 1984], Chinese restaurant processes [Aldous al. 1985 and Dirichlet processes [Ferguson] 1973
An important result of exchangeable data are Finetti's theorems (de Finetti 1931 which show any exchangeable sequence can be represented as a mixture of conditionally i.i.d. data. Building upon the work de Finetti [193T], Guo al. [2023a] observes that exchangeable but not i.i.d. data possess extra conditional independence relationships compared to i..dd data. This enables:
3

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(a): Structural Causal Model
(b): ICM generative process
i.d.
Ux
Uy
exch.
y
Observational
do(X x)
Observational
do(X = =
: Ux
= Y: f(,
i p(0), db p(v)
X~ 8(X = x)
: U()
X p(x)0), ~ p(ylz,
Y~ ~ p(yl,,
Figure 3: An illustration of differences in what the do-operator does between structural causal model (a) and an ICM generative process (b). In the observational phase, SCMs (a), where the dotted plate indicates i.i.d. sampled, illustrates that fixed assignment of Ux and Uy leads to fixed observable values X and Y; on the other hand, for ICM-generative processes where exch. is an abbreviation for an exchangeable process. fixing v does not fix X and Y. instead, means sampling from a fixed distribution. Because SCM fails to characterize (CM-generative processes, we define the operational meaning of do-interventions ICM-generative processes assigning d-distribution to the intervened variables and substituting the corresponding values in the remaining distributions.
unique causal structure identification (Theorem in [Guo et al.
2023a and
statistical formalization of ICM principle (causal de Finetti theorems in Guo et al. 2023a
In contrast previous work's focus structure identification, this work studies causal effect identification and estimation in exchangeable data. Markovian models here mean there are no unmeasured confounders for each tuple of random variables observed in exchangeable sequence.
To start with, we introduce the necessary terminologies inherited from Guo et al. 2023a]].
Definition 2 (ICM generative process) We say data is generated from an ICM generative process with respect to DAG G if the sequence of random variahle arrays {X:n] is infinitely exchangeable
and satisfies X..ms) I. ND ND inDim (N for all i E [d] and n E N, where PA, denotes parents of node X,, ND; denotes the corresponding non-descendants and ND, denotes the set of non-descendants excluding its own parents. By causal de Finetti theorems, is equivalent to say the joint distribution of the sequence can be represented
N d
P(X.NNI = X. = /
p(ri Iaa
(0,)
(4)
7=1
where vs probability measures.
X1,1
X2,1
X3.1
In this work, we assume that any probability measure has corresponding density function. Following Definition 2 an immediate result is any distribution /' generated by an
ICM generative process Markov ICM( Definition in Appendix A defines formally what we mean by an ICM operator on DAG G. Here we illustrate an example of (CM(G) and the use of our notation. Consider DAG :== X1 + X₂ > X3 with variable indices. Two data tuples are generated under an ICM generative process with respect to G. Fig. 2 shows the Markov structure compatible with the above ICM generative process.
X1.2
X2.2
Xa.2
Figure 2: An example of ICM(g): Two data tuples generated by ICM generative process with respect to g == X1 + X2 + Xs, where X is the i- th variable in the n-th position and gray means latent variables.
3 Causal Effect in Exchangeable Data
We first motivate our study of causal effects exchangeable data by noting exogenous variables are different from causal de Finetti parameters. Traditional causal effect in i..dd processes is defined as

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graph surgery with respect to SCMs (cf. (ection2.II. Such definition does not transfer its operational meaning to that in exchangeable processes because SCMs fail to characterize ICM generative processes. In this section, we first define what causal effect means and show their differences in properties compared to that of i.i.d. processes. We proceed to illustrate in the simplest possible case pair of random variables, and then provide general statement for the multivariate case.
Figure 3 illustrates how SCMs fail to characterize ICM generative processes in the bivariate example. In SCMs. fixed assignment of values to exogenous variables Ux and Uy uniquely determines the values of all the observable variables X and Y. This is not the case for ICM generative processes. fixed assignment to causal de Finetti parameters 9 and z only restricts the observable variables X and Y to sample from fixed distributions but does not uniquely determine their values. Therefore, do-operator commonly defined as graph surgery in SCMs demands a new operational meaning in ICM generative processes. Definition defines that do X = x)) in ICM generative processes means assigning the sampling density p(x () to 5(X- )) Doing so, we have:
i.i.d. generative process :P(Y = y do x)) - p(yla, (0) - P(Y = yy(x), 10 E I
(5)
ICM generative process :P(Y y do z)) = /
plylz (/ ) P(Y y|
(6)
Despite being identical in expression, causal effect has different implications in i.i.d. and ICM generative processes, Under i.i.d., the randomness captured in /' is driven only by the randomness in exogenous variables Uy. Under the ICM generative process, the randomness in P is driven both by pay| 15) and the randomness in the causal de Finetti parameter p(v)). is well-known that ii.d is a subcase of exchangeability. Here we observe that the causal effect expression follows the same pattern, i.e., the causal effect expression in i.i.d. also a subcase of that under exchangeability whenever p(() = 8( = vo)).
Equipped with this operational meaning of intervention, we consider the set of feasible interven- tion targets. Here we first clarify what we mean by the data-generating process. Data generated from i..dd. ICM processes refers to a sequence of random variables. For example, in the bivariate case, the sequence is (X1, YI), (X2, Y),.... Often one omits position indices in i.i.d. because:
i.idd. generative process: P(X1, YI,.... NN,YN ind II P(X Y.)
ide [P(X,Y)]
The first equality is due to independence and the second equality is due to identical distributions. One thus does not differentiate the position indices in i.i.d. P(X, characterizes the joint distribution fully by Eq. 7 Intervention is thus defined only on X and Y rather than X,, and Y,. However, in the ICM generative process, P(X, Y) cannot fully characterize the joint distribution. An example application of Definition 2 in the bivariate case gives:
ICM gen. process P(11, VI...... IN UN) / / II Lu () (8)
Eq. 8 shows that one can no longer omit the position indices in ICM generative processes, as P(X.Y cannot fully characterize the joint distribution. Intervention ICM generative processes thus should be considered at the level of X,, and Y,.. Definition presented below, formalizes the concept of causal effect in ICM generative processes. In particular, both intervention sets and target variables of interest can be any random variables observed in the sequence.
Definition 3 (Causal Efffec in ICM generative processes) Let X and Y be two disjoint sets of variables generated by an ICM- BeM--gnerative process. For each realization x of P(y]do(x)) in the ICM-generative process denotes the probability Yf =y induced by assigning pl= 0,) = 8(Xin = Iimn) in Eq. 4). VX E X, and substituting X = x in the remaining conditional distributions.
Although marginal distribution P(X,Y Y) cannot characterize the joint distribution in Eq. (8) due to mutual dependence of X,, Y, with causal de Finetti parameters 0 and db we note variables in

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different positions share identical marginal distributions. Mathematically, data generated under an exchangeable process shares identical marginal distributions: P(X,, Y,,) P Ym). vn & m. This also means that identical interventions performed on variables in different positions result in , Vn * m. Similar to i.i.d., quantities such as P(Y]do(X x)) are thus well-defined in ICM generative processes. Corollary in ICM generative processes.
the same post-interventional distributions: P(Y, X, = x)) P(Ym. / Xm = x)),
[ proves the multivariate equivalent of identical marginal intervention effects
Corollary (Identical marginal post-interventional distributions) Let P be the distribution for some ICM generative process. Let and J be two disjoint subsets in [d] == [1....,d.. Denote X1.1 = {Xim i 1} and similarly for X3; Then,
P(X.m do((x:n - x)) = pm do( = x]), Vn o m
(9)
ie., identical interventions on variables in different positions share the same marginal post- interventional distributions. See Appendix Bor the proof.
One thus omit the notation of position indices when appropriate as supported by Corollary and
as illustrated in Figure Block A. In the next section, we address a focal problem in causal effect estimation: causal effect identifiability in ICM generative processes.
3.1 Causal Effect Identifiability in ICM generative processes
Continuing with the bivariate example Fig. (b), consider X1, Y1, X2, Y2 generated under an
ICM generative process with respect to G == X Y. Suppose one performs hard intervention on X1,
ic., do
.). Applying traditional truncated factorization developed for i.i.d. data (Eq. (2) yields:
P( Vı, 22, (X =)
(10)
However, the independence between (X,, Yi) and (X2, Y2) in iidd generative processes does not hold in ICM generative processes. fact, in ICM generative processes, applying Definition 3 gives:
P(11, U2, (2 = =)) / plyi L )) 22, )) (1) (1d())))p((222)))
(11)
Eq. (1] does not equal to Eq. (10) whenever p())) * 8( = vho) for some v/n- Thus, for data gener- ated under ICM generative processes, new theorem for truncated factorization is required. See Theo-- rem for the statement for general multivariate distributions and Appendix C for detailed derivation.
Theorem (Truncated Factorization in ICM generative processes) For a given graph G, let P be the probability distribution for data generated under an ICM generative process with respect G and let p be the corresponding density. The post-inted post-interventioaal distribution after intervening on X has density given by:
p(x:,1,.... X..,. (X x)) II p(x=) N. pa ((-NA) x
II p(x: pa [[N)
x=
(12)
iglx
where Ix == (i: X X} denotes the set of variable indices being intervened on and N; : {n Xin X denotes the set of position indices corresponding to variable index in the intervention set X and N.) denotes the set of positive integers less than or equal to N excluding values in N
Theorem presents a procedure for computing the joint post-interventional distribution using ire-interventional conditional distributions when intervening any set of variables in the Markovian model. This demonstrates that causal effects are identifiable in Markovian models under ICM generative processes. Note that the traditional truncated factorization i.i.d. processes is again special case of Eq. 112) just as i.i.d. processes are special case exchangeable processes.
3.2 Conditional Interventional distributions
The fundamental difference between i.i.d. and ICM generative processes lies in the violation of the independence condition inherent i.i.d.. Consequently, interventional distributions computed by conditioning observations differ between the two. Specifically, causal effect of do(Xi i) on Yi

