A Causal Framework for Decomposing Spurious Variations
Authors: Drago Plecko
Elias Bareinboim
Venue: NeurIPS 2023 poster (2023)
Keywords: Causal Inference
Confounding
Fair and Explainable AI
TL;DR: We introduce a novel set of tools that allow for decomposition of spurious (confounded) variations between variables $X$ and $Y$ in general Semi-Markovian models.

Abstract: One of the fundamental challenges found throughout the data sciences is to explain why things happen in specific ways, or through which mechanisms a certain variable $X$ exerts influences over another variable $Y$. In statistics and machine learning, significant efforts have been put into developing machinery to estimate correlations across variables efficiently. In causal inference, a large body of literature is concerned with the decomposition of causal effects under the rubric of mediation analysis. However, many variations are spurious in nature, including different phenomena throughout the applied sciences. Despite the statistical power to estimate correlations and the identification power to decompose causal effects, there is still little understanding of the properties of spurious associations and how they can be decomposed in terms of the underlying causal mechanisms. In this manuscript, we develop formal tools for decomposing spurious variations in both Markovian and Semi-Markovian models. We prove the first results that allow a non-parametric decomposition of spurious effects and provide sufficient conditions for the identification of such decompositions. The described approach has several applications, ranging from explainable and fair AI to questions in epidemiology and medicine, and we empirically demonstrate its use.

A Causal Framework for Decomposing Spurious Variations
Drago Plecko and Elias Bareinboim Department of Computer Science
Columbia University
ap31149collumiaa.eii coltsssccoumba....om
Abstract
One of the fundamental challenges found throughout the data sciences to ex- plain why things happen in specific ways, through which mechanisms certain variable X exerts influences over another variable Y. In statistics and machine learning, significant efforts have been put into developing machinery to estimate correlations across variables efficiently. In causal inference, a large body of lit- erature is concerned with the decomposition of causal effects under the rubric of mediation analysis. However, many variations are spurious in nature, including different phenomena throughout the applied sciences. Despite the statistical power to estimate correlations and the identification power to decompose causal effects, there is still little understanding of the properties of spurious associations and how they can be decomposed in terms of the underlying causal mechanisms. In this manuscript, we develop formal tools for decomposing spurious variations in both Markovian and Semi- Markovann models. We prove the first results that allow a non non-parane con-aarmettric decomposition of spurious effects and provide sufficient conditions for the identification of such decompositions. The described approach has several applications, ranging from explainable and fair AI to questions in epidemiology and medicine, and we empirically demonstrate its use.
Introduction
Understanding the relationships of cause and effect is one of the core tenets of scientific inquiry and the human ability explain why events occurred in the way they did. Hypotheses on possible causal relations in the sciences are often generated based on observing correlations in the world, after which a rigorous process using either observational or experimental data is employed to ascertain whether the observed relationships are indeed causal. One common way of articulating questions of causation through the average treatment effect (ATE), also known as the total effect (TE), given by
Fly do(x1)] - Fly do(zo)).
where do(-) symbolizes the do-operator [9. and Zo. are two distinct values attained by the variable X. Instead of just quantifying the causal effect, researchers are more broadly interested in determining which causal mechanisms transmit the change from X to Y. Such questions have received much attention and have been investigated under the rubric causal mediation analysis [ [2 10
Often, however, the causal relationship may be entirely absent or account only for part of the initially observed correlation. In these cases, the spurious (or confounded) variations between X and Y play a central role in explaining the phenomenon at hand. Interestingly, though, tools for decomposing spurious variations are almost entirely missing from the literature causal inference
'The only previous work which considers decompositions of spurious effects is [6] However, this work considers the specific case of the covariance operator, and no claims are made about the general setting.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Phenomena in which spurious variations are of central importance are abundant throughout the sciences. For instance, in medicine, the phenomenon called the obesity paradox signifies the counter- intuitive association of increased body fat with better survival chances in the intensive care unit (ICU) [ While the full explanation still unclear, evidence in the literature suggests that the relationship is not causal [5] i.e., it is explained by spurious variations. Spurious variations also play central role in many epidemiological investigations [13]. In occupational epidemiology, for example, the relationship of exposure to hazardous materials with cancer is confounded by other hazardous working conditions and lifestyle characteristics 44, and such spurious variations themselves may be the target of scientific inquiry. Quantities that measure such spurious variations (or a subset thereof) are called spurious effects in this paper.
Spurious variations are key in applications of fair and explainable Al as well. For instance, consider the widely recognized phenomenon the literature known as redlining [S 77 in which the location where loan applicants live may correlate with their race. Applications might be rejected based on the zip code, disproportionately affecting certain minority groups. Furthermore, in the context of criminal justice [8], the association of race with increased probability of being classified as high-risk for recidivism may in part be explained by the spurious association of race with other demographic characteristics (we take closer look at this issue in Sec. 5). Understanding which confounders affect the relationship, and how strongly. is an important step of explaining the phenomenon, and also determining whether the underlying classifier is deemed as unfair and discriminatory.
These examples suggest that a principled approach for decomposing spurious variations may be useful addition to the general toolkit of causal inference, and may find its applications in wide range of settings from medicine and public health all the way to fair and explainable AI. For concreteness, in this paper we will consider the quantity
Y
P(y |x)
Ply do(x))
Figure 1: Exp-SE representation.
Ply x) - Ply do(x)),
which we will call the experimental spurious effect (Exp-SE, for short). This quantity, shown captures the difference in variations when observing X vs. intervening that X = xr which can be seen as the spurious counterpart of the total effect. Interestingly, the Exp-SE quantity is sometimes evoked in the causal inference literature, i.e.
graphically in Fig. [
Ply x)) Ply do(x)) 0
(2)
is known as the zero-bias condition 2 Ch. 6]. This condition allows one to test for the existence of confounding between the variables X and Y. A crucial observation that, in many cases, the quantity itself may be of interest (instead of only its null), as it underpins the spurious variations.
Against this background, do not exist in the literature. Our goal in this manuscript is to fill in this gap, and provide a formalism that allows for non- parameeettiic decompositions of spurious variations. Specifically, our contributions are the following:
note that tools that allow for decomposing the Exp-SE quantity currently
(i) We introduce the notion of a partially abducted submodel (Def. T). which underpins the inference procedure called Partial Abduction and Prediction (Alg. (akin to Balke & Pearl 3-step procedure ( Ch. 7)). Building on this new primitive, we prove the first non-parametric decomposition result for spurious effects in Markovian models (Thm. [].
(ii) Building on the insights coming from the new procedure, we prove the decomposition result for settings when unobserved confounding is present
models) (Thm. 3)
(iii) We develop sufficient conditions for identification of spurious decompositions (Thm 2
2 Preliminaries
We use the language of structural causal models (SCMs) as our basic semantical framework 99. A structural causal model (SCM) is tuple M := (V, U, F, P(w)). where V. U are sets of endogenous (observables) and exogenous (latent) variables respectively. F is set of functions fv;, one for each
V, E V. where V; + fv; (pa(V,), Uv,) for some pa(V)) S V and Uv; S U. P(u) strictly positive probability measure over U. Each SCM M is associated to causal diagram G [ over

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Uz
Uz
1
Uxz
Uxz 2
Uy
Uy
Ux
X
X =
(a) Causal diagram correspond- ing to the SCM in Ex.
(b) Extended representation of Ex. latent variables in red.
(c) Diagram of Ex. do(X x) intervention.
under
Figure 2: Graphical representations of the SCM Ex.
the node set V where Vi + V, V is an argument fv,. and V; <-- Vj if the corresponding Uv, Uv, are not independent (21. model with no bidirected edges is called Markovian, while model with bidirected edges is called Semi-Markovian. An instantiation of the exogenous variables U = is called a unit. By Y, (u) denote the potential response of Y when setting X x for the unit which is the solution for Y(u) to the set of equations obtained by evaluating the unit in the submodel Mg. in which all equations in JF associated with are replaced by X = x. slight abuse of notation, we also replace Y = y with just whenever the former is clear from the context. We next introduce an important inferential procedure for solving different tasks causal inference.
2.1 Abduction, Action and Prediction
The steps of the abduction -alioonnnattton---iee aaction-prediction method can be summarized as follows:
Algorithm 1 (Abduction, Action and Prediction 19) Given an SCM (F. P(u),, the conditional probability P(Yc = e) of counterfactual sentence "if were C then Y", upon observing the evidence E e, can be evaluated using the following three steps:
(i) Abduction update P(u) by the evidence e to obtain P(u e).
(ii) Action modify FF by the action do(C). where C is an antecedent of Y, to obtain Fc.
(iii) Prediction use the model (Fo, P(u e)) to compute the probability of Yc.
In the first step, the probabilitics of the exogenous variables U are updated according to the observed evidence E Next, the model M is modified to submodel MM.. The action step allows one to consider queries related to interventions or imaginative, counterfactual operations. In the final step, the updated model (Fc, P(u. e)) is used compute the conditional probability P(yc e). There are two important special cases of the procedure. Whenever the action step is empty, the procedure handles queries in the first, associational layer of the Pearl's Causal Hierarchy (PCH, (2). Whenever the abduction step is empty. but the action step is not, the procedure handles interventional queries in the second layer of the PCH. The combination of the consider queries in all layers of the PCH, including the third, counterfactual layer. In the following example, we look at the usage of the procedure on some queries.
steps, more generally, allows
Example 1 (Abduction, Action, Prediction). Consider the following SCM:
X + fx(xxX Uxz)
(3)
F:
Z fz(UU,UXXx
Y fx,
(5)
with PUUx, Uxz, Uz, Uv) the distribution over the exogenous variables. The causal diagram of the model is shown Fig. 2a with an explicit representation of the exogenous variables in Fig. 2h
We are first interested the query P(y (x) in the given model. Based on the abduction--rediction procedure, can simply compute that:
Ply 2)) 1
1)) 2 =)) = P(Ux, Ugz x)).
(6)
where the first step follows from the definition of the observational distribution, and the second step follows from noting the independence Uz, Uy 11 Ux, Uxz, X. the abduction step, we can compute the probabilities P(u,, Uzz x). In the prediction step, query P(y x)) is computed based Eq. 6
3

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Based on the procedure, we can also compute the query P(Ux) (see Fig. 2c
P(yz) - 1 (Y Uazy Us Uy) y) P(u).
(7)
where the first step follows from the definition of an interventional distribution, and the second step follows from noting that Yz does not depend on Ua- In this case, the abduction step void, since we are not considering any specific evidence E = e. The value of Y (xx, Uza, Ug Uyy can be computed from the submodel Mg. Finally, using Eq. [ww can perform the prediction step. We remark that
1(Y (xx, Uzz U2 Uy) = =y 100=<10 An(Y(ux: Uoss Ug Uy) = P(uz I, Uazs Ug Uy):
(8)
Viz
by the law of total probability and noting that X is a finermmiistticuuncciii of Uz Maz. Thus, P(y also admits alternative representation
P(yx)= Σ (Y Mxzi Mgs Ug) y) Pu, Uxzi Uu Ugy P( U2 Uy)
(9)
Σ 1(Y(4)= P(u Uwa ) P(uzz Usi Uy);
(10)
where Eq. [0]follows from using the independencies among U and X in the graph in Fig. 2b We
from revisit the representation in Eq. 10 in Ex.
3 Foundations of Decomposing Spurious Variations
After getting familiar with the abduction-action iatiutioon-attiioo-rriiiiioooo procedure, our next task is to introduce procedure that allows to decompose spurious effects. First, we define the concept of partially abducted submodel:
Definition (Partially Abducted Submodel). Let U1, S be a partition of the exogenous variables. Let the partially abducted (PA, for short) submodel with respect to the exogenous variables U1 and evidence E e be defined as:
==
u1, E)).
(11)
In words, in the PA submodel, the typically obtained posterior distribution P(u is replaced by the distribution P(uz e). Effectively, the exogenous variables U1 are not updated according to evidence. The main motivation for introducing the PA model is that spurious variations arise whenever we are comparing units of the population that are different, realization dating back to Pearson in the 19th century (. To give a formal discussion on what became known as Pearson's shock, consider two sets of differing evidence E = and E = e'. After performing the abduction step, the variations between posterior distributions P(u e) and P(u | e') will be explained by all the exogenous variables that precede the evidence F. In PA submodel, however, the posterior distribution P(ur) U1, e) will differ from P(u)) U1,e") only in variables that are in U2. while the variables in U1 will induce spurious variations. Note that if U1- = U, then the PA submodel will introduce no spurious variations, a point to which we return the sequel.
We now demonstrate how the definition of a PA submodel can be used to obtain partially abducted conditional probabilities:
Proposition (PA Conditional Probabilities). Let P(Y = y E = e denote the conditional probability of the event Y = conditional on evidence E = e, defined as the probability of = in the PA submodel -Ue.E= (i.e., the exogenous variables U1 are not updated according to the evidence). Then, we have that:
P(Y = E = =.)) P(U = E = e, u.))
(12)
3.1 Partial Abduction and Prediction
Based on the notion of a PA submodel, we can introduce the partial-abduction and prediction procedure:

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Uz.
Uz
Ux
21
Z2
21
Z2
Uy
Ux
Ux
(a) Causal diagram corresponding to the SCM in Ex.
(b) Extended graphical representation of the SCM in Ex, latent variables in red.
Figure 3: Graphical representations of the SCM in Ex.
Algorithm (Partial Abduction and Prediction). Given an SCM (F, P(u)), the conditional prob- ability P(Y y E eui) ee)) of an event Y upon observing the evidence in world where variables U1 are unresponsive to evidence, can be evaluated using the following two steps:
(i) Partial Abduction update P(u) by the evidence obtain (wn, w2) is a partition of the exogenous variables u,
un, e), where
(ii) Prediction use the model P(n)) P(uz u1, e)) to compute the probability of =
In the first step of the algorithm, we only perform partial abduction. The exogenous variables U2 are updated according to the available evidence E = e, while the variables U1 retain their original distribution P(u1) and remain unresponsive to evidence. This procedure allows to consider queries in which only subset of the exogenous variables respond to the available evidence. We next explain what kind of queries fall within this scope, beginning with an example:
Example 2 (Partial Abduction and Prediction). Consider the model in Eq. We are interested in computing the query:
P(y x uu
Σ (( = P(urzs Uz) Mz)) By Unzz Mar,
(13)
Σ (Y P(uzz, Uz P( P(ux Marz Un,
(14)
Σ (u) = P(uas Un Uy P(ux I Ugas Uay U2y
(15)
where the first step follows from Prop. 7 and the remaining steps from conditional independencies between the U variables and X. Crucially, the query yields the same expression in Eq. [T0 that we obtained for P(yz) in Ex. 7 Therefore, the conditional probability P(y uuss.Us) in a world where Uxz, Uz are unresponsive evidence is equal the interventional probability P(y=)-
As the example illustrates, we have managed to find another procedure that mimics the behavior of the interventional (do(X X .)) operator the given example. Interestingly, however, in this procedure, acttion-rrediction we have not made use of the submodel M, that was used in the abduction-acte watoon-acttn---ote prediction procedure.
We next introduce an additional example that shows how the new procedure allows one to decompose spurious variations in causal models:
Example 3 (Spurious Decomposition). Consider an SCM compatible with the graphical repre- sentation in Fig. 3b (with exogenous variables U shown explicitly red), and the corresponding Semi-Markovian causal diagram in Fig. 3a We note that, based on the partial abduction -rreccioon-rreictioon procedure, the following two equalities hold:
Ply x)) = P(y | a.2)
(16)
P(yx) = P(y I (onn))Ua)2)
(17)
which shows that
(x) Ply x²)- P(y Uess
(18)
The experimental spurious effect can be written as a difference of conditional probabilities y x in world where all variables U are responsive to evidence vs. world in which Uxz, Uxz,
5

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22
2
21 Z1
Za
P(y 20)
Ply | 1
Ply x
Ply Mixx
(a) Exp-SE Uix (v).
(b) Exp-SE Lx (Ux Uax (w)
Figure 4: Graphical representation of how the Exp-SE effect is decomposed Ex.
Procedure
SCM
Queries
Abduction- Abduution-Preediction
(F, P(u E))
Layer
PretionPredictton
(Fzz P(u))
Layer
Abduction-AAcioo-Preciiee
(Fa, P(M E))
Layers 1, 2, 3
Partial Abduucioon-Predictoon
(F, P(u1) E))
Layers 1, 2, 3
Table 1: Summary of the different procedures and the corresponding probabilistic causal models.
unresponsive to evidence. Furthermore, also consider refinement that decomposes the effect
Exp-SE = P(y x²)) P I' Ply [2/00)) -P Unor Uuaz)
(19)
variations of UnaI
variations of Uniz
allowing for additive, non-parametric decomposition of the experimental spurious effect.
The first term in Eq. 19 shown in Fig. Sa encompasses spurious variations explained by the variable -xz,- The second term, in Fig. 4b
encompasses spurious variations explained by Ux
For an overview, in Tab. I we summarize the different inferential procedures discussed so far, indicating the structural causal models associated with them.
Non-parametric Spurious Decompositions
We now move on deriving general decomposition results for the spurious effects. Before doing so, we first derive new decomposition result for the TV measure, not yet appearing in the literature (due to space constraints, all proofs are given Appendix[A]
Proposition 2. Define the total variation (TV) measure as TV (y) P(y x1) - P(y 2o). and the total effect TE as TE (y) P(y.) - P(yno). The total variation measure can be decomposed as:
TV (y) = TEro. (v) (Exp-SE,, (v) Exp-SE (y)).
(20)
The above result clearly separates out the causal variations (measured by the TE) and the spurious variations (measured by Exp-SE terms) within the TV measure. The seminal result from [00 can be used further decompose the TE measure, In the sequel, we show how the Exp-SE terms can be further decomposed, thereby reaching full non-parametric decomposition of the TV measure.
4.1 Spurious Decompositions for the Markovian case
When using the definition of a PA submodel, the common variations between X, Y can be attributed to (or explained by) the unobserved confounders UI...., Ut. In order to do so, we first define the notion of experimental spurious effect for set of latent variables:
Definition 2 (Spurious effects for Markovian models). Let M be Markovian model. Let Zi.... Za be the confounders between variables and Y sorted any valid topological order, and denote the corresponding exogenous variables U1...., Uv. respectively. Let Z(a) -Zi..... 2,} and

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2-11 (Z }.). U1g and U-] [b] are defined analogousty. Define the experimental spurious effect associated with variable Us+1 as
Vig Up+t (v) = P(u xltaa= P(y | Exp-SEz
(21)
The intuition behind the quantity Exp-SF EXp-SF--u M (y) can be explained as follows. The quantity Ply =/11a) captures all the variations in Y induced by observing that X apart from those explained by the latent variables UI,.... U, which are fixed priori and not updated. Similarly, the quantity Pyy | (1( captures the variations in Y induced by observing that X x, apart from those explained by UI.... Ue, U1+1. Therefore, taking the difference of the two quantities measures the variation in Y induced by observing that X
that is explained by the latent variable U4+1.
Based on this definition, we can derive the first key non non--aramen parametric decomposition of the experimental spurious effect that allows the attribution of the spurious variations to the latent variables Uy:
Theorem 1 (Latent spurious decomposition for Markovian models). The experimental spurious effect Exp-SE (y) can be decomposed into latent variable- speciffc contributions as follows:
k-1
k-1
Exp-SE (v)
Σ Exp-SE HM (n) (v) = Σ
Ply 2011) P(y x M.tt)).
(22)
An illustrative example of applying the theorem is shown in Appendix B.1 Thm. allows one to attribute spurious variations to latent variables influencing both X and Y The key question when such an attribution, as shown in Eq. 22 can be computed from observational data in practice (known as an identifiailily problem (9). In fact, when variables are added to the PA submodel in topological order, the attribution of variations to the latents U; is identifiable, as we prove next:
Theorem 2 (Spurious decomposition identification in topological ordering). The quantity P(y x x211)) can be computed from observational data using the expression
P(y x 214) = =EP(y P(zx) x)),))
(23)
rendering each term of decomposition Eq. 22 identifialle from the observational distribution P(e).
We discuss in Appendix B.2 B.2why a decomposition that does not follow topological order of the variables Ug is not identifiale.
4.2 Spurious Decompositions in Semi-Markovian Models
In the Markovian case, considered until now, there was one-to-one correspondence between the observed confounders Z& and their latent variables U:. This, however, is no longer the case in Semi-Markovian models. In particular, it can happen that there exist exogenous variables U, that induce common variations between X, Y. but affect more than one confounder Z,. We are interested in U, C U that have causal (directed) paths to both X, Y. described by the following definition:
Definition 3 (Trek). Let M be an SCM corresponding to a Cemi-Markoviads model. Let G be the causal diagram of M. A trek in G (from X to Y) is an ordered pair of causal paths (gi. 9r) with common exogenous source U, E U. That is. 9t is a causal path U, + .... + X and gr is a causal path U, - - Y. The common source U; is called the top of the trek (ToT for short), denoted top(gi. 9r) A trek is called spurious if gr is a causal path from U₃ to Y that is not intercepted by X.
When decomposing spurious effects, we in fact interested in all the exogenous variables U, that lie on top of a spurious trek between X and Y. It is precisely these exogenous variables that induce common variations between X and Y. Using any subset of the variables that are top of spurious treks, we define a set-specific notion of a spurious effect:
Definition 4 (Exogenous set-specific spurious effect). Les OTT S be the subset of exogenous variables that lie on top of spurious trek between X and Y. Suppose A, B C USTOT are two nested subsets of USTOT, that is A C B. We then define the exogenous experimental spurious effect with respect to sets A, B as
Exp-SEA..11) = P(y x² Ply ))
(24)
7

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Un
UBla
UB
Une
Uno
Ply a*))
Plyl x
Figure 5: Quantity Exp-SEA..B (v) as a graphical contrast. Dots indicate arbitrary observed confounders along the indicated pathway.
The above definition is analogous to Def. but spurious treks. Def. [ supports partial abduction of exogenous variables that are not on top of spurious trek, but we are seldom interested in these since they do not induce covariations of X, Y. The quantity Exp-SEA.BB (y) is presented as a graphical contrast in Fig. In particular, the set of tops of spurious treks USToT is partitioned into three parts (UA, UB A. U BC The causal diagram in the figure is informal, and the dots (..-) represent arbitrary possible observed confounders that lie along indicated pathways. On the h...s. of the figure, the set UA does not respond to the conditioning X x, whereas UB AAA UB do. This is contrasted with the r.h.s., in which neither UA U UBA respond to X = 2, whereas Upe still does respond to the X = conditioning. The described contrast thus captures the spurious effect explained by the tops of spurious treks in UR,4-
fixing different subsets of the tops of
Analogous to Thm. [ we next state a variable-speciff decomposition of the spurious effect, which is now with respect exogenous variables that are top of spurious treks:
Theorem 3 (Semi-Markovian spurious decomposition). Let UsTor = (U1, Un) S U be the subset of exogenous variables that lie on top of a spurious trek between X and Y. Let Un denote the variables U1..... ......,,. U₂ (U[0) denotes the empty set 0). The experimental spurious effect Exp-SE can be decomposed into variable variable-specific contributions as follows:
R-1
Exp-SE
Σ 1=
Exp-SE W/u
Σ
Ply x²x) - P(y (4(441).
(25)
An example demonstrating the discuss the question identification. We begin by discussing how annotate the exogenous variables given a aami-Markovian causal diagram:
decomposition is given in Appendix B.3 We next
Definition 5 (Top of trek from the causal diagram). Let M be a Semi-Markovian model and let G be the associated causal diagram. set of nodes fully connected with bidirected edges called a clique. A maximal clique C; is such that there is no clique C², such that C₃ ? C. The set of variables Usr can be constructed from the causal diagram in the following way:
(I) initialize USTOT= 0.
(II) for each maximal clique C., consider the associated exogenous variable Uc, pointing to each node in the clique; if there exists spurious trek between X and Y with a top in Uc., add Uc. to USTOT.
After defining the explicit construction of the set USTOT, we define the notion of the anchor set: Definition (Anchor Set). Let ................... Uj U be subset of the exogenous variables. We define the Ui) as the subset of observables V that are directly influenced
anchor set AS(U,...., UU) of (U, by any of the U.S.
AS(U, Up -U ch(U.))
(26)
Another important definition is that of anchor set exogenous ancestral closure:

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SCM M
Causal Diagram G
Z
Z1 - B
Z2 -- +
72
Z3 + B(0.3 + Z2)
4
Z3
X + B(0.2 7 A2 + 2z
Y + B + 0.2X + 1 + >2 + >zZ3)
X
Y
Table 2: SCM and causal diagram for the Synthetic example.
Definition 7 (Anchor Set Exogenous Ancestral Closure). Let U, CU be a subset of the exogenous variables. Let AS(Us) denote the anchor set of Us. and let an (AS(U)) denote all exogenous variables that have a causal path to any variable in AS(U.). Us is said to satisfy anchor set exogenous ancestral closure (ASEAC) if
U. =n an
(27)
Based on the above, we provide sufficient condition for identification in the Semi-Markovian case: Theorem 4 (ID of variable spurious effects in Semi-Markovian models). Let U, C USTOr- The quantity (ly 2Ua) is identifiable from observational data P(V) the following hold:
(i) X $ AS(U,). & AS(U,)
(ii) U, satisfies anchor set exogenous ancestral closure, U, (AS(U))
Some instructive examples grounding Defs. 7 and Thm. can be found in Appendix B.4 In words, the conditional expectation of Y given X in the partially abducted submodel w.r.t. a set U, identifiable whenever (i) neither X nor Y are elements of the anchor set of U, and (ii) the set U, satisfies the anchor set exogenous ancestral closure. Thm. Thm.4pprovel provides sufficient, but not a necessary condition for identification. An additional discussion of the conditions given in Appendix C We hope to address in future work an algorithmic way for identifying spurious effects in full generality.
Experiments
We apply framework synthetic example (called Synthetic A) with known ground truth, summarized in Tab. where the SCM M and the causal diagram & are given. The source code for the experiment can be found in jour repository For this example,
set the parameters A = A2 = Aa = 0.2. We then vary each parameter X; E 0.2) (while keeping the other two parameters fixed), which changes the value of the effect associated with latent variable U,. The effects associated with each U,, E {1, 2, are computed based on the decomposition Thm.
Exp-SE" (y) == SEp (y) Ply x0) 1 PPy 111)
(28)
Exp-SE (w) = Exp-SE / (01 (w) Ply =11)- ) P(y x UU)
(29)
Exp-SE (y) := Exp-SE
(v) P(y U.) - P(y UU).
(30)
The key task is to compute the ground truth values of P(y xu(4 for different values of i. According Def. we want obtain the conditional distribution of given X x but subject to not updating Uig according to the evidence X = Based on the true SCM, this can be done efficiently using rejection sampling as follows:
(1) Take samples from the SCM M in Tab. 2
(2) For all samples E N. with (k) such that
(31) re-sample the part of the unit (k) that is not included in Un (e.g., Un (U, U2), latent (k) (k)
((u # x,
(k) ui u2 are not re-sampled but uz is) and replace (k) with this new sample.

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Direction
Directon 2
Directon
Eflect Vallue
0.01
0.00
0.00
0.05
0.10
0.15
0.20
0.00
0.05
0.10
0.16
0.20
0.00
0.05
0.10
0.15
0.20
Coefficient Size
Effect
Exp SE
Exp - SE
Exp SE"
Figure 6: Experimental results on the Synthetic A example. Lines indicate the estimated values, dots the ground truth obtained from the SCM using rejection sampling, and the 95% confidence intervals are indicated with color. As expected, increasing the A5 coefficient increases the spurious effect associated with the latent variable Uz.
(3) Evaluate the mechanisms F of M for all units (k),
(4) If there exists a sample with X ((u()) * go back Step (2)
Return the mean of the Y variables Σx= Y())
Notice that the described procedure gives samples from the distribution P(y x*x). J. The values of UjJ are sampled only once and are not updated after the initial sampling. Other values in U, however, sampled anew until their values are such that they Therefore, the procedure guarantees that U[g do not respond to the evidence, whereas the complement
compatible with the evidence X
of UPj does, allowing to compute P(y x²4()) ) and in turn the expressions in Eqs. 28 30 The
effects are also estimated from observational data based on the identification expressions Thm.
Fig. demonstrates that the SCM-based ground truth matches the estimates based on Thm.
Conclusions
In this paper, introduced general toolkit for decomposing spurious variations in causal models. In particular, we introduced a new primitive called partially abducted submodel (Def. [T] together with the procedure of partial abduction and prediction (Alg. 2 This procedure allows for machinery
for decomposing spurious variations Markovian (Thm. I and Semi-Markovian (Thm. 3) models. Finally, we also developed sufficient conditions for identification of such spurious decompositions (Thms. [ 4). and demonstrated the approach empirically (Sec.5). T The main limitation of our approach the need for fully-specified causal diagram, which may be challenging in practice. However, from a fully specified graph and the data, our tools for decomposing spurious effects give a fine-grained quantification of what the main confounders are. As is common in causal inference, the granularity of the obtained knowledge needs to be matched with the strength of the causal assumptions (in this case, specifying the causal diagram). Conversely, in the absence of such assumptions, fine-graine gna-ggai quantitative knowledge about these effects cannot be obtained in general (2), and we hypothesize that precise quantification of spurious effects is not attainable in the absence of causal diagram. Finally, we discuss another technical solution that may alleviate some of the difficulty of causal modeling. Recently, cluster diagrams have been proposed [I] in which one can consider groups of confounders (instead of considering each confounder separately). and thus the specification of causal assumptions becomes less demanding (due to clustering, number nodes in the graph is smaller). However, causal decompositions a way to choose different level of granularity for settings where domain knowledge may not be specific enough to clicit full causal diagram.
described in this paper can still be applied to cluster diagrams. This offers
10

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References
[1] I. V. Anand, A. H. Ribeiro, J. Tian, and E. Bareinboim. Causal effect identification cluster dags. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37(10), pages 12172-12179, 2023.
[2] E. Bareinboim, J. D. Correa, D. lbeling, and T. Icard. On pearl's hierarchy and the foundations of causal inference. In Probabilistic and Causal Inference: The Works of Judea Pearl, page 507-556. Association for Computing Machinery, New York, NY, USA, 1st edition, 2022.
[3] R. M. Baron and D. A. Kenny. The moderator-mediatoo variable distinction in social psycho- logical research: Conceptual, strategic, and statistical considerations. Journal of personality and social psychology, (6): 173, 1986.
[4] H. Checkoway, N. Pearce, and D. Kriebel. Research methods in occupational epidemiology, volume 34. Monographs in Epidemiology and, 2004.
[5] A. Decruyenaere, Steen, K. Colpaert, D. D. Benoit, J. Decruyenaere, and S. Vansteelandt. The obesity paradox in critically ill patients: causal learning approach to a casual finding. Critical Care, 24(1):1-11, 2020.
[6] V. Hainer and I. Addhon-Hainerovvá Obesity paradox does exist. Diabetes care, 36 (Supplement (impllemenn_2)S827-5333333 2):8276-S281, 2013.
[7] Hernandez. Redlining revisited: mortgage lending patterns sacramento 1930-2004. International Journal of Urban and Regional Research, 33(2):291-313, 2009.
[8] J. Larson, S. Mattu, L. Kirchner, and J. Angwin. How we analyzed the compas recidivism algorithm. ProPublica (5 2016), 9, 2016.
[9] J. Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, New York, 2000. 2nd edition, 2009.
[10] Pearl. Direct and indirect effects. In Proceedings of Seventeenth Conference on Uncertainty in Artificial Intelligence, page 41 4-1- San Francisco, CA, USA, 2001. Morgan Kaufmann Publishers Inc.
[11] K. Pearson. Iv. mathematical contributions to the theory of evolution. -v. on the reconstruction of the stature of prehistoric races. Philosophical Transactions of the Royal Society of London. Series A, Containing Pupers of Mathematical or Physical Character, 1(192):169-244, 1899.
[12] J. M. Robins and S. Greenland. Identifiability and exchangeability for direct and indirect effects. Epidemiology, pages 143-155, 1992.
[13] K. J. Rothman, S. Greenland, T. L. Lash, et al. Modern epidemiology, volume 3. Wolters Kluwer Health/llippincott Williams & Wilkins Philadelphia, 2008
[14] T. VanderWeele. Explanation in causal inference: methods for mediation and interaction. Oxford University Press, 2015.
[15] Y. Zenou and N. Boccard. Racial discrimination and redlining in cities. Journal of Urban economics, 48(2):260-285. 2000.
[16] J. Zhang and E. Barcinboim. Non-parametri path analysis in structural causal models. In Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence, 2018.
This research was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation. We would like thank Inwoo Hwang for providing very useful comments on the paper during the NeurIPS conference.
Acknowledgements

---

Supplementary Material for A Causal Framework for Decomposing Spurious Variations
The source code for reproducing the experiments can be found in our code repository
A Theorem and Proposition Proofs
A.1 Proof of Prop.
Proof. Note that TV and TE are defined as:
TV xo (v) = Pvy | x1) - P(y xo)
(32)
TE (v) P(yx.) P(yxo).
(33)
We can expand the TV measure in the following way:
TV 20,21 (v)
P(y x1) P(y (0)
(34)
Ply σ1) P(yx2) + P(yx.) ) Ply x00)
(35)
Ply x1) P(ye.) P(ye)) - P(yzo) + P(yza) Ply x0)
(36)
Exp-SE,, (3)
TExO (v)
Exp-SE (v)
TE (v) + Exp-SEz, (v) Exp-SEzo (v),
(37)
showing the required result.
A.2 Proof of Thm. 1
Proof. Note that
k-1
Exp-SE: SE
9RI (1)
Σ
Ply |2011 Ply 20(-11)
(38)
is a telescoping sum, and thus we have that
SEp Un (1+1)))))))) (1)
Σ
Plu 10 P(y so
(39)
i=0
Ply aa0) - -Py (2/11))
(40)
P(y xx) - P(uz)
(41)
Exp-SE,,(y).
(42)
completing the proof of the theorem.
A.3 Proof of Thm. 2
Proof. Notice that fixing a specific value for the variables (U1,... Uk) (.1,..., ux) also gives unique value for the variables (Zi, Zx) = (2),..., 2x). Therefore, we can write
PΦy [w)
Σ P(wn) P(y 2 up)
(43)
P(wa) x ua) Zpl (4a1)
(44)
MC4
P(u (1 (1(Z))((1)))))) zn) x, 2(1)
(45)
((p)))(()) xx, 21)
(46)
= Σ P(vl xx 22,2) P(e-p
pp I xx 21. 214 P(=)g)-
(47)
12

---

The above proof makes use of the fact that the exogenous variables U, are considered in the topologiaal ordering the decomposition in Eq. [2 since in this case fixed value uij implies fixed value of 2|j- However, when considering decompositions that do not follow topological ordering, this is not the case, and we lose the identifiability property of the corresponding effects, as shown in the example in Appendix B.2
A.4 Proof of Thm. 3
Proof. The proof is analogous to the proof of Thm. [ the only difference being that there is no longer 1-to-1 of the latent variables U, with the observed confounders Z. Rather, each U, may correspond to one or more Z, variables. However, we still have that
k-1
Exp-SE W
11+4 (v) = Σ
P(y x011)-- P(y ()(e-))
(48)
0
a telescoping sum, and thus we have that
SE, Unn (1422))))
Σ Plu 10 11) P(y I se
(49)
i=0
P(y (x0) -Pyy 22(11)
(50)
P(y x)) - P(Uz)
(51)
Exp-SE (y);
(52)
completing the proof of the theorem.
A.5 Proof of Thm.
Proof. Let Ura be the set of exogenous variables not updated according to evidence, and suppose that (i) X, Y & AS(UrA):: (ii) UPA an (AS(UPA)). Note that
P(y (
(def)
Σ
P(uPA)P(r x. upa)
(53)
UPA
Σ
P(ura)P(y x, ZAS) xr, UPA),
(54)
APA.FAS
where Zs is the anchor set of UPA and the second line follows from the law of total probability. Consider any exogenous ancestor of Zss. denoted by Uz. By condition (ii) of ancestral closure, Uz must be in UPA. Therefore, UPA contains all exogenous ancestors of Ass. Consequently, fixed value of up also implies a value of ZAS, labeled EAS- This means that
P(ZAS x, upu) = 1(ZAs(uPA) = =As))
(55)
Next, suppose there is an open path from Ura to Y when conditioning on X, ZAS, labeled Ur
Z, Z Z + + Y. By definition of the anchor set, ZAS must contain the first variable on this path, Z., and Z, is different from X.Y. Consider first the case with the arrow from Z, Y. When conditioning on ZAS, this path closed since Z, E ZAS, yielding contradiction. Consider then the second case with the arrow incoming into Z., UPA > Z, < Z + ... + Y. Since Z points to Z., Zu differs from X, Furthermore, by anchor set exogenous ancestral closure, the exogenous variable of Zu, labeled U.. must also be in UPA. Hence, ZAS contains Z and Z cannot be a collider on this path, so conditioning on Z blocks the path, again yielding contradiction. We conclude that no open path between UPA and Y exists when conditioning on Z X. Therefore, it holds that
outgoing, UPA. + Z, +
Ply x, UPA, FAS) = P(y x. 2AS).
(56)
Finally, by plugging in Eqs.
55
56 into Eq. 54 we obtain that
P(y x UPA)=
Σ
P(uPA)
x, ( = 2AS)
(57)
MAARAS
13

---

U1
U2
21
Z2
X
Figure 7: Markovian causal diagram used in Ex.
with explicitly drawn latent variables U1, U2.
ΣΣ
P(y
(A
Σ P(usa))(ZAs(upaa= 2AS) 2A8)
(58)
248
P(aas) by definition
Σ Ply x, (As)
(59)
RAS
therefore witnessing identifiability of P(y xlPA) and completing the proof.
B
Examples
B.1 Markovian Decomposition Example
Example 4 (Latent variable attribution in Markovian model). Consider the following SCM M
21 + B(0.5)
(60)
Z2 B(0.4+
(61)
M
X +
+ 0.2Zz)
(62)
Y X+ +2.
(63)
and the causal diagram Fig. 7
We wish to decompose the quantity Exp-SE (y) into the variations
attributed to the latent variables U1, U2. Following the decomposition from Thm. 7 we write
Exp-SE (y an) FFyy (i)
Ely #1
U1
(64)
Ui contribution
+ Eyy x M1)
Ely a Uz.U2)
U2 contribution
We now need to compute the terms appearing in Eq. 64 In particular, we know that
Ely x 32)) BE(y do(x1))
(65)
1 IE(Z) do(z))) + E[2 do(xi))
(66)
+ E ( + E(Z,) = 0.5++ 2.
(67)
Similarly, we can also compute
Ely 21) - P(Z 11
P(Z. 21))
(68)
where P(Zı x1) can be expanded as
P(Z = 1 x1) P(Z x
(69)
P(X=1)
P(Zı -1 1, =1
P(Z = =1, 1,22 0)
(70)
P(X=1)
0.5 0.7 0.7 * 0.4 0.5
0.62.
(71)
0.5
The value P(Zz 1 ||(r) is computed analogously and also equals 0.62, implying that E(y 21) 1 0.62 0.62 2.24. Finally, we want compute E(y x 2//1), which equals
x1)
Ely ( = 1 P(Z,
4 + P(Zz 2
(72)
14

---

21 Z2
21 Zz
22
x
x
Y
x
Y
Ply ())
pMsx (y 1x)
PMn (y)
PM (2)
(a) Exp-SEz (v).
(b) Exp Stppseei.teee (U1 (v).
Figure 8: Graphical representation of Exp-SE effect decomposition in Ex.
By definition, PIZ = 1 = P 1) = 0.5. For P(Z2 1 write
P(Z2 = 2 x Σ P(Z-2 = 211
(73)
1
P(Z2 = 1, 1, 1,
1,21 =0 P(Z2 1 =1,
(74)
2
P(X = 1, =)
= -1X=1,27 =0)
0.21
0.21
~= 0.68,
(75)
2
10.31
0.31
implying that Ey x = 2.18.
Putting everything together. we found that Exp-SE2 0,U1 (y 121) ExppStEy..U |
Exp-SE (y 27
(76)
=0.24
from U
foo
The terms appearing on the r.h.s. of Eq. 76 are shown as graphical contrasts in Fig. On the left side of Fig. h U1, U2 are responding to the conditioning X x, compared against the right side where only U2 is responding to the conditioning X = x. the second term, in Fig. Bb, on the left only U2 responds to X I, compared against the right side in which neither U, nor U2 respond to X conditioning.
B.2 Non-topological Counterexample
Example (Non-identification of latent spurious decomposition). Consider two SCMs M M2.
Both SCMs have the same set of assignment equations F, given by
Z1 t U1
(77)
Z1
i
1
Z1
if 2
Z2
(78)
F
1
if =3
0
if =4 4
X - (Z) AUx) V (Ze A Ux2) VUx
(79)
Y - X + Z1 + Zz,
(80)
and the causal diagram given in Fig. 7 The two SCMs differ in the distribution over the latent variables. particular, for Mi we have
UxI Ux2, Ux ~ Bernoull((0.5)
(81)
PM. :
1 1
Un Multinom (4 (0,
4' 2 1),
(82)
for M2
U1, Ux, Ux, Ux ~ Bernoulli
(83)
(M)(U))
U2 ~ Multinom 4, 1, (
4' 2 4 0)).
(84)
That is, the only difference between PM: (U) and PMa (U) how U2 altains its value. In fact, check that the observational distributions (V) and MMa(y) are the same. However,
15

---

21
Z2
X
Figure 9: Causal diagram appearing in Exs. 6
when computing EM (y a) have that
EM (y U2)
(85)
E EM =0 0.93,
(86)
showing that the quantity EE (y/ a 2 is non-identifiabe.
The example illustrates that even in the Markovian case, when the variables are not considered in topological order (in the example above, the variable U2 was considered without the variable U1 being added first), we might not be able to identify the decomposition of the spurious effects.
B.3 Semi-Markovian Decomposition Example
Example 6 (Semi-Markowe) spurious decomposition). Consider the following SCM M:
Zi + U1 Aux
(87)
Z2 U2 U2x
(88)
X + Ux A (Uix U2x)
(89)
P()(
Y+ + Z1 Zz
(90)
U1. U2, U1x, U2x Ux lid Bernoulli(0.5).
(91)
The causal diagram G associated with M is given in Fig. top of spurious trek are U1x. U2x. Therefore, following the decomposition from Thm. 3 we can attribute spurious variations to these two variables:
9 The exogenous variables that lie on
Exp-SE, (y) x1) (y mn) E(y 2
UIX)
(92)
Uix contribution
E
(ix) E(y 1
Usx U2x
U2x contribution
We now compute the terms appearing in Eq. 92 In particular, we know that
Byy a U1x.U2x)
= [Fyy do(x1)) = ( do(x,)) + [E do(x:))
(93)
= +1+1E(2) + (2) = 0.25 0.75 2
(94)
Similarly, we can also compute
Ely m1)) = P(Z = =1 1 P(Zz xx);
(95)
Now, P(Z 1 m) P(Z,=1,2.) and we know that X 1 if and only if Ux and P(a1)
Uix V U2x = 1, which happen independently with probabilities 1/ and 4. respectively. Next, Z1 1,X 1 happens if and only if Ux 1, 1,x and U1 1, which happens with
probability 8. Therefore, we can compute
P(Z = x1) =
8
(96)
3
Furthermore, similarly compute that Ze 1, =
happens if either Ux 1, 12x or
Ux = 1, U2x = 0, U2 = 1, U1x = which happens disjointly with probabilities 4' 16. respectively. Therefore.
1 1
+
1 16
P(Z₂ = x1)
(97)
6
16

---

Putting everything together we obtain that
1
5
13
Eyy 21) 1
+
(98)
3
Finally, want compute Ey [1 1xx1), which equals
Ely 21 [1x) 1+ 1 | 11 + P(Zz 1 x
(99)
Now, evaluate these expressions, we distinguish two cases, namely (i) UIx and (ii) Uix
In the first case, P( 11) = and P(Zz | x1) = 4. In the second case, P(Z1 x1 = 0 and P(Z2 = 11) = Therefore,
compute
1
Pu Pix= (21 x1) + PU Pux=0 [11 = 1
P(21 = 1 1
4 (100)
2
1
P(Z2 = | = PUIx=I (Z2 x1) + P (Z [1i) = (101)
2
which implies that Fyly 11 [UXx
17 8. Finally, this implies that
Exp-SE (1 x1) = Exp-St U.x (u x1) Exp-SE wil.UxxxUxUxxua Uux)
Xxx x1)
(102)
t from Usx
= from U2x
The terms appearing on the r.h.s. of Eq. 102 are shown as graphical contrasts in Fig. On the left side of Fig. 4p. U1x, U2x are responding to the conditioning X I, compared against the right side where only U2x is responding to the conditioning X = In the second term, in Fig. 4b. on the left only U2x responds to x, compared against the right side which neither Uix nor U2x respond to X = x conditioning.
B.4 Semi Markovian Identification Examples
Example 7 (Spurious Treks). Consider the causal diagram Fig. 7 In the diagram, latent variables U1, U2 both lie on top of spurious trek because:
- + 24 +Y is a spurious trek with top U1
X Z- Ze - Y is spurious trek with top U2-
There are also other spurious treks with U1 on top, such as X + Zi + U, U1 -2 +
Example 7 (continued U Urror construction). We continue with Ex. Band the causal graph in Fig. and perform the steps follows:
(i) initialize UxToT -.
(ii) note that {x, Z.} create maximal clique, since:
(a) they are connected with bidirected edge and thus form a clique,
(b) (X, 71, 72) do not form clique, due to the bidirected edge 7 +--+ Zz not being present,
(c) (X, Zı, Y). (X, Zi, Y) do not form a clique, since is not incident to any bidirected edges,
(d) thus, the clique {X, 31) is also maximal.
Let the variable U be associated with this clique, pointing to X, Zi, and note that U x lies on top of a spurious trek between X,Y:
(iii) similarly, (X, Zg) also create a maximal elique, associated with the variable Ux2, pointing to X, Zz. that lies top of a spurious trek between X. Y,
(iv) the node Y also forms maximal cligue and is associated with the variable Uy that does not lie on top of a spurious trek (it does not have a path X).
Therefore, we have constructed the set USTOT LUIx,UxX!
17

---

7
73
7,
Z2
X
X
Y
(a) Causal diagram in Ex. 8
(b) Causal diagram in Ex. 9
Figure 10: Causal diagrams Exs.
8
Example (continued anchor set). For the set USTOT = {U1x. U2x) associated with the causal diagram in Fig. | the anchor sets can be computed as follows:
AS(U1x) = {
(103)
AS(U2x) {X, Z2).
(104)
AS(U1x, U2x) = (xx, Zz}-
(105)
Example (continued anchor set exogenous ancestral closure). Consider the causal diagram in Fig. 9
With respect to the diagram, we have that
an (AS(Ux))) = an " (1) (Uix,U2x]:
(106)
an (AS(UX)) an th (X, 22) (U1x,U2X)
(107)
an AnS(AAS(UUL U2x an (X, Z1, Z2) (Uix,U2x,,
(108)
Therefore, 1Uix, } satisfies anchor set exogenous ancestral closure, whereas Uix and U2x do not, since for instance Uix has X in its anchor set, but X has U2x as its ancestor.
We now consider an example of effect identification based on Thm.
Example 8 (Thm. 4 Application). Consider the causal diagram Fig. 10a Consider the query Eyy
at ) associated with partially abducted submodel in which the noise variable U12 determining
the values of Zi, Ze is not updated according to evidence. Based on Thm. 4
we verify that
(i) X, Y are not in the anchor set AS(U12) {Z,, Z2};
(ii) an (AS(U12)) an (Z1, Z2) U12 meaning that U12 satisfies anchor set exogenous ancestral closure (ASEAC).
Therefore, the query E(y (12) ) is identifiable from observational data. To witness, expand the query as:
E y
U12) 1
Σ
P(u12)E(y 21,, 21, (12)
(109)
Σ
P(u12)E(() 22,, = 21, Z2(u12) 22)
(110)
Σ
P(112)
01, = z1, Z2(11) =22)
(111)
Σ
Eyy 11 21, 22) Σ (Z(12) 21, Z2(u12) 22)
(112)
81
u12
Ely |1 21, 22) P(21, z2)
(113)
21.22
Σ $1
Ely æ1, 22, 23) P() 22) P(za (1,211222 22),
(114)
providing an identification expression from observational data.
C Discussion of Thm.
Thm. 4 introduces sufficient condition for the identification of quantities under partial abduction
(Def. Here, we discuss why some of the conditions the theorem are necessary, and provide an example that further elucidates the theorem's 's scope.
18

---

Example 9 Nonnndeeniificttinn in Markovann Models). Consider the causal diagram in Fig. 10b and consider two SCMs M2 constructed as follows. Both SCMs have the same set of assignment equations F, given by
[1 f >4 0
21
(115)
if 4
Z2 + U/2
(116)
fUxx E [1,5]
F ==
Z2
Ufx E [2,6]
X
(117)
1- Zz
Ufx E [3,7]
if E {4,8]
Y + Z1 V Zz.
(118)
The two SCMs differ in the distribution over the latent variables. In particular, for Mi we have
U2 ~ (emoolll(()0)
(119)
(U):
Uix Multinom (8,
100
25
))
(120)
anr M2
U2 ~ Bernoull((0.6)
(121)
(Ma(U)):
Uix ~ Multinom (8, 1, (0, 4' 0,0, 1 1
4' 4 0)),
(122)
That is, the only difference between pMi (U) and pMa(U) (U) is in how U1x attains its value. Fur- thermore, we can verify that M1 and M2 generate the same observational distribution, given by following distribution table
Z
Z2
X
P(Z,, Z2, X)
0
0
0
0.10
0
0
0.10
0
0.15
1
0
0.15
0
0
1
0.10
0
1
0.10
1
1
0.15
1
1
0.15
and Y given simply as deterministic function of Z1. Ze. Now, suppose we are interested in computing the conditional probability Y given X Fy in a partially abducted submodel where U1x does not respond to evidence. The quantity By Uixx can be computed as
8
E 1
Mix)
Σ
P(u1 EEy F1, w1x)
(123)
=1==1
8
Σ
P(u1= E(y F1, U1v, P(22 I1, U1z)
(124)
#1==1
Σ P(un))
Ii, uy 11 zgl,
(125)
#1
where P(Z₂ (1) for Wi E (1,4,5,8) P(Zz = 11, u1z) = for uix {3, 7).
and P(Z2 = 1 æ1, (1z) = f uix {4,8). Evaluating the expression in Eq. 125 for the two
19

---

SCMs yields
BMM (y | l
31
(126)
40
3
M (y I x
(127)
4"
x demonstrating that the quantity Eyy 2 ) is not identifiable for the diagram in Fig. 10b
The above example provides some intuition about why the variable cannot in the anchor set of the variables UP that are not updated according to evidence. Very similarly, analogous example demonstrating that the variable cannot in the anchor set of Up PA can also be constructed.
20


Reviews:
[{"review_id": "cr4fzUuFqk", "reviewer": "Reviewer_dHJ7", "summary": "The manuscript studies the decomposition of spurious variations. Specifically, a tool is developed for the non-parametric decompositions in both Markovian and Semi-Markovian models.", "questions": "1. What could be the limitations of the proposed tool?", "limitations": "I didn't find a discussion on the limitations.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The considered problem is practical. As stated in the introduction, the related tools are almost entirely missing in the literature (though I'm not totally sure about it).\n\n2. The theoretical framework is rather complete.", "weaknesses": "1. The empirical evaluation seems to be not strong enough. The proposed tool has only been applied in one specific case. The generality and stableness of the method could be better illustrated by more types of evaluation, perhaps including some simulations.\n\n2. Since causal inference in the presence of latent confounders is a well-studied problem, it will be helpful if more discussion on the considered task and the other related ones can be conducted. This can provide additional insight and broaden the understanding of the subject matter.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort in providing the comments. We respond point-by-point to the concerns raised. In the strengths, the reviewer states we are solving a practical and important problem, and provide a complete theoretical framework. We kindly ask the reviewer to reconsider the gap between the current literature and the tools proposed in our paper, based on the clarifications provided below. \n\n---\n\n[W1: Empirical Evaluation] Thanks for the suggestion, and for giving us a chance to clarify this further. We would also like to emphasize the generality of the theoretical framework proposed. We handle both the Markovian and the Semi-Markovian cases in the non-parametric setting. Furthermore, we now also added two synthetic experiments, to test the method on a setting where the ground truth is known. Please also see global response [G1].\n\n---\n\n[W2: More discussion on considered task] We would like to clarify the issue. While there are indeed methods in the literature for handling latent confounders, the estimation target is almost invariably a causal effect or its conditional / mediated versions. That is, no method addresses the specific challenge investigated in this manuscript, that is, how to decompose spurious variations. Notwithstanding, the problem of understanding spurious variations is quite pervasive in applied sciences, such as epidemiology, and in other contexts such as explainable/fair AI. For instance, exposure to a harmful toxin and the development of a disease may be confounded by other work-related hazards. The newly proposed method allows one to quantify which hazards confound this relationship, and how strong they are, which may be quite important for epidemiologists.  \n\nThe types of questions our manuscript addresses have not been explicitly considered before. However, the type of reasoning about the strength of the confounders has mostly been done for linear models (where confounding can be associated with the estimated coefficients) or through variable importance. Therefore, our work can be seen as providing the first, non-parametric generalization of these basic types of reasoning, which decomposes the spurious effect quantities. In this sense, we believe it fills in an important gap, complementing the existing literature on the estimation of causal effects, and providing another tool for more precise and fine-grained causal analyses.\n\n---\n\n[Q1: Limitations] Please see global response [G2]. In particular, the discussion on cluster DAGs shows how some of the causal assumptions may be relaxed."}, {"review_id": "70wmqgfyxt", "reviewer": "Reviewer_VtmC", "summary": "This paper provides a new framework for decomposing spurious effects and its identification conditions. In addition, they proposed a nonparametric method for decomposing the spurious effect under sufficient conditions.", "questions": " See the weaknesses above.", "limitations": " See the weaknesses above.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- In general, this paper is well-motivated. \n- It provides a procedure to decompose spurious effects. \n- The theoretical results in this paper are sound. ", "weaknesses": "- The paper could clarify the difference between \"spurious effect\" and \"spurious variations\", or state if they are used interchangeably. More explanation of how these two terms are defined would help the reader understand if they refer to the same concept.\n- More details could be provided on how the procedure for Semi-Markovian spurious decomposition is implemented. In particular, explaining the steps for removing the effects of all exogenous variables would make the approach clearer.\n- Comparing the proposed method against existing methods on estimating causal effects could highlight an advantage. Since the proposed method can estimate redundant effects, the resulting causal effect estimates may be more accurate compared to current approaches. This comparison could emphasize the improved performance enabled by accounting for spurious variations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort in providing your feedback. We respond to each point in the sequel and would be happy to provide further clarification if you feel suitable. \n\n---\n\n[W1: Spurious Effect vs. Variation] Thanks; we acknowledge that in the paper the distinction between the two is not entirely clear. We now add the following clarification in the Intro:\n\n“A spurious effect is a quantification of spurious variations (or a subset thereof)”.\n\nThus, for a spurious effect, we usually have some kind of quantity in mind, for example, the experimental spurious effect (Exp-SE) in the l.h.s. of Eq. (2). Spurious variations are a more broad concept and represent co-variations induced by the latent variables in the SCM, for example, a specific value of a latent $U_z$ that affects both $X$ and $Y$ and induces a correlation between them. Hence, an effect is usually tied to a quantity, whereas variations are a more broad term. In the text, in most places, these two notions could indeed be used interchangeably (although we do not think of them as identical, as described above).  Please let us know if this helps clarify the distinction.\n\n--- \n\n[W2: Details of Semi-Markovian procedure in practice] Thanks for mentioning this. Indeed, some of the examples related to Theorem 4 have been pushed into the appendix due to limitations of space. Our key proposal here is to bring some of these examples into the main text. In particular, in Example 7 (currently Appendix B.4), we ground the definitions of an anchor set, and also anchor set exogenous ancestral closure. We believe that adding this will increase the transparency of the proposed definitions, and make more explicit the types of assumptions that are needed for decomposing spurious effects in the Semi-Markovian setting. We thank you for this suggestion!\n\n--- \n\n[W3: Comparing against existing methods] Thanks for bringing up this point. In fact, for estimating the Exp-SE, one needs to estimate $P(y \\mid x)$ (which is easy for a binary $x$), and also $P(y_x)$, which is a causal effect. Therefore, the approach in the paper builds on methods for estimating causal effects, and is not really a competitor in this sense. It would be indeed nice to have a result saying that a decomposition of the Exp-SE quantity can improve the estimation of causal effects. For concreteness, Pearl was able to show that in a special case, a causal (not spurious) effect can be estimated in pieces and then composed so as to obtain a more efficient estimator of this effect [1] ([2] may also be interesting in this context). However, there are challenges of how to do this regarding spurious effects and under more general causal diagrams, which we hope to investigate further in the future.  Again, here our goal is mostly to introduce spurious effects that were not even defined, and also to understand the decomposability and other key properties of this quantity. The natural subsequent question after having this foundational step solidified is to investigate estimation properties, both from a statistical and computational perspective. \n\n[1]  J. Pearl, \"What is Gained from Past Learning\" UCLA Cognitive Systems Laboratory, Technical Report (R-472), March 2018. Journal of Causal Inference, Causal, Casual, and Curious Section, 6(1), Article 20180005, March 2018. https://doi.org/10.1515/jci-2018-0005\n\n[2] J. Hahn and J. Pearl \"Precision of Composite Estimators\" UCLA Cognitive Systems Laboratory, Technical Report (R-387), September 2011. Working paper. "}, {"review_id": "Iq4bbu2mB7", "reviewer": "Reviewer_G3e3", "summary": "The authors tackle the problem of decomposing the source of variation in causal analysis.  In many real-world problems, there are many factors that can introduce spurious correlation between a treatment and outcome, and breaking down the contribution of each variable to that spurious correlation can be useful in many fields where an explanation is important.  The authors propose a methodology for both Markovian and semi-Markovian models, based on the 'abduction-action-prediction' method.  It allows for the application of evidence to only a subset of exogenous variables, which then allows us to separate the variation from each exogenous variable.", "questions": "Is there a list of all of assumptions required for your method?  In Section 4.1, you say that the decomposition needs to follow a topological order of the variables U_i, but are there others?", "limitations": "In terms of social impact, this method is more likely to have a positive social impact, since causal inference is often applied in domains where spurious variation is present but where the decisions made can have significant effects on people's lives (e.g., the COMPAS example)", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper is well-written and well-motivated.  The problem it seeks to address is important, and the introduction does a good job at setting the theoretical foundation and describing the problem.  For the most part, the terminology is introduced at a good pace, making the overall narrative easier to follow. I appreciate the authors interspersing examples with figures, since without them, the paper would likely bog down with all the math.  Overall, I found this a compelling approach to an important problem.", "weaknesses": "I think, for such an equation-heavy paper, the notation could be made clearer/more explicit in sections.  For example, Proposition 1 is the first time that we see the $e^{U_1}$ notation, which it seems like means 'this evidence is applied to everything except for $U_1$'.  That's a fairly counter-intuitive notation for such a thing, since generally a subscript or superscript implies that we're using those variables in someway, rather than explicitly excluding them (I realize you need to use them to exclude them, but conceptually, the notation is non-obvious for me).  If I'm not misunderstanding what this notation means, then a more explicit definition before Proposition 1 would be helpful, given how central it is to the rest of the paper.\n\nAs another notation point, Theorem 2 uses the notation $z_{-[i]}$, but until now, the $Z_{[i]}$ notation has meant, as defined in Definition 2, \"all $Z_i$'s, from 1 to i\", and it's not obvious how this translates to a negative.\n\nThe COMPAS demonstration is great, but since it clearly lacks ground truth, additional synthetic experiments would be helpful to help demonstrate both the correctness of the theory and how this method can be used in practice on other types of data.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort in sharing his/her thoughts and comments. We were quite glad an encouraged by the fact that the reviewer appreciated the main strengths of our work. We respond point-by-point and hope to have addressed the raised concerns.\n\n---\n\n[W1: Clearer notation] Thanks for this suggestion; the reviewer’s summary “evidence is applied to everything except for $U_x$” precisely captures the intuition behind the operator (i.e., there is no misunderstanding). However, regarding the choice of superscript / subscript – we have little flexibility here, as we see it. The subscript operator in the causal inference literature is already taken, and it has been used heavily to indicate (and index) interventions. Therefore, the superscript notation is the only option available to us (note that, in general, one may apply different parts of evidence to different latent variables, and thus the indicator for latent variables needs to be specific for each part of the evidence). \n\nStill, we definitely agree on the centrality of such operator, and thus re-write the proposition slightly to make explicit how the operator is defined:\n\n> Let $P(Y = y \\mid E = e^{U_1})$ denote the conditional probability of the event $Y = y$ conditional on evidence $E = e$, defined as the probability of $Y = y$ in the PA submodel $\\mathcal{M}^{U_1, E=e}$ (i.e., the exogenous variables $U_1$ are not updated according to the evidence). Then, we have that:\n\\begin{align} \n    P(Y = y \\mid E = e^{U_1}) = \\sum_{u_1} P(U_1 = u_1)P(Y = y \\mid E = e, U_1 = u_1).\n\\end{align}\n\nWe hope this addresses your concern but, please, let us know. Additionally, we wonder if the notation\n\n$$ x^{\\underline{U_{[i]}}}$$\n\nwould be better (the underline indicating the evidence is not updated). We are happy to introduce such a change if the reviewer feels strongly about this (although the reviewer will, just like us, note a slight trade-off between making the notation more bushy vs. more informative). \n\n--- \n\n[W2: $Z_{-[i]}$] Thanks for spotting this glitch, which is an omission on our side and not defined anywhere. We now mention this explicitly in Definition 2. Specifically, the $-[i]$ notation corresponds to the complement, that is: \n\n$$Z_{-[i]} = \\{Z_{i+1}, \\dots, Z_{k}\\}.$$ \n\n---\n\n[W3: Synthetic Experiments] Please, see global response [G1]. We now add synthetic experiments with a known ground truth!\n\n---\n\n[Q1: Assumptions for the Method] Thanks for asking this, it is indeed a good question. We distinguish three different considerations to try to clarify this point:\n\n(i) Firstly, a fully specified causal diagram is needed for the procedure. This is now also discussed in the limitations paragraph of the Discussion.\n\n(ii) Regarding the topological ordering of variables $U_i$ (which follows from the topological ordering of the $Z_i$) – this part is necessary for the identifiability of the spurious decomposition. However, for defining the decomposition itself, it is not strictly necessary. In other words, there may be a decomposition that does not follow a topological ordering, but we will not be able to compute it from the data. However, the one that does follow a topological ordering will be computable, as shown in Theorem 2. We now add a remark on this in the text!\n\n(iii) A similar consideration is also present in the Semi-Markovian case; here, however, we do not invoke a topological ordering, since there is no one-to-one correspondence with the observed variables. However, the very same intuition is behind the condition called anchor set ancestral closure (Definition 7). That is, for a specific order, as shown in Theorem 4, we will be able to compute the decomposition (otherwise, the decomposition can still be defined, but it cannot be uniquely computed from the data).\n\nWe appreciate the thoughtful review and hope these answers help clarify the required assumptions and the provided results. Please let us know if there are any further questions!\n\n"}, {"review_id": "AFdxWRqOJI", "reviewer": "Reviewer_1quh", "summary": "This paper considers the problem of decomposing spurious variations in both Markovian and Semi-Markovian models, which are potentially useful to many real applications. A non-parametric  decomposition of spurious effects is given given and identification issue is also discussed under suitable conditions.", "questions": "The paper is very clear in both motivation and technical part. As the first work on this problem with a general setting, the content of the present version is sufficient for an acceptance. \n\n", "limitations": "It seems that there is no discussion on limitations. Developing more identification theories is worth future investigation, as also mentioned by the authors.", "rating": 7, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- decomposing spurious variations can be potentially useful to many real applications\n- Presentation is very good with well-structured preliminary and examples\n- Proof is complete and concise", "weaknesses": "It seems that there is no discussion on limitation of the developed theory. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort in providing his/her feedback. We were encouraged by the fact that the reviewer appreciated the main strengths of the manuscript. We respond point-by-point to the raised concerns.\n\n\n[W1: Limitations of the developed theory] Thanks for noting this; Please see global response [G2] for a detailed discussion of the limitations, which is now added to the Discussion in the paper! "}, {"review_id": "jZUqzeIG6a", "reviewer": "Reviewer_EguC", "summary": "The paper proposes a novel theoretical framework for decomposing spurious causal effects. For this purpose, the authors introduce the concept of partial abduction and prediction. Here, only a subset of the latent variables in a given structural causal model (SCM) is updated when conditioning on observed evidence. The authors use this concept to derive nonparametric decompositions of spurious effects and provide identifiability results both for Markovian and semi-markovian SCMs. The paper ends with an empirical example using the COMPAS dataset.", "questions": "- It seems like the decomposition in Theorem 1 depends on the chosen topological ordering of the observed confounders $Z_i$. Consequently, would the \"spurious contribution\" of confounder $Z_i$ change when selecting a different ordering? This would seem counterintuitive for a framework that aims at decomposing the overall spurious effect into individual \"spurious contributions\".\n\n", "limitations": "I suggest including a section on limitations. Here, the authors should particularly stress the assumptions necessary for the identification of spurious decompositions from observational data. Furthermore, a discussion on these assumptions in the context of the empirical example in Section 5 should be included.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The paper addresses a relevant problem: decomposing the spurious effect in the presence of multiple confounders. This has potential applications in e.g., causal algorithmic fairness.\n\n- Novelty: I am not aware of any other work done in this regard. The authors develop a novel theory for defining such decompositions based on Pearl's SCM framework.\n\n- The paper is technically solid and rigorous. Definitions are clear and precise, all assumptions are stated transparently, and proof and illustrative examples for theoretical results are provided.", "weaknesses": "- My only main concerns are regarding the applicability of the proposed framework in practice. To my understanding, the full (semi-markovian) causal graph needs to be known (including the spurious arrows between the observed confounders) in order to apply the decomposition result from Theorem 3. This may be the case for simple examples like the COMPAS dataset. However, in many applications, confounders are potentially high-dimensional and the causal graph is unknown.\n\n- I think the paper would benefit from expanding the experiment section, i.e., by including a second example dataset.  \n\n\nMinor points:\n\n- Eq. (7): I suggest using the notation $p(Y_x = y)$ so that $y$ on the right-hand side of the equation is well-defined. \n- Typo in Definition 7: path causal path\n- Proposition 2: I suggest moving the definitions of TV and TE from Appendix A1 to the main paper.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort in sharing his/her thoughts and comments. We respond point-by-point to each of the concerns raised. \n\n---\n\n[W1: Required causal assumptions]  Indeed, you are correct that a fully specified graph is needed for Theorem 3. We also agree that this may be challenging to obtain in practice. However, one also needs to consider that fine-grained spurious (or causal / mediated) effects can be inferred from such a fully specified graph (and data of course). In particular, our tools decompose the spurious effect, and give a fine-grained quantification of what the main confounders are. As is common in causal inference, the granularity of the obtained knowledge needs to be matched with the strength of the causal assumptions. Conversely, in the absence of such assumptions, fine-grained knowledge usually cannot be obtained. For instance, it has been shown in the literature in the result known as the causal hierarchy theorem  [1, Theorem 1], which shows that no claim can be made about higher level counterfactuals from lower level data (observational or experimental) without causal assumption, which we consider through the language of causal graphs. \n\n\nHaving said that, there is a possible relaxation of our approach, based on the recent development of cluster diagrams [2], which offers some formal way of trading off knowledge versus quantitative predictions. For cluster DAGs, one can consider groups of confounders, and thus the specification of causal assumptions becomes easier and less demanding. However, causal decompositions can still be applied to cluster diagrams. This provides a way to choose a different level of granularity for settings where domain knowledge may not be specific enough to elicit a full causal diagram.\n\n[1] Bareinboim, Elias, et al. \"On Pearl's Hierarchy and the Foundations of Causal Inference.\" In Probabilistic and Causal Inference: The Works of Judea Pearl, ACM, Special Turing Series, pp. 507-556, 2022. \n\n[2] Anand, Tara V., et al. \"Effect identification in cluster causal diagrams.\" Proceedings of the 37th AAAI Conference on Artificial Intelligence. 2023. \n\n--- \n\n[W2: Adding experiments] Thanks for the suggestion; please see the global response [G1]. We now add experiments with a known ground truth.\n\n---\n\n[W3: notation $p(Y_x = y)$] The reviewer is indeed correct that this is a slight abuse of notation. However, this notation is standard in the graphical approach to causal inference, which makes us afraid that deviating from it may hurt the legibility to causal readers familiar with it. (For example, this is the notation in place at least since the Causality book, Pearl, 2000.)  Still, to address the concern, we added an explicit remark saying there is a slight abuse of notation, and explaining the $Y = y$ is sometimes just replaced with $y$.\n\n---\n\n[W4: Minor points] Thanks for all of these. We have fixed all three!\n\n---\n\n[Q1: Topological Ordering] Thanks for raising this interesting point. In fact, the decomposition may change when the topological ordering is changed. However, this should not come as a major surprise, since the decomposition for path-specific causal effects has exactly the same property (i.e., depends on the ordering). However, in our setting, the topological ordering can also be seen as the \"natural\" ordering, and it is also the only identifiable one. Additionally, one may also consider that the decomposition is unique (independent of ordering) for the additive case (this is again similar to the path-specific causal effects). Let us know if we can elaborate further on this note.\n\n---\n\n[L1: Limitations] Thanks, please see global response [G2] which describes the discussion we added. Furthermore, regarding the necessary assumptions for identification, our key proposal is to bring some of thee examples from the appendix into the main text. In particular, in Example 7 (currently Appendix B.4), we ground the definitions of an anchor set, and also anchor set exogenous ancestral closure. We believe that adding this will increase the transparency of the proposed definitions, and make more explicit the types of assumptions that are needed for decomposing spurious effects in the Semi-Markovian setting."}]

Author rebuttal:
The authors would like to sincerely thank all the reviewers for this paper. The main strengths and novelty were clearly appreciated, and the questions raised were quite useful for us to revise and improve the paper.

In our response, we index all weaknesses with W, questions Q, and limitations L. We do not fully cite reviewer's questions (due to character limit), but try our best to provide a caption for each W/Q/L.

Here, we would like to provide two global responses, which are then cited in the individual responses as well.

--- 

[G1: Extending the Evaluation] As several reviewers suggested, synthetic experiments where the ground truth is known could be a useful addition. We have conducted such experiments, and we provide an example in the pdf that accompanies our review response (an additional experiment for the Semi-Markovian case will also be added).

Synthetic A experiment explanation (see pdf): For this example, we set the parameters $\lambda_1 = \lambda_2 = \lambda_3$ in the described SCM. We then vary each parameter $\lambda_i \in [0, 0.2]$. This changes the value of the effect associated with latent variable $U_i$. We compute the ground truth values for the spurious effects based on the true SCM, using rejection sampling. We also estimate the values using the estimator described in Theorem 2. The plot in the accompanying pdf shows the result.

In particular, we see that the effects associated with each $U_i$ change as the $\lambda_i$ increases. Furthermore, we see that the associated estimates for the effect are correct, and the ground truth values fall within the 95% confidence interval (CI). Therefore, this confirms the correctness of our approach. 

--- 

[G2: Discussion of the Limitations] The following discussion is added to the main text, to clarify the possible limitations of the proposed approach
> “The main limitation of the approach proposed in the paper is the need for a fully-specified causal diagram, including all the bidirected edges that indicate confounding. Specifying such a graph may be challenging in practice. However, one also needs to consider how much knowledge can be ascertained from a fully specified graph and the data. In particular, our tools decompose the spurious effect, and give a fine-grained quantification of what the main confounders are. Such knowledge can be quite powerful and informative in practice. As is common in causal inference, the granularity of the obtained knowledge needs to be matched with the strength of the causal assumptions (in this case, specifying the causal diagram). Conversely, in the absence of such assumptions, fine-grained quantitative knowledge about these effects cannot be obtained in general. For instance, the causal hierarchy theorem  [1, Theorem 1] shows that claims about counterfactuals from observational or interventional data cannot be made without causal assumption, as usually specified through causal diagrams. We hypothesize a similar result, that is, a precise quantification of spurious effects is not attainable in the absence of a causal diagram.

>Furthermore, another technical solution is possible that may alleviate some of the difficulty of causal modeling. Recently, cluster diagrams have been proposed [2], in which one can consider groups of confounders (instead of considering each confounder separately), and thus the specification of causal assumptions becomes easier and less demanding (i.e., due to clustering, the number of nodes in the graph is smaller). However, causal decompositions as described in this paper can still be applied to cluster diagrams. This offers a way to choose a different level of granularity for settings where domain knowledge may not be specific enough to elicit a full causal diagram, while a causal analysis still needs to be undertaken.

>Finally, we also mention that the identification criteria for Semi-Markovian spurious decompositions provided in Theorem 4 are not complete (i.e., they are sufficient but not necessary). We hope to address this question in future work, and provide an even stronger identification criterion, or prove the completeness of the current one.“ 

Meta-review: The paper presents a novel problem formulation for understanding spurious variations and has some useful theoretical results to that end. I agree with the reviewers that the problem is quite important, and while there are many empirical papers, this paper makes a unique contribution in studying the types of spurious variations formally. 
I encourage the authors to incorporate the reviewers' feedback in their final version, especially a discussion on applicability in practice and limitations.
Decision: Accept (poster)