Towards identifiability of micro total effects in summary causal graphs with latent confounding:  extension of  the front-door criterion
Authors: Charles K. Assaad
Venue: Accepted by TMLR ()

Abstract: Conducting experiments to estimate total effects can be challenging due to cost, ethical concerns, or practical limitations. As an alternative, researchers often rely on causal graphs to determine whether these effects can be identified from observational data. Identifying total effects in fully specified causal graphs has received considerable attention, with Pearl's front-door criterion enabling the identification of total effects in the presence of latent confounding even when  no variable set is sufficient for adjustment. However, specifying a complete causal graph is challenging in many domains. Extending these identifiability results to partially specified graphs is crucial, particularly in dynamic systems where causal relationships evolve over time. This paper addresses the challenge of identifying total effects using a specific and well-known partially specified graph in dynamic systems called a summary causal graph, which does not specify the temporal lag between causal relations and can contain cycles. In particular, this paper presents sufficient graphical conditions for identifying total effects from observational data, even in the presence of cycles and latent confounding, and when no variable set is sufficient for adjustment.

Published in Transactions on Machine Learning Research (05/2025)
Towards identifiability of micro total effects in summary causal graphs with latent confounding: extension of the front-door criterion
Charles K. Assaad
charles assaad@inserm..
Sorbonne Université, INSERM Institut Pierre Louis d' d'Epidémiologie de Santé Publique P75012, Paris, France
Reviewed on OpenReview: https: // openeeiew. net/ forum? id 5S7V
Abstract
Conducting experiments to estimate total effects be challenging due to cost, ethical con- cerns, or practical limitations. As alternative, researchers often rely on causal graphs to determine whether these effects can be identified from observational data. Identifying total effects in fully specified causal graphs has received considerable attention, with Pearl's front- door criterion enabling the identification of total effects in the presence of latent confounding even when no variable set sufficient. for adjustment. However, specifying a complete causal graph is challenging in many domains. Extending these identifiability results to partially specified graphs is crucial, particularly in dynamic systems where causal relationships evolve time. This paper addresses the challenge of identifying total effects using specific and well-known partially specified graph in dynamic systems called a summary causal graph, which does not specify the temporal lag between causal relations and can contain cycles. In particular, this paper presents sufficient graphical conditions for identifying total effects from observational data, even in the presence of cycles and latent confounding, and when no variable set is sufficient for adjustment.
1 Introduction
Causal questions arise when we seek to understand the effects of interventions, such as asking, 'If we administer today hypertension treatment to a patient with kidney insufficiency, will the kidncy func- tion (represented by the creatinine level) improve tomorrow?'. These questions, often referred to total causal effects simply total effects (Pearl et al. 2000), denoted P((Creatinineaamrrva = do stsppppeeenssiotelaaa h)) where the do() operator denotes intervention. They differ from associa- tional relationships, Pr(Creatindneeemmrros = c H Hypertensiontoday = h), as they isolate the effect of an intervention, disregarding other influencing factors such as sodium intake or protein intake or stress level. Experimentation known as the traditional approach across various fields to estimate the total effect of in- terventions free from confounding bias (Neyman al. 1990). However, conducting experiments not always feasible due to cost, ethical considerations, or practical limitations. Consequently, scientists often resort estimating effects of interventions from observational data. This process relies specific assumptions and typically involves two sequential steps: identifiability and estimation (Pearl) 2019). The identifiability step refers to the question of whether the total effect of interest can be uniquely determined from the available data and the assumptions made about the causal model. This step usually also include finding a way to express the total effect in terms of the observed data distribution, i.c., using a do-free expression. On the
In nonparametric setting, the total effects specific functional of Cmmrr =e dol goreteecctnns h)) different values of However, for simplicity, the total effect often reffered as Pr(Creatininel do H (iggetennsionnddd 1)).

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other hand, the estimation step refers to the process of calculating the value of a total effect, once it has been identified, from finite observational data using statistical methods. This paper focuses on the first step.
Graphical models, provide a framework for identifying total effects from causal graphs which encode variables as vertices and causal relationships as arrows, allowing researchers to visualize and analyze complex causal structures (Pearl et al. 2000). Identifying total effects in fully specified non-temporal causal graphs has been a subject of considerable attention (Pearl 1993b] 1995 Spirtes et al., 2000 |Paal al. 2000 Shpitser & Pearl 2008 Shpitser et all 2010). Adjusting for covariates is one method among several that allow to identify causal effects and the back-door criterion (Pearl 1993b) is one of the most known methods that allow us to find covariates using causal graphs. [Paar] (1993a 1995) has provided examples where no set of variables is sufficient for adjustment, yet the causal effect can still be consistently estimated through multi-stage adjustments. Pearl's front-door criterion offers a graphical method for identifying total effects despite the absence of an adjustment set due to latent confounding, assuming the causal graph is a directed acyclic graph (DAG) or an acyclic directed mixed graph (ADMG). Initially criticizod for its limited practical application, this criterion has recently gained recognition and is now employed in epidemiooogy (Inoue al.
2022 Piccininni et al. 2023).
The above identifiaiility methods are directely applicable to fully specified temporal graphs Blondel et al. 2016) which represent causal relations in dynamic systems where causal relationships evolve over time. How- ever, constructing a fully scpecified temporal graph requires knowledge of all causal relationships among observed variables, which is often unavailable, especially in many real-world applications. However, experts may know that one variable causes another without knowing the exact temporal lag. For example, under- standing the transmission of SARS-CoV-2 from younger to older individuals, and vice versa, can help define interventions most likely to reduce the number of deaths. Indeed, it has been shown that younger adults tended to be highly infected during the first wave of the pandemic, while older individuals faced higher risk of death if infected (Carrat et al. 2021 Lapidus et al. 2021 Glemain et al.) 2024). Considering sufficiently large time intervals (several wecks) as in repeated scrosurvcys, like in Wiegand et al. (2023), it is not clear if the number of new infections in one age group during a time interval (incidence) may be influenced by incidence in the other age group during the same interval. Incidence in an age group can also be influenced by incidence during the previous time interval in any age group. Therefore, constructing fully specified causal graph is difficult. In such cases, partially specified causal graphs can be useful. A very well known and useful partially specified causal graph is the summary causal graph (SCG) which represents causal re- lations without including temporal information, i.e., each vertex represents a time series. Both medical and cpidemiological examples given above can be represented by an SCG with a cyclic relationship representing the interplay between creatinine and hypertension the first example and between the two age groups in the second example.
Recently, there has been new interest in extending identifiability results to partially specified graphs Eichler & Didelez (2007: 2010); Maathuis & Colombo| (0013) Perkovic (2020); Wang et al. (0223 Anand et al. [Assaa)
[2023); ; Ferreira & Assaad (2024); [Assadd et al. (2024) Ferreira & Assaad (2025). Most of these partially specified graphs represent Markov equivalence classes, where the partial specifications differ conceptually
from those in SCGs, For instance, in these graphs, partial specification typically manifests as undirected edges or edges with specific endpoints indicating uncertainty about the orientation. Additionally, each vertex in these graphs corresponds directly to an observed variable, maintaining straightforward one-to-one rela- tionship. As a result, the exlensions of identifiability results for these graphs (Maathuis & Colombo 2013; Perkovic 2020 Wang et al. 2023) are not applicable to SCGs. Another important type of partially specified graph is the cluster graph, which represents causal relationships between clusters of variables rather than individual variables. This means that, unlike graphs representing Markov equivalence classes, the skeleton of a cluster graph does not correspond to the skeleton of true causal graph. SCGs are a specific type of cluster graphs, where each cluster represents a time series. Most works extending identifiability results to cluster graphs have focused on extreme multivariate cases, where the goal is to identify the total effect of one entire cluster on another entire cluster (Anand al.] 2023 Ferreira & Assaad 2025). The few studies that consider the total effect of single variable within a cluster on another single variable within different
cluster have been conducted on SCGs. For example, under the assumptions of no instantaneous relations and
no latent confounding between two timepoints within the same time series, Eichler & Didelez (2007) 2010
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demonstrated that both the back-door and front-door criteria are applicable in SCGs. Assaad et al. (2023) further established that in the absence of latent confounding, the total effect remains identifiable through adjustment in acyclic SCGs, even when instantaneous relations are present. More recently, Assaad et al. (2024) provided graphical conditions for identifying total effects via adjustment in SCGs that incorporate both cycles and instantaneous relations, still under the assumption of no latent confounding. Finally, inde- pendently and concurrently of this work . Reiter et al. (2024) investigated the identification of total effects in the frequency domain using the SCG (referred to as a process graph in their work) under the assumption of linearity; however, unlike this and previous works (Eichler & Didelez 2007; 2010 Assaad et al. 2023 2024), they did not explore specific graphical conditions within the SCG for identification.
The previous paragraph underscores the novelty of this work: in the setting where a fully specified temporal causal graph is not available, this paper is the first to address the challenge of nonparametricaaly and graphically identifying total effects of one variable within a cluster on another variable in a different cluster when having access to an SCG while allowing instantaneous relations, cycles, and latent confounding (even between two timepoints within the same time series). It shows that the standard front-door criterion (Pearl 1993a; 1995 and its previous extension to time-series (Eichler & Didelez 2007 2010) are not sound when applied to SCGs in the general setting considered in this paper. Nevertheless, by leveraging the front-door criterion, this work introduces sufficient conditions to identify total effects from observational data using SCGs in scenarios where latent, confounding prevents identifiability through standard adjustment methods.
The remainder of the paper is organized as follows: Section 2 introduees necessary terminology and tools and it formalizes the problem. Section 3 demonstrates that the standard front-door criterion and its previous extention to time series are unsuitable when applied to SCGs. Section 4 presents the main result of this paper and Section 5 discusses several examples of non-identiffability. Finally, Section 6 concludes the paper.
2 Preliminaries and problem setup
This section, first introduces some terminology and tools which are standard for the major part and then, formalize the problem that this paper is going to solve. Let us start by defining the causal model that is considered.
Definition 1 (Discrete-time dynamic structural causal model (DTDSCM)). A discrete-time dynamic struc- tural causal model is tuple .M- (L. V, F,F, where L = ULL | T'e [L, d]. e Ito, (mar]) is a set of ezogenous variables, which cannot be observed but affect the rest of the model. V = U(V i E d.) such that vi E [1,d], V (Vi t [to: is set of endogenous variables, which are observed and every Vi E V is functionally dependent on (directly caused by) some subset of 1 U VVVV where Vst = (V j E [1, <t}. F is a set of functions such that for all Vü E EV, for is a mapping from Lod and a subset of VeV( to Vτ.. P(I) is a joint probability distribution over L.
A DTDSCM implicitly assumess that an effect cannot precede its cause. This assumption is explicitly stated as follows.
Assumption 1. Consider DTDSCM M (L, (L,V.F, P(O)). Suppose Vi is an endogenous variable which is functionally dependent on 'W C VAVi), i.e., Vi = (.)... For all V E wul it is assumed that t' t.
Furthermore, stationarity is assumed.
Assumption 2. Consider DTDSCM M (L.V,F, vf fo e E fri = foh.
Assumption 2 have two important implications. First, it entails that if Y. for f' W,-1, then Vt' E [to + 1, tmoz, Ye fd (Xr-1, (Wrr) Furthermore, it allows us to fix the maximum temporal lag between a cause and an effect, denoted as 7max- The contributions of this paper are relevant whenever multivariate time series-whee each V' such that i [1, d represents a time series with a single observation per temporal variable V multivariate spatio-temporal series -where each V such that i E [1, d] is a spatio-temporal series with multiple observations exist for each temporal variable V2 (as seen in certain types of cohort studies) are considered. Tn the former case, if Assumption 2 is violated, identifying unique total
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X
X
Wi
Wi
Y
FT-ADMG
(b) FT-ADMG
(c) SCG
X
X
W.
W.
FT-ADMG 3.
(e) FT-ADMG
(f) SCG
X
X
W
W.
(g) FT-ADMG 5.
(h) FT-ADMG
SCG
Three SOGs and six FT-ADMGs, where Tmax 1, such that FT-ADMGs and compatible with SCG 1, FT-ADMGs 3 and 4 are compatible with SCG 2, and FT-ADMGs 5 and are compatible with SCG 3. Each pair of red and blue vertices represents the cause and the effect of interest and green vertices are those that interpect all paths from the cause to the effect of interest. In SCG and SCG 2, {W) satisfies Definition 7 for the total effect Pr(y (do(x;-1)) However, {W} does not satisfy Definition in SCG 3 for the total effect Pr(yy]do(z;... since Cycles(X, (X,9*) + 0 and y +
Figure 1:
effect becomes ill-posed. This is because this case assumes dynamic system with single multivariate observational time series, meaning that variables preceding a given variable V within the same time series yi would act as substitutes that the total effect varies over time, making estimation impossible. Furthermore, in this case, additional as- sumptions such as those proposed in Eichler & Didelez (2010) are required for the estimation of the total effect. However, in the latter case, where multiple observations of each temporal variable Vž are available, these additional assumptions are not necessary. The distinction between these two cases is considered to be beyond the scope of this work. Therefore, I will not explicitly differentiate between them and for clarity and brevity, 1 will refer to the set of variables yi simply as a time series throughout the remainder of this paper.
different realizations of that variable. Violating this assumption would imply
It also supposed that the DTDSCM can be qualitativly represented by full-time? directed acyclic graph (FT-DAG), commonly known as a full-time causal graph (Peters al. 2013). To check the identifiability of
2Te 'full-time" underscores that the graph represents the entirety dynamic system over (from to fru.s). range, suggesting that any segment full-time DAG depicted in a figure should
This representation stretch
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total effect, it is standard to first transform the FT-DAG to full-time acyelic directed mixed graph (FT- ADMG) [Richardson] 2003) which is naturally derived from FT-DAGs with latent variables via an operation called latent projection (Tian & Pearl 2002). The FTT-ADM is supposed to be a DAG with bidirected edges representing latent confounding. If all variables representing latent confounding become observed then the FT-ADMG becomes a FT-DAG. Figures Ta
1g and The present six different FT-ADMGs. In our setting, FT-ADMGss can also be obtained directly from the DTDSCM, as shown below.
Tb Id
Definition 2 (Full-Time Acyelic Directed Mixed Graph). Consider DTDSCM M. The full-time acyclic directed mixed graph (FT-ADMG) g = (V, induced by M is defined in the following way:
E = (Xx E² == [xp+++++ Y, MYr,, Y, EV such that 2 J.
WY Xx E such that Y= :- [x L") in M and & CVDD).
where E = ==EUEE
FT-ADMG notions For an FT-ADMG 9, a path from X to Y, in G is sequence of distinct vertices (Xe....,YY Y2) in which every pair of successive vertices is adjacent. A directed path from Xy to Y is path from Xe to Y, in which all edges are directed towards Y, in G, that is X2. -.... + Ye. A back-door path between X2r and Y, is a path ween Xy and Y, with an arrowhead into Xy in G. If Xe - Y, then X2 is a parent of Yg If there is a directed path from Xe to Yú, then Xy is an ancestor of Y,, and Y; is a descendant of Xr. A vertex counts as its own descendant and as its own ancestor. The sets of parents, ancestors and descendants of Xy in G are denoted by Par(Xe, 9), Anc(Xy, 9) and Desc(Xp, 9) respectively. If path x contains X₁- - We <- Y2 as a subpath, then We is a collider on и. A path x from X₁ to Y, is active given a vertex set W, with Xp, Y2 & W if every non-collider on π is not in W, and every collider on и has a descendant in W. Otherwise, W blocks т. A set of vertices W intercepls all directed paths from X₁r to Y, if every directed path from Xe to Yt passes through at least one vertex in W. Lastly, each vertex in an FT-ADMG is called a temporal vertex or a micro vertex.
The lotal effect (Pearl al., 2000 between two micro variables is writlen as P(Y, = _ =))-)). Y1 corresponds to the response and do(X,_y xz-z)) represents an intervention (as defined in Pearl et al. (2000) and Eichler & Didelez (2007 Assumption 2.3)) on the variable X at time t t-17 y, with 2 0. Unlike Yimax (which represents the maximum possible lag between cause and effect), y simply specifies the lag of interest for the query posed by the user. In the remainder of the paper, y is considered to be in {0, 7max). ) and, with a slight abuse of notation, P(Y, = Ye do