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given X2, Y2 differs when computed under i..dd. (Eq. (3) and ICM generative processes (Eq.
(44
This difference arises because (X,, (X2, Y2) holds i.d but not in ICM generative processes.
i.i.d. generative processes :P(Y1 (do((; - f), X2 Y2) = P(Y1
(13)
ICM generative processes iYi do a), X2, 12) = P(Yi X2, Y2)
(14)
Fig. (c) depicts this example of intervention via graph surgery on (CM(g) where the operational meaning of edge deletion is explained in Figure [ We observe that conditioning on the collider node 12 in the ICM generative process renders Yi y X2 Yg. Lemma provides the corresponding general statement for the multivariate case when conditioning on other observations. See Appendix D for
the proof.
Lemma (Intervention effect conditioned on other observations) For a given graph G, let be the distribution for the ICM generalive process with respect to G. Let X be the intervention set. Assume X In = {Xim Vi E II. Let C [N] such that n & and [-I] denotes [d] (I Then,
P P(X-Im/do x), xs
II P(X X PA (n))) XIm=X ig
(15)
A similar argument also applies to the intervention effect conditioned on observations of experimental results performed on other tuples of random variables in the sequence. Appendix(E discusses in detail. Here we show the implications of conditional interventional distributions via causal Pólya model.
Causal Pólya Urn Model Imagine an urn with left and right compartments. The experimenter puts white balls and / black balls in each compartment. At each step n, one ball is uniformly drawn from the left and one ball uniformly drawn from the right. The chosen two balls the order of left and right are then placed in dark chamber unobserved by the experimenter. A hidden mechanism reads the color of the two balls and outputs X,,, Y, to the experimenter. The mechanism outputs X,, 1 whenever the n-th left ball is black else X, = 0 and Yn = 1 whenever the left and right balls disagree in color. After observing X, and Y,. the experimenter puts the original balls back in the corresponding compartment and add ball of the same color as X,, in the left and add ball of the same color as Zn := (1 - Xx)) Yn + (1 -1-33 (1-Y))= * Xn in the right.
Causal de Finetti application The causal Polya urn model is a real-world illustration of causal de Finetti thereom in its bivariate form. The joint distribution of observed sequence P(11, (a.1..1...... Zn, Vn) for all n N can be perfectly modelled as the RHS of Eq. with two variables and X
/ P(Un 2n, ))((n ()p(()p(())ddss
where p(0), p(t) are Beta distributions and p(yn In, 1), p(z, 0) are Bernoulli distributions. Appendix[i Faonndxxxxpooooo provides a detailed amalysis of how causal Pólya Umn Model satisfies both exchangeability and certain conditional independences conditions. Therefore causal de Finetti theorems apply the causal Pólya urn model in the sense the joint observational distribution can be equivalently modelled as ICM generative processes in bivariate form. Next, we study how intervention propagates in this model.
Discussion For a given n, if Xm, = 1, Ym for many m < n, then it is more likely X,, = Yn. 0 there are more black balls are added to both left and right urn compartments. Consider an intervention:
imagine an agent replaces the left ball in the dark chamber to be a white ball for the n-th draw, do(X,, 0), then if X,, 1, Y. 0 for many < n, it is more likely that Y, = 1 more black balls are placed in the right compartment and X,, is fixed to be a white ball by intervention. This illustrates a practical example of the aforementioned conditional post-interventional distributions. After the intervention, the observer, ignorant of what happens in the dark chamber, proceeds as normal with the replacement process. The intervention thus changes the dynamics of causal Pólya urn model. We consider causal Pólya model interesting as it illustrates through simple game-like settings that observables (X,, Yn) satisfying Yi, 1 Xx XX are indeed driven by the independent mechanisms hidden from the observers. This is akin to how Nature hides causal mechanisms from observers and scientists can only reason through observables.
3.3 Rules of compact representation of causal effect
From Theorem I any interventional effect (YY)))(X) can obtained from marginalization. How- ever, in practice, marginalization many variables is computationally expensive. Further, Theorem
7

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Trquurres observations and measurements of all joint variables which is resource- resource-intensive in practice. Such problem on the observability of variables is more explicitly posed as out-of-variable problem in Guo et al. 2023b]. In this section, we present rules that allow simplification of causal expressions.
Lemma 2 (Intervention effect on differing positions) Given a graph G and let P be the distribu- tion for the ICM generative process with respect to G. Let X and Y be two disjoint sets such that X is the intervention set and Y is the target set. Let Nx := {n X E X3 be the set of position y be the set of position indices being targeted to observe.
indices being intervened, and similarly Ny Assume Nx n nNy = 0. Then,
P(Y | do = -x)x=x))=PP P(Y),
(16)
Lemma (Intervention effect within the same position) Given a graph 9 and let P be the distri- bution for the ICM generative process with respect to G. Let X be the intervention set such that consists only X where n. E S S [N] and I C [d] is a set of variable indices. Let Y be the target set such that it consists only XJ where E S [N] and J S [d]. Note InJ 0. Then,
P(Y dx = x)) = EP(Y xx.pxx)
(17)
pax
where PAx denotes the parent set of intervened variables X.
Lemma 2 shows that the interventional distributions on target variables of interest are unchanged when only intervening on variables in differing positions other than the target variables. Lemma 3 shows for arbitrary causal queries when acting on a consistent set of variables across positions (i.e., the intervention set X consists of X1.. where each position n shares the same set of variable indices I being intervened on), the cost-interventional distribution can be estimated by only observing and measuring Y, X, PAx. where PAx denotes the parent set of X. This is consistent with the parent adjustment formula under i.i.d. generative processes, with the additional dependence structure across variables in differing positions. See Appendix aad for detailed statements and proofs.
4 Causal Effect in Multi-envirrnment data
In Section 3 we establish the identifiability of causal effects given the graph G and the distribution P generated by an ICM generative process. Note under i.i.d. generative processes, causal effect identification hinges on the knowledge of causal diagram G. Here, we show that generalizing causality to an exchangeable framework does not necessarily mean we have less ability to perform graphical identification and effect estimation. In fact, with an unknown graph, ICM generative processes allow one to identify graph and causal effects simultancously. In other words, data adhering to the ICM principle and generated under exchangeable process alone is sufficient for both graphical and effect identification.
Theorem (Causal effect identification in ICM generative processes) Denote Y,X be two dis- joint subsets of observable variables in {X..) Then P(Y do(X x)) is identifiable given the distribution P from the class of distributions generated from ICM generutive processes. Here ideetifiability means the causal query can be computed uniquely from P.
Informally, Theorem 2 says that for ICM generative processes, both causal graphs and causal effects can be identified tanuuulle This derived from a concatenation of Theorem 5 in Guo et al.) [2023a] and Theorem [ in the current work. Theorem 5 in Guo et al. [2023a] shows for ICM generative processes, causal graph identifiable and Theorem I shows the computation, and thus identifiability, of causal effects in ICM generative processes given graph.
To see how to apply Theorem 2 in practice, we build connection between exchangeable and grouped or multi-cnvironmect data and propose the Do-Finetti algorithm. In the causal literature, grouped data refers data available from multiple environments, each producing (conditionally) i. observations from a different distribution, which are related through some invariant causal structure shared by all environments. We can interpret multi-environment data through the lens of exchangeability as follow: In each environment e, we observe exchangeable random variables X [N.j. where X denotes the d- th random variable observed at n--th position in environment Ne denotes the number of observations
8

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HBM So-ived
Cauesl Effeet MSE (he lower te bather
0233
Graph Accurnay (the Highter balte)
Numberof envrronments
Number of annvrromments
Figure 4: Our method's (do-Finett)) performance in simultaneously identifying DAG (right) and causal effect estimation (left), compared to the i.i.d. algorithm (i.i..) and corresponding methods with known true DAG (do Finetti with true DAG and i..d with true DAG) in bivariate setting. Left shown are the mean and standard deviation of MSE compared analytic solutions for each method aggregated over 100 experiments. Right shows the accuracy of identifying the correct underlying DAG for each method. Do-Finetti identifies unique causal structures and achieves near-perfect causal effect estimation.
from environment e. Data across environments are independent and identically distributed in the sense that the distribution of [[]]] and X [[N]] is identical for all N min(Ne, Ner). Each environment thus provides finite marginal of an i. i.d copy of the same exchangeable process.
The Do-Finetti algorithm combines Causal-de-Finert algorithm developed in Guo et [203aa and the truncated factorization for ICM generative processes developed in Section 3 of this paper. provides proof-of proof-of-concep of-concept algorithm verify that multi-environment data generated under the independent causal mechanism principle alone via exchangeable process can simultaneously estimate the causal effect and causal graph. See Algorithm Ii Appendix[]for details of the procedure.
5 Experiments
We construct synthetic datasets according causal Pólya model (cf. Section 3.2 and demonstrate that Do-Finetti algorithm can estimate causal effects and graphs simultaneously y. We compare with the standard method in i.i.d. processes (i.e., PC algorithm and truncated factorization Eq. ( and show empirically that the traditional truncated factorization cannot estimate causal effects in our setting. We then perform ablation studies attribute errors either graphical misclassification or effect estimation.
Let superscript denote variables generated in environment The mata-generating process for
X + Y, for example, as follows:
~ Beta(a, 8), t6 ~ Beta(a, 3) X X Ber(O"), Y Ber( ① X.
where + denotes xor operation and X Y denote variables generated at the i-th position in environment e. We collect two pairs of random variables across all environments and run do-Finetti algorithm compute the post-interventional distributions with randomly initialized intervened variable and its corresponding values as in Eq. (22). We repeat the experiment for 100 times and report the mean squared error loss between predicted and analytic solutions across varying number of environments. Fig. 4 shows i.i.d. fails to estimate causal effects, giving high estimation error in ICM generative processes. Even with knowledge the true DAG and infinite data, i.i.d. with true DAG never achieves analytic solutions (Fig. 4a). On the other hand, do-Fineti achieves near-zero causal effect estimation error, meaning correct DAG identification and correct effect estimation. Appendix details exact experiment setups.

---

6 Discussion
Causality and exchangeability. Causal effect estimation in randomized controlled trials (Rubin 2005) relies on the exchangeability assumption between the controlled and treatment group. More recent work [Dawid] [02T] introduces a
framework for causality and uses pre-treatment and post-treatment exchangeability as foundational assumptions on external data used to solve decision problem. [Jensen et al. (2020) studies exchangeability in the context of object conditioning. It shows object conditioning, due to exchangeability, mitigates latent confounding and measurement errors for causal inference. [uo et al. [2023a] characterize independent causal mechanisms in exchangeable data via causal de Finetti theorems. Reizinger et al. (0024) proposes relaxed conditions for causal discovery in ICM generative processes and shows exchangeable non-i..d. data is the key to both structure and representation identifiability. This work extends on causal de Finetti and studies causal effect identification and estimation in ICM generative processes.
Causal effect estimation in multi-environment data. Bareinboim and Pearl (2016) study the transportability of causal effects for populations under different experimental conditions. [Peters] and Meinshausen (0016) quantify causal effect estimation and its uncertainties through exploiting invariance of causal mechanisms. Jaber et al. (2019) develop a complete algorithm for causal effect identification in the Markov equivalence class of causal graphs which was previously deemed impos- sible to identify further without additional assumptions in the i.i.d. framework. In the present work, we show ICM generative processes, naturally arising in cnliimenvironment data, allow simultaneous causal structure recovery and effect identification. Further, we illustrate via a causal Pólya urn model that exchangeable process, though related, is not limited to only multi-environment data.
Causality in non-i.id. data. A classical problem in causality is the study of interference, motivated by real-world complex interdependencies within subjects. For example, in social networks and infectious disease, treatments on one subject affects outcomes of other subjects. [Sherman| [2022] provides general identification formula of causal estimands under dependent data when graphs contain unobserved confounders. Oggurn and VanderWeele (2044) provides a comprehensive review on causal inference under interference and studies how to determine what variables must be measured for computation of causal estimands. Zhang et al. 2023) provides an elegant study on modeling uncertain interaction using linear graphical causal models. Maier [2014] studies causal discovery in relational data. This work focuses on a particular type of non-i.i.d. data, namely exchangeable data. In contrast with classical setting where assume aprior known causal graph and develop causal estimands from thereon, the series of causal de Finetri work focus on empirical inference of causality from exchangeable non-i.i.d data. This we hope paves the way to uncover governing laws of Nature from empirical observations.
Real-world Applications. Exchangeable data offers an expressive and realistic representation of complex structured relational data, which often appears in clinical studies [Bowman and George 1995]. microarray gene expression data [Qin| 2006), in high-dimensional inference tasks for images Korshunova et al., 2018 2020), 3D point cloud modeling [Yang et al. 2020] to topic modeling [McAuliffe and Blei 2007]. While it is beyond the scope of the present work, we hope that our work, through building theoretical framework of interventions in exchangeable data, may ultimately help study and understand naturally occurring mechanisms in scientific domains.
7 Conclusion
We study causal effects and prove a generalized truncated factorization for an important class of exchangeable generative processes. We show for conditional interventional distributions that other ob- servations are relevant to the causal query for exchangeable but not for i.i.d. data. We introduce causal Pólya urn models and demonstrate in in-practice interventions in exchangeable data. We develop Do- Finetti algorithm that performs simultaneous DAG and effect estimation from multi-environment data.
It is exciting to start to understand the complexities of non-i.i.d. data, and much is left to do: from developing algorithms that perform at scale to understanding counterfactual queries in exchangeable contexts. Appendix details limitations and broader impact. Going beyond i.i.d. has been major bottleneck when applying machine learning to real-world applications, and exchangeable data offers a doable and realistic next step.
10

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Table of Contents
Graphical Terminology
14
B Proof of Corollary |
14
Proof of Theorem
15
D Proof of L
16
Conditioned interventional effect on other experiments
17
E.1
Derivation of Eq. 39
17
Causal Pólya Urn Model
18
Exchangeability of causal Polya urn model
18
F2
Conditional independences of causal Pólya urn model
18
F3
Application of causal de Finetti thcorems
19
Lemma and its proof
19
Lemma and its proof
20
Do-Finetti Algorithm
21
Proof of Theorem 2
21
K Experimental Details
22
K.1 I Analytic solution
22
Limitations and Broader Impacts
23
13

---

A Graphical Terminology
A graph 9 consists of vertices V and edges E. The set of vertices is often denoted as {X .X.......X. ). We say a pair of vertices is connected with a directed edge X; - X,. The set of edges E contains set of pairs (X; + X; X,, X, E V). If there does not exist sequence of edges such that X; - ................................ - X; for some X; E V, then G is acyclic. Whenever X; - X; E E, then X, is the child of X; and that X; is the parent of Xj in G. One use PA, denote the parents of X; and CH; to denote its children. A sequence of vertices XI,..., X& forms directed path in G if, for every i 1, 1,...,h, k; we have X; X One say that X is an ancestor of Y in G, and that Y is descendant of X, if there exists a directed path XI...., X with X1 = X and X₂ = Y. We write DE; as descendants of X; and ND; as non-descendants of X; and ND; as non-decendants of X; excluding its parents.
Both graph and probability distribution encode conditional independence relationships. Given a probability distribution P. we say X IL YJW, if P(X W) P(XJW). In graphical terms, we detect the set conditional independence
a DAG through d-separation.
Definition (d-separation) A path p is d-separated by a block of node Z if and only if one of the two conditions holds: 1) p contains chain X; + Xm - X2 a fork X, t- Xm + X₃ s.s. Xm E Z: 2) p contains X; + Xm +X,, X s.t. the middle node Xm & Z and DE m. of Z. We then say Z d-separates X and Y it blocks every path from node in X to a node in Y.
Definition 5 (I-map) Given a probability distribution P, I(P) denotes the set of conditional inde- pendence relationships of the form X L Y Z that hold in P. Given a DAG G, I(g) denotes the set of conditional independence assumptions read-off via d-separation.
Definition 6 (Markovian and Faithful) A distribution is Markov to a DAG G ifI(G) _ I(P).
A distribution P is faithful to G ifI( S I(G).
Next proceed to define what is ICM(G). through explicating the ICM operator.
Definition 7 (ICM operator on DAG 9) Let U be the space of DAGs with vertices XI,..., Xa- Let V be the space of acyelic directed mixed graphs (ADMG (Richardson) 2003)) with vertices {(X+x) where E [d]. n N. A mapping F from U to V is an ICM operator if F()) satisfies 1) F(G) restricted to the subset of vertices {X .dmm] is DAG G. for any E N; 2) X + Xim whenever n + m for all i [d]: 3) there are no other edges other than those stated above, We denote the resulting ADMG as ICM and PA im denotes the parents of Xig and similarly
& for the corresponding non-descendants.
B Proof of Corollary 1
Definition 8 (Exchangeable arrays) An array of size d contains variables (Ximy..., dmm) where Xam denotes the d-th random variable in n-th array. Such an array is denoted as X A finite sequence of size d arrays is exchangeable, if for any permutation map n,
= =P(X, P(.,......,,..
(18)
Next we show exchangeability implies identical marginal distributions. This is standard result in exchangeable sequences (de Finetti 1931], here we include it for completeness.
Theorem (Exchangeable implies identical marginal distribution) Let P be the distribution for some exchangeable processes, where X.,...., X.:N are exchangeable arrays. Then for any n + m:
(19)
Proof of Theorem Without loss of generality, assume a < m. Let π [N] + [N] the
permutution mapping such that xi)) i, Vi m, and nt m(n) m. (())
Then, by definition of exchangeabillryy
P(X X.:1..., X
X X.m Xam X X::N)
(20)
P(X X;:1, X X X Xmm X.n..... XN,N x..N
(21)
Integrating out every values X.yn; m. + n, we have P(X P(X.mm x..) = P(X x:n) for all values Xyyn Therefore, X.m and X., shares identical marginal distribution.
14

---

Corollary (Identical marginal post-interventional distributions) Let P be the distribution for some ICM generative process. Let and J be two disjoint subsets [d] == {l....,d} Denote X1. == (Xig ; {] and similarly for X Then,
P(X do(Xim
1m =)) = pm do(X
= =x)), Vn o m
(9)
ie., identical interventions on variables in different positions share the same marginal post- interventional distributions. See Appendix Bfoor the proof.
Proof of Corollary By Theorem 3 P (m)) Vn + m. One can thus drop the notation of position indices when considering X for some TL as X.; from identical P. Due to truncated factorization in i...d. case (Eq. 2 any post-interventional distributions can be represented as pre- interventional distributions. Thus post- inteeveteveennnll distributions in P(X :n. X (oo(X; 1:n x)) uniquely determined by and is P(X.;. X:mn |do( X I.m = x)) Since identical, then so is the inst-interventioall distributions. Marginalizing out irrelevant variable indices that are not
contained in I JUJ J gives the desired result.
C Proof of Theorem 1
Notation Let X denote the random variable corresponding to i-th variable index and n-th position index for data generated under ICM generative processes. Write X == ; xd:)) and
X.jNN] (Xa;1.-- Xd:N). ). Define UP:: :;: (X,+1:1-.... Xd;), which contains all random variables that have higher variable index value than i, i.e upstream of node i.
Theorem (Truncated Factorization in ICM generative processes) For a given graph G, let P be the probability distribution for data generated under an ICM generative process with respect to G and let p be the corresponding density. The post-interventional distribution after interveming on X & has density given by:
p(x:II,... ..... Xx:N/do x))
II p(X.(-N, pa ((NN.) xx
II p(x.,[M) pa <dix
((N)
x=
(12)
where Ix == (i: E X) denotes the set of variable indices being intervened on and N; := {n Xim X) denotes the set of position indices corresponding to variable index in the intervention set X and [-N:] denotes the set of positive integers less than or equal to N excluding values in N
Proof of [1 Without loss of generality, we reorder the variables according to reversed topological ordering. i.e., a node's parents will be placed after this node. Note a reversed topological ordering is not unique, but it must satisfy node's 's descendants will come before itself.
By chain rale, P(X. P(X.,....,X.. X..N) = II. [ & By Theorem 7 (Causal conditional de Finelli) in Guo et al. (2023a), since P is generated under some ICM generative process, there exists latent variable 0, with suitable probability measure v such that
/ II p pa 0.)
((X:IN|UPHII)
(22)
/
NN
II
p(z ppag
km O))p(0.)d0
(23)
In this work, assume for every probability measure there exists a corresponding probability density associated with it and thus the second equality follows.
Note by the second condition in causal de Finetti theorem multivariate version in Guo el al. (202aa) we have
X s:jl s|n)
where PA; selects parents of node i and ND; selects non-descendants of node in G. ND, denotes the set of non non-descendants of node excluding its own parents. Then the naive application of the conditional independence on P(X ([M] UP
H[M) gives:
P(X UP s(N) P(X.N)) PAAN))
15

---

Combining gives:
N II n=
P(xe]N]lPPstt = P(X Ina (IN)
p(aem pa Ox)p(O:)dO:
(24)
Applying Eq. 24 on the chain rule decomposition of the joint distribution,
"(X.i...,,X.. II P[X PA
P ((N))
(25)
Inheriting Definition
When E Ix. intervention on X does not change the probability distribution but just enforces
consistency PA with values in &, i.e..
PA 9 i[M)) do(X x
Vi Vi Ix
(26)
When E Ix, intervention on X affect variables only contained in the subset of X
Let
Xin E X.N, Then p( I iin 0:) = 8(Xi;n = Pign). Aggregating together,
P(X Ipa 1:NN]: do(X x))
II 6(Xim = =m
p( 2 Cn 0 O0), nE[-N.]
II
(27)
mEN:
{
P P(x (Da
when pa consistent with X, Vn E N; else
(28)
P(xsI-N.lpad .N.) x X= Ix
(29)
Combining Eq. 26 and 29
X do = 8))
II
P(X.) N.
PA (-NN.)
II ig1x
P(X.NN)
PAG PAGI
x=
(30)
iElx
D Proof of Lemma 1
Lemma (Intervention effeet conditioned on other observations) For a given graph G, let be the distribution for the ICM generative process with respect to G. Let X be the intervention set.
Assume X X1n = {Xim Vi I}. Let [N] such that n o S and [-]] denotes [d] I. Then,
P(X Im (o X1mm = x). x
II P(X X PA PA I
(15)
Proof of Lemma
By Theorem 7
P(X su(n) do 1X fm =x))
II P(X, is PA
II (u(()) PA (U Ixx
Xxm=x
(31)
By Lemma since n ES,
P(X.ssdo(Xo = x)) P(X.s) = II P PA iss
II
SPA
(32)
iET
igt
By do-operator can be used as normal conditional probability,
P(X (u(() |d (X = =x))
P(X-Ixn|do(X) = X.)
(33)
P(X,s do X Xin = =)))
Nle P(X; PA (. llg P P(X,, PA (l) m
Iher P(X.ss 9
Iligs P(X:ss PA
(.s)
(34)
P(X P(X (S) PA. ((((()))))
II
x
(35)
P(X.ss|AA
16

---

By P is the distribution for some ICM generative process, it satisfies the conditions of causal de Finetti theorem (multivariate) version:
X T 2D
PA s:[m]
where PA, selects parents of node and ND, selects non den-descendants of node i in G. By exchangeabil- ity, one can use arbitrary set other than [n], here use S. Applying above conditional independence, we have
X;s 1 PA PA
as PA G C C
km Therefore,
P(X: (s.) P ss PA (ss(n))
(36)
With properties of conditional probability,
((X) 9 (.SU(n) Xin= P(Xim Xiss PA (sS(()))) x
(37)
P(X, PA
The result follows.
E
Conditioned interventional effect on other experiments
Under i.i.d. generative processes, if different interventions are placed on the system, then one often considers that data coming from different environments and observing one experiment result cannot provide extra information about other experiments performed positions of the sequence. For example, consider sequence of random variables generated by i.i.d., i... (X1, Y1, (1),((x Y2, Z2).... ... such that Xn. - Yn + Zn for all n. Then, i.i.d. generative processes :P(Yi]do(X, x), do = y), 22) =
random variables in different
= x))
(38)
due (X,,Y, Z1) 1 (X2, Y2, Z2). Therefore interventions on inconsistent intervention targets across the i.i.d. generated sequence can be equivalently considered as mixture of data sampled from different environments, i.e., identical copies of SCMs with different hard-in sanrdintervenioos..
However, this is not the case in ICM generative processes. Continuing the previous example, and let the sequence be data generated from ICM generative process. Then,
ICM generative processes : :P(Y 91 (o do(Y, = y), Zz = 22)
/ p(v1 x. 8z (p()))(())))) 2222 y)
(39)
Eq. 39 explicitlly shows that knowing the intervention effect Yg acting on Zz helps one to infer the causal de Finetti parameter oy and thus the intervention effect of X1 on Y1.
E.1 Derivation of Eq. 39
Consider a DAG g == X + Y - Z and a distribution generated by the ICM generative process
corresponding to (g). By truncated factorization in Theorem | we have:
P(Z1 = 21, Yi = y1, X1 7/ = 22, Y22 = y. X₂ =22|do( X1 -), do(Y22 û)) 6(X, = 110===)ppplllxx00000 )d0x / plunl 0y by (8(Y2 = o)p(Or)
(40)
/
p(zz 10. Oz p (n, Oz
)d02
:=p(a))
(41)
p((2) / p(ezl0 02) 12
(42)
/ p(ze. 0200) vi 02 d20
= =p(#2)
(43)
The merge of p(=22p( Oz (0x) = p(22. 0z 9) is due to 0z is independent of Y,, for any n as Z,
is always a collider in the path. The merge of p(z1 111 Oz )(y) ( p(21, y1 ( Oz) is due to 2, (0,, Y,. Marginalizing away X2, Z1 and note PiZz = 22 |do{ X1 = 2), do = g))
P(Zz 22 do =x)) by Lemma 2 We have the desired result.
17

---

F
Causal Pólya Urn Model
E. Exchangeability of causal Pólya urn model
We first want to show exchangeabilityy of the pair of the random variables (X,, Y2) in the sequence
X1, Y1, X2, Y2,-.., Xn, Yn that are generated via causal Pólya model. Assume n pairs of
(Xe, Y,) are observed: out of n balls we observe 71 times that X, = 1. Y, = 1. f2 times that X, =,,Y, = O, ny times that X,, 0, Y, = and n4 times that X,,
0, Y,
0 where
n1 + n2 + n3 + 144 = n. Note for the first time the number of balls in the urn for left and right compartment is a + each, for the second draw a + 8 1 each, and for the k-thh draw + B k
each. The probability that we draw (X, = 1, Yn = 1) pairs first, followed by (Xn = 1, - 0) second, and then (X,, 0,
1) third, followed by (X,, = 0, Y, = 0) last is:
= 1
1, ....,, Xn = Y, =0)
(44)
= a x
B
a a+1 x B+1
a B1 1
B+ -1
x
x x
X
a a
a+ B
a+ B+1 a+3+1
a + B + n1 -
a B 121
(45)
a +++0
a
+ n1 + 112 1
a' + 112 1
X
X
X X
X
(46)
Q B + 71
a + B + 11
11 + 112
Q + + n1
+ 112
a + 122
8 n13
a a+ 122 + 11/
x
x
x ... x
x
Q B + f1 + 12
a a 3 + n1 +
a + 8 + n1 + 1/2 + n233 -
0+ 3 + n1 + 12 + n23
(47)
B + n3
8 + n2
8+ n2 + 144 -1
B + n1 + - 1
X a B + 11 n22 + n13
x x
X
a + 3 + m + 122
n13
0 + 3 + n
a' + 3 + n
(48)
For any permutation of the order T: : [n] + [n)]. the denominator will not change as each step, we put an additional ball in each of the left and right compartments of the urn regardless of the color of the ball observed.
If we observe j-th black ball, i.e., X,, = 1 in the left compartment and 1-th Y, (X, in the right compartment such that Z,, = 1 at step m, then the numerator in the probability is + 33-1 1) x (a- I 1-11 1) with the denominator as (a + 3 + m - 1) x (a + 3 + - 1). Therefore, for any sequence ær; ......, Nw: Un if X; = 1 occurs m. times and Z; =1 (1 - X.) x Y + (1 Y2) X, occurs mz times, then the final probability will always be equal to
P(X, 11, Y1 = 11, ,,,,...,,,
Xn = Fn Yn = yn))
(49)
Il (a +i 1) 11=-mm (8 +i 1) IT (a (a+ 1II1 1 m-
j=1
(50)
IT= (a + 3 + 1: 1) (a k
(a m-+ 1))(3 + n - m1 -
(a + m2 - 1)1(13 +
m2 1)1( + 3 - 1)!
(a (a- 10(33-))((a-)) 1)133)1)11(++=--1-1-
(a-1)!(B - 8 1)!
(51)
F22
Conditional independences of causal Pólya urn model
For any n E N, we show Y(n) I Xx+1 Xjn). i... P(M X(m)) = X[m))=((M()) ((n))
By conditional probability is the division of two joints, we have
P(X, 21, Yi 3l.............................. Xn.
En, Yn yn)
P(Yn = "n...., Yi = 31 I Xn= ,n...., 11= =x=]=
P(X: = 31, Xn, = xn)
(52)
la (a +i 1)
(a (1+
(B+j-1) (a +
" my (3 (8+i-1 1)
(a + 3 + 6-11 1)
(53)
m2 (a +1- 10- (x
(54)
I + 88+
18

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To compute the joint probability of
P Fi, Y1,Y1 = y1, y1,....
Xn = n = ynn Xn+1 Fn+1)
(55)
my (a+i || 1 1) (a j- 10--1)111 1 ( -
(56)
Th=1 (1+ - (a 3 + --
(a + m1)d + (5+ (0-m11 L
(57)
a' a+ ++
Dividing P(X1 F1 Xn In, Xx+1 21+1) leads the same result.
E3
Application of causal de Finetti theorems
According to causal de Finetti theorems, given the causal Pólya urn model satisfies both exchange- ability and certain conditional independence relationships, there must exist two unique independent prior functions that allow the joint distribution to be factored into two independent products. Here we provide explicit representation. Let B(; or, B3 denotes beta distribution with parameter a, B and 1o+B-111 0 (1 (1-0) 0)8-1
B0; ax B) = to- 11!
P(X1 21; Vi = y1....; Xn = In, Yn Vn! g 21 (1 0) (n-
(58)
En, 2x) B(0; a, 3)
(59)
/
2 E. 111 =n mn + yn men (1 (1- -) (-0)--E11
Blu a, B)
(60)
(a B 1)! 2 / (aatIn 11 (1 f)m- 1
(61)
la (a-13 1)!
jatIn, (1 ~ En, =n deb where (1 - In,) Un + (1 - yn)
(62)
note Σ In mi, Σe 2n m2, by conjugate priors with bernoulli and beta distributions
(63)
(a + B 1)!
2 133 + n - m---m)]((aa- l' m2)l m2)
(64)
- 11-1) - 1)!
T(a ((
T(a + + +B++)
(a + mi 11)((3 + - 1))((a 3 1)! (a + m2 1/m2-)1(B--)) + n
m2 1)1(( + 3
1)!
(a-1))(B 1)](a 3+n
(a 1)1(3 1)]((a + 1)!
(655
G Lemma 2 and its proof
Lemma 2 (Intervention effect on differing positions) Given graph G and let P be the distribu- tion for the ICM generative process with respect to G. Let X and Y be two disjoint sets such that X is the intervention set and Y is the target set. Let Nx ;= {n X E X3 be the set of position indices being intervened, and similarly Ny be the set of position indices being targeted to observe. Assume Nx n Ny = 0. Then,
P(Y =
(16)
Proof of Lemma 2 Given an ICM generative process, the sequence is exchangeable by definition. By causal de Finetti's theorems, the joint distribution can represented as
p(x N:N) f ... p(x
1
(...dd().... dv(Oa)
(66)
II nENY
p(x 1........................ 0a) II
oix.. (.).d(()))......oo..))))))).))). de (0
ngNY
(67)
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The post-interventional distribution is:
p(x;11 ........ X;,N do =x))
(68)
p(x |d( = x), 01........ 0a) II
o(x,m//o(X = 01.0....... ...... 0x)dv(0
(
neNy
n
(69)
II nENY
p(x:m 10..l00................... Oa) II ngNY
p(x:: do = 0,)dv(())) dv(Oa)
(70)
The second equality follows from Definition 3 on the operational meaning of do-operator in ICM generative processes: since Nx n NYv there is overlapping variables between X.m. and X. for all n Ny, which leads to no change of probability distribution.
Next, we proceed to integrating out every variable in for & Ny. Due to do-operation replaces intervened variable's 's conditional distribution as detta-distributtonss and keep the rest conditional distributions while enforcing consistency over the intervened set of variables, the post-interventional distribution is fact a conditional distribution by Theorem with enforced consistency. Thus integration on all variables lead to summation 1.
J...
/
II ngNY
/do
,))),,..,.. =1
(71)
Therefore,
p(x:Ny |oox/do((X =x))
II
pxx 0 , (.)....,,)d((ii)......
(72)
BENY
-((:NN)
(73)
Marginalizing every other variable in N, except the target variables in Y, the result follows.
H Lemma and its proof
Lemma (Intervention effect within the same position) Given a graph G and let P be the distri- such that
bution for the ICM generative process with respect to G. Let X be the intervention consists only X where n. E S C [N] and I C [d] a set of variable indices. Let Y be the target set such that it consists only XJ; where
[N] and J C [d]. Note InJ 0. Then,
P(Y do = x)) = & x.pa))
(17)
pax
where PA x denotes the parent of intervened variables X.
Proof of Lemma The proof follows a similar idea as in the case of i..dd. duta, here we include for completeness. Note the joint distribution be written (by argument in Proof of Theorem 77
N) = = PA
(74)
Then when intervening on X which shares the identical set of variable indices across different samples, do-operation means assigning P( PAA,N) to distriibitiibtion for all i E I. This can equivalently expressed division of conditional distributions:
P(X
when X = =x
P(x = x))
TTe P(x.N PARN)
(75)
else
Next we proceed to show
II P(N.N) P(XIN|PAIN) -
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Without loss of generality, we can always order XIN according to reversed topological ordering such that all X; non-descendants will be placed after the variable index i. Note if X; is the non-descendant of x, then so is PA If not, suppose some variable in PA is a descendant of X; then X, is a descendant of that variable, so X; is descendant of X,, contradiction. This means after re-ordering due to NNN L N |PANN where ND = PARAAON U X S in this case, we have:
P(XIN PAIN)
II P( PAIN X SEN) II P(X ( iEI
(76)
Then,
P(X.N) P(PAX)
N| = x)) = P(x, PAx)
(77)
Integrating on every variable other than those contained in Y, we have
doY = x)) = Σ P(Y P(pax)
(78)
pax
I Do-Finetti Algorithm
Do Finetti algorithm recovers graphical estimation and causal effect estimation simultaneously
through collecting multi-environment data, that is marginal copies of an exchangeable process. The algorithm combines Causal-de-Finees Finetti algorithm developed in Guo al. [203a and the truncated factorization for ICM generative processes developed in Section of this paper.
Algorithm "Do-Finetti" Algorithm: causal effect estimation ICM-generative processes
Input: For E, we have (Xima:..., X ( where X denotes the i-th variable observed nith position in environment Let N denote the number of observations in each environment e. Assume N 2 2. Let X and Y be two disjoint subsets of variables such that X, Y S ".:.....).dd)))) J.N I==
Output: P(Y = yddo=
Step 1: Run Causal-de- iieeFiietri Algorithm on multi-ennvionmmnt data and return estimated Ś.
Step 2: Estimate the probability distribution from grouped data P(X........ X.,N,, e.g., histograms for discrete variables or kernel density estimation [Simonoff] [996] for continuous variables.
Step 3: Marginalize out variables not included in X U Y according the truncated factorization for
ICM generative processes with G = و (see Eq. 12 or below):
p(x.I,...,,,, x:: =)) = II
p(x p( pa G (NN.))
II
p(x.NN) pa (N))
lg1x
Step 4: Return P(Y x
Proof of Theorem 2
Theorem 4 (Unique graphical identification theorem [uuo et al.) 2023a Consider the set of distributions that are both Markov and faithful ICM (G), denoted as E()). Then,
E(Gi) = E(G)) if and only if G2
(79)
Theorem (Causal effect identification in ICM generative processes) Denote Y, be two dis- joint subsets of observable variables in (X.,)) )ne. Then P(Y do = x)) is identifiable given the distribution P from the class of distributions generated from ICM generative processes. Here
identifiability means the causal query can be computed uniquely from P.
Proof of Theorem Denote Pe be the set of distributions that is Markov and faithful to the DAG G. Consider a statistical model P = {PG G is a DAG). We say G is identifiable from P if G + Pg one-to-one, i.e., Pc, Pc, if G1 Gz. This directly follows from Theorem in Guo et al.) 2023aj. As G is identifiable from P theoretically, given the puir P. G, by Theorem 7 in this paper, we know the truncated factorization over all observable variables. Through marginalization for arbitrary target set Y and arbitrary intervention set X, the causal query is uniquely computable.
21

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K
Experimental Details
The data-generating process is described as below for each of the bivariate graph:
0 Beta(a, {), 1/ Beta (3)
X == Ber((*) ), Y = ()r(b)) + X
Y - X : = Ber(i") ② Y Y, = Ber(
X T Y : == Ber((*), X == (er
where @ denotes xor operation and X;, Y denotes variable generated at i-th position in environment and set = 1,3 = 3.
The experiments can be reproduced using single laptop with CPUs with a time estimate within minutes. The code is building on top of Guo et al. [2023a] under license CC-BY 4.0. For each experiment, we randomly generate data corresponding to one of the three graphs. We randomly choose the intervened variable X = X1. Yi, X2 or Y2 and randomly determine the intervened value & to be 0 or 1. Our goal is to estimate the post-interventional distribution 1(X1, Y1, X2, Y2 Vo(oo = &)) as accuracy as possible. Our mean squared error loss is the predicted Í with analytic P summing over all possible enumeration of X1, Y1, X2, Y/ values. We use PC algorithm for i.i.d. graph identification and truncated factorization as in Eq. for effect estimatinon given estimated DAG. Similarly, we run Do-Finetti algorithm. We compare with the analytic solutions that can be derived due to Beta- bernoulli distributions are conjugate priors towards each other. Section K. shows an example of derivation. To clearly see errors in MSE loss in causal effect estimation is not purely driven by graphical misclassification, we provide true oracle DAG to each of the method - Co--Fiietti---Fee dag and -ID-w-true-dag- - and observe that even with inifinte data, IID can never achieves near-zero causal effect estimation errors as shown in Fig. 4a This means the traditional truncated factorization fails to characterize causal effects in ICM generative processes.
K.1 Analytic solution
We choose Beta and bernoulli distributions in particular due to they are conjugate prior properties to allow us calculate easily their analytic solutions. Note the distributions can be arbitrary and we can approximate the integration via standard methods, c.g. histograms or kernel density estimations. Taking the example of X + Y generated under the data-generating process described in Experiment section, then by truncated factorization in ICM generative processes we have:
P(Yi)do(z1); X2, Y2) = P(Y X2, X2)
(80)
P(Y1 Y2 XX2)
(81)
P(Y2 X2)
The second equality is due to ICM generative processes properties Y2 1 X2, Note by causal de Finetti, we know:
P(u,, Vel = ff plusles, ((b())
(82)
Given Bernoulli distribution, we can re-write the assignment function Y. = Ber (ve) @ X in terms of probability distribution, where = (1 x) y
y) and p(y)a (r v) o (1 きجزー
Note p(b) follows a Beta distribution such that p(v) = (1-0)- The marginal likelihood from Bla,g)
manipulation gives (2) .2)). where aN -En nn +a, N En nn + Bla,3)
Putting all together,
P(u (do(t)))) = Bla + 21 + 22, 2 21 22 +2+ 3))
(83)
B(a + 22, 1 + 3 - 22)
Recall B(a, B) (()) and [(n) (n (1).. Rearranging gives: T(++3)
P(y = 0/do(11 = 0), 22 0.12 0) B(a, 2 + B)
+2
x r(a+8+1)
(84)
B(a, 1+ B)
T(a + T(++3+2 3
T(8+1)
B+1
4
= 0.8
(85)
a ++++11
22

---

Similarly, P(y1)do(z1) P(v1 m1) = B(a + 2, 3 -1). Then, P(y1 0fdo(z1 =0) when a 3 3.
[(a)
I[+++t])
L Limitations and Broader Impacts
Limitations This work has taken first steps in formulating what intervention means in exchangeable data satisfying the ICM principle. However, much left to do: from understanding interventions to counterfactual queries, from Markovian models to semi-Markovian models. We note there is
a whole world of possibilities the traditional causality view - e.g., [Karlsson and Krijthe 2024 perform confounder detection in multi-enviromment data and from the view representation learning where [Reizinger al.) 2022] show that nonlinear ICA under certain assumptions can perform causal discovery in a de Finetti framework. We hope this work opens up possibilities to connect to interventional representation learning and the branches of causality research.
formalizing causality in exchangeable data settings: both from
Broader Impacts This paper is foundational research that studies causal interventions under exchangeable contexts. It is intended to advance the field of machine learning and through studying causality, this work intends to provide control and understanding over the world and the current increasingly more opaque and 'black-box' models to safeguard potential harms for society. However, we note, every innovation has potential to misused with malicious intent. For example, we foresee that once offering more control and understanding. users have the potential to have more precise manipulation. This paper at its current status still presents a huge gap between theory and practice.
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Reviews:
[{"review_id": "suZrEV8p1w", "reviewer": "Reviewer_Jkyk", "summary": "The paper formalizes the observational and interventional distribution under the ICM generative process, of which iid is the special case. It provides an identifiability result for the causal effect given that the causal graph is known. Then, it shows that both the causal graph and the causal effect can be identified simultaneously.", "questions": "1. Definition 3: We should also break the edge from the de-finnetti parameters to the intervened variable, right? Or do we not need any graphical operations?\n\n2. I am slightly confused by the statements in Lines 153-154 and Line 197. They seem to contradict each other. Is it due to conditioning on x1  and x2 that Eq10 and eq11 are not equal since, in ICM, they are not iid?", "limitations": "Yes, the authors have addressed the limitation in the Conclusion section and Appendix L.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "1. Problem: The problem is important as it will bring the causal effect estimation literature closer to real-world scenarios. \n\n2. Theory: The theoretical results are strong, especially Theorem 2, which shows that both the causal graph and the effect can be estimated simultaneously. I have not checked the proofs, though. \n\n3. Experiment: The experiment on the simulated data verify the theoretical claim.\n\n4. Presentation: The paper is well-written and easy to follow. All the notation and definitions are clear.", "weaknesses": "1. Experiments: I understand the main purpose of the work is to establish the theoretical foundation of causal effect estimation for exchangeable data, but it would be interesting to apply the method to some real-world datasets (not necessary for the rebuttal).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and their recognition of the importance of relaxing the i.i.d. assumption, a general problem in the ML community. Please see below answers to the  questions: \n\n> Definition 3: We should also break the edge from the de-finnetti parameters to the intervened variable, right? Or do we not need any graphical operations?\n\nYes, we need to break the edge from the de-Finetti parameters to the intervened variables when performing an intervention. This definition aims to clarify different implications when performing graph surgery on SCM and ICM processes. \n\n> I am slightly confused by the statements in Lines 153-154 and Line 197. They seem to contradict each other. Is it due to conditioning on x1 and x2 that Eq10 and eq11 are not equal since, in ICM, they are not iid?\n\nLines 153-154 state that IID is a special case of exchangeability whenever p(ψ) = δ(ψ = ψ0), and line 197 states that the causal effect differs whenever p(ψ) does not equal δ(ψ = ψ0). These statements are consistent in that IID is a special case of ICM when $ p(\\psi) = \\delta(\\psi = \\psi_0) $. However, our focus is on causal effects for the ICM exchangeable non-IID case, where $p(\\psi) \\neq \\delta(\\psi = \\psi_0)$. We show that the causal effects in the ICM exchangeable non-IID case (Eq. 11) differ from those in the ICM IID case (Eq. 10) due to the dependency among observations.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qLHeMXNdtP", "reviewer": "Reviewer_H65Y", "summary": "The paper studies causal effect identification and estimation in exchangeable data. The main result here is theorem 1, which shows that causal effects are identifiable in ICM generative processes.", "questions": "- In the experiments, it seems to me that the model generating the synthetic dataset is different from that described in section 3.2. In particular, in the experiments, X_i is sampled from a Ber(theta) and hence P(X_i = 1) = P(X_2 =1) = ... = theta, whereas if I understood the model in 3.2, then the probability P(X_n =1) will be much greater than P(X_1 =1) if for example all X_m =1 for all m < n. Are they actually different? Or did I misunderstood? And how can the model described in 3.2 be represented by equation 4? (I read F.2 but it seems to me that equation 51 follows the model in Section 5).\n\n- In the experiments, can the authors elaborate on the IID baseline? Do you run the algorithm on the \"full\" graph G which have nodes X_1 Y_1 X_2 Y_2? I assume this is what's been done as it is the fairest baseline, but I'm not sure. Appendix K seems to imply that and the main paper did not make it clear.\n\n- In the description of ICMs, the author mention the expression: \n> Causal mechanisms are independent of each other in the sense that a change in one mechanism P(Xi | PAi) does not inform or influence any of the other mechanisms P(Xj | PAj) \nWhat would be a concrete example where such condition is violated?", "limitations": "It has been addressed.", "rating": 7, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The paper provides a great framework to think about interventions in exchangeable data. Starting from what interventions should be considered (Definition 3) to identifying a procedure for computing the post-interventional distributions.\n- The paper presentation, at least in the first part, was simple and intuitive. I always found myself asking a question and then find it being answered in the next paragraph. However, probably due to space constraints, this did change in the latter parts of the paper.", "weaknesses": "- The latter parts of the paper is rushed and left me confused. For example, it is unclear  how causal de Finetti theorems apply to the Causal Pólya Urn Model, Theorem 2, and the entirety of section 4 is very rushed and I have struggled to understand what theorem 2 say exactly. \n- I have felt that the algorithm could have taken more of real-estate in the presentation of the paper. Also, it is unclear how the graph structure is learned in the algorithm\n- This is more of a nit pick, but the appendix contains a few typos and is in a worse state in general than the main text. For example, the use of index i in equation 51, 53, ...", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and their appreciation of our work. We hope to clarify their questions below:\n \n> The latter parts of the paper is rushed and left me confused. For example, it is unclear how causal de Finetti theorems apply to the Causal Pólya Urn Model, Theorem 2, and the entirety of section 4 is very rushed and I have struggled to understand what theorem 2 say exactly.\n\nWe apologise and will try to clarify. Appendix F shows the causal Pólya urn model can be equivalently modelled as in the causal de Finetti theorem, i.e., $\\int \\int \\prod_i p(y_i | x_i, \\psi) p(x_i | \\theta) p(\\theta) p(\\psi) d\\theta d\\psi$, where $p(\\theta), p(\\psi)$ are beta-distributions and $p(x_i | \\theta), p(y_i | x_i, \\psi)$ are Bernoulli distributions. This is a bivariate version of equation 4 when we only consider two variables X and Y. We will include a more detailed discussion on Appendix F in the main text for the next version.\n\nTheorem 2 says that for ICM generative processes, both causal graphs and causal effects can be identified simultaneously. This is in contrast to an i.i.d. process, where causal effect identification often requires access to aspects of the causal graph which itself is not identifiable from observational data, and thus it is assumed that the causal graph is provided in addition to the observational data. \n\n> Also, it is unclear how the graph structure is learned in the algorithm\n\nFor learning the graph structure, we refer to Algorithm 1 in Guo et al. 2024 [1], as the present paper focuses on the study of causal effects. We will make this point clearer in the next version.\n\n> This is more of a nit pick, but the appendix contains a few typos and is in a worse state in general than the main text. For example, the use of index i in equation 51, 53, ...\n\nThank you for pointing out the typos, we will correct them in next version. \n\n> In the experiments, it seems to me that the model generating the synthetic dataset is different from that described in section 3.2. In particular, in the experiments, X_i is sampled from a Ber(theta) and hence P(X_i = 1) = P(X_2 =1) = ... = theta, whereas if I understood the model in 3.2, then the probability P(X_n =1) will be much greater than P(X_1 =1) if for example all X_m =1 for all m < n. Are they actually different? Or did I misunderstood? And how can the model described in 3.2 be represented by equation 4? (I read F.2 but it seems to me that equation 51 follows the model in Section 5).\n\nThe model described in 3.2 can be represented by equation 4, because Appendix F.2 shows that the joint distribution $P(x1, y1, x2, y2, …) $ in the causal Pólya urn model can be modelled as  $\\int \\int \\prod_i p(y_i | x_i, \\psi) p(x_i | \\theta) p(\\theta) p(\\psi) d\\theta d\\psi$, where $p(\\theta), p(\\psi)$ are beta-distributions and $p(x_i | \\theta), p(y_i | x_i, \\psi)$ are Bernoulli distributions. This corresponds to equation 4 as this is the equivalent bivariate version where X is the parent of Y, and $\\theta, \\psi$ are statistically independent $\\theta_i$’s in equation 4. Therefore as equation 51 follows the model in section 5 (as the reviewer suggested) and it is the representation for the causal Pólya urn model (due to the arguments above and in Appendix F.2), we argue that it is the same as the one described in section 3.2. We hope it clarifies things and thank you for going into the Appendix. \n\n> In the experiments, can the authors elaborate on the IID baseline? Do you run the algorithm on the \"full\" graph G which have nodes X_1 Y_1 X_2 Y_2?\n\nThe IID baseline is taking into account the full graph x1, x2, y1, y2 and treats the variables as $(x_i, y_i) \\sim_{i.i.d.} (X, Y)$. This means there are no bi-directed edges connecting X1, X2 and Y1, Y2. The causal effect estimand for the i.i.d. case is analogous to Eq. 10. The experiment is designed to show that the truncated factorization developed for i.i.d. data indeed does not apply for exchangeable non-i.i.d. data, hence the need for the generalised truncated factorization introduced in Theorem 1. \n\n> In the description of ICMs, the author mention the expression:\nCausal mechanisms are independent of each other in the sense that a change in one mechanism P(Xi | PAi) does not inform or influence any of the other mechanisms P(Xj | PAj) What would be a concrete example where such condition is violated?\n\nThis condition will be violated when we decompose a distribution into non-causal conditionals. For example, suppose that for weather stations, altitude (A) causes temperature (T) but not vice versa, i.e. building a greenhouse effect on top of a mountain will not increase the height of the mountain. In that case, P(T | A) and P(A) will be independent causal mechanisms that can be changed independently in the generative process, but P( A | T) and P(T) will not be. This is described in more detail for instance in Peters et al., Elements of Causal Inference. The causal de Finetti theorem formalises ICM mathematically: suppose $\\theta$ represents a mountain and $\\psi$ represents seasons. With random measurements given fixed mountain and season, we have $T_i, A_i$. The causal de Finetti theorem says $A \\to T$ characterizes by $T_1 \\perp A_2 | A_1$. Equivalently, it means $P(T_1 | A_1, A_2) = P(T_1 | A_1)$, i.e. knowing the altitude of measurements in other locations will not help the prediction of the temperature measured at location 1. If $T \\to A$, causal de Finetti theorem says $A_1 \\perp T_2 | T_1$, equivalently expressed as $P(A_1 | T_1, T_2) = P(A_1 | T_1)$. However we know it does not hold as if $T_1 = -10$, and $T_2 = 30$, then one can infer it is likely to be in hot season and given observed a low temperature $T_1$, one could infer $A_1$ will be high in altitude.\n\n[1] Guo, S., Tóth, V., Schölkopf, B. and Huszár, F., 2024. Causal de Finetti: On the identification of invariant causal structure in exchangeable data. Advances in Neural Information Processing Systems, 36.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JvI7L31OZz", "reviewer": "Reviewer_nbBj", "summary": "The paper generalizes the traditional iid settings in casual inference to exchangeability settings by de Finetti theorem, and proposes a new model named the casual Polya urn model to illustrate the new scheme and to catch more relationship. The experiments show when the number of environment is less than 5000, the new schemes performs well.", "questions": "Besides above, 5. In casual inference problems, is it easy to identify the exchangeability, especially for the real data?", "limitations": "Besides above, 6. Some typos even in reference; for example, the first reference is not well-written. 7. Could see more complicated structure between theta psi and X. In the paper, psi is independent of X. 8. Typos, like ‘Nature’ should be ‘nature’.", "rating": 7, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "First of all, I am sorry that I do not know much about the casual inference. But the paper uses exchangeability instead of iid settings, which seems an improvement.", "weaknesses": "1. Aldous 1985 shows many (not all) conclusions in iid can be naturally transformed into those under exchangeability. So the theoretical improvement seems not much. \n2. Only simulated experiments for the casual inference problems. \n3. De Finetti Theorem is ‘iff’. So it is inappropriate to use methods based on iid settings on the exchangeable but not iid data to compare. \n4. For the experiment, what about the larger number of environment? It seems the original one performs better.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for taking the time and address the questions below.\n> Aldous 1985 shows many (not all) conclusions in iid can be naturally transformed into those under exchangeability. So the theoretical improvement seems not much.\n\nIndeed, many i.i.d. results transfer to the exchangeable case, however, the situation is more complex. Aldous (1985) presents a lecture on exchangeability in probability theory, covering topics such as de Finetti’s theorem, its consequences, extensions, analogues, distributions invariant under group transformations, and exchangeable sets. However, Aldous (1985) does not focus on causality. Our work shows that causal effect estimation in certain types of non-IID data (specifically those meeting an ICM assumption) enables us to draw conclusions that cannot be inferred from IID data alone. \n\n> Only simulated experiments for the casual inference problems.\n\nThe experiments are designed to show that the truncated factorization formula developed for i.i.d. data fails to apply for exchangeable non-i.i.d. data even given knowledge of the true graph. Fig. 4 shows even with near infinite data, the dotted blue line (i.i.d. with the true graph) cannot reach 0 mean squared error in contrast to do-finetti with the true graph. The simulated experiment thus merely creates a controlled setup to demonstrate that conclusions in i.i.d. (e.g. truncated formula) fail to apply for exchangeable non-i.i.d. data. \n\n> De Finetti Theorem is ‘iff’. So it is inappropriate to use methods based on iid settings on the exchangeable but not iid data to compare.\n\nWe agree. The standard i.i.d. methods aren’t appropriate. Maybe we have not expressed this well, and a statement akin to the above would help put the experiment in perspective? \n\n> For the experiment, what about the larger number of environment? It seems the original one performs better.\n\nWe are confused with the statement. In Figure 4 (both left and right), we observe that do-Finetti algorithm outperforms the i.i.d. baseline in the large number of environments. The left plots show that do Finetti achieves near zero mean squared error in causal effect estimation compared to the i.i.d. baseline which has high errors. The right plot also shows that do Finetti simultaneously identifies the correct graph in contrast to the i.i.d. baseline with low graph accuracy. \n\n> 5. In causal inference problems, is it easy to identify the exchangeability, especially for the real data?\n\nWe thank the reviewer for the question. There might be empirical tests one could run based on permutations and there exists some work for testing exchangeability in real-world data, e.g., [1], [2]. Likely, this may not be a closed question, especially in terms of causal inference and exchangeability. Though we’d hope to leverage the above work as a promising first step to study it for future directions. \n\n> 6. Some typos even in reference; for example, the first reference is not well-written.\n\nWe apologise and will thoroughly go over this for the revision. \n\n> 7. Could see more complicated structure between theta psi and X. In the paper, psi is independent of X. \n\nThis could be interesting future directions, however we see this as a starting point, and we found ICM assumption was ideal for us in that \nIt allows us to derive non-trivial results (Theorem 2) \nIt is an assumption which is common in the causality community [3]. \n\n> 8. Typos, like ‘Nature’ should be ‘nature’.\n\nHere we deliberately choose the capital letter “Nature” to show respect and reverence towards the governing laws of the universe. This though common in the literature, we acknowledge it is more a personal preference. \n\nOverall, we thank the reviewer for taking the time and if we adequately addressed your concerns over \n* theoretical improvement (exchangeable non-i.i.d. data reveals important properties current causal literature does not cover), and \n* fair experimental comparisons and results (i.i.d. methods fail to apply for exchangeable non-i.i.d. data and hence demands a new causal effect estimation function for exchangeable non-i.i.d. data and do Finetti algorithm offers a solution),\n\nwe invite the reviewer to consider raising the score. \n\n\n[1] Vovk, V., Gammerman, A., Shafer, G. (2022). Testing Exchangeability. In: Algorithmic Learning in a Random World. Springer, Cham. https://doi.org/10.1007/978-3-031-06649-8_8\n\n[2] Aw, Alan J., Jeffrey P. Spence, and Yun S. Song. \"A simple and flexible test of sample exchangeability with applications to statistical genomics.\" The annals of applied statistics 18.1 (2024): 858.\n\n[3] Schölkopf, Bernhard, et al. \"Toward causal representation learning.\" Proceedings of the IEEE 109.5 (2021): 612-634.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Reviewers were agreed in their support for accepting this paper, praising the topic, the strength of the results, and the clarity of presentation (particularly in the early parts of the paper).  The authors should revise the later parts of the paper, consistent with the comments of reviewer H65Y, including providing more detail on how the causal de Finetti theorems apply to the Causal Pólya Urn Model, Theorem 2, Section 4, and more clearly noting the reliance on Algorithm 1 in Guo et al. 2024. The authors should also fix the various minor issues (typos, math notation) mentioned by various reviewers.

The authors should also provide readers with better pointers to a fairly substantial amount of prior work on structural causal models in non-iid settings (e.g., Sherman 2022, Zhang et al. 2023, Jensen et al. 2020, Ogburn et al. 2014, Maier 2014). Perhaps the authors are only citing work that explicitly references exchangeability, in which case they should concentrate on that segment of the work that does that (e.g., Jensen et al. 2020, Ogburn et al. 2014). Note that these are only examples, and that the authors should do a more substantial literature review to find relevant prior work. 

References

Jensen, D., Burroni, J., & Rattigan, M. (2020). Object conditioning for causal inference. Uncertainty in Artificial Intelligence (pp. 1072-1082). PMLR.

Maier, M. E. (2014). Causal Discovery For Relational Domains: Representation, Reasoning, And Learning. Doctoral dissertation, University of Massachusetts Amherst.

Ogburn, Elizabeth L, Tyler J VanderWeele, et al. (2014). Causal diagrams for interference. Statistical Science 29.4, pp. 559–578.

Sherman, E. (2022). Observational Causal Inference For Network Data Settings. Doctoral dissertation, Johns Hopkins University.

Zhang, C., Mohan, K., & Pearl, J. (2023, August). Causal Inference under Interference and Model Uncertainty.  Conference on Causal Learning and Reasoning (pp. 371-385). PMLR.
Decision: Accept (oral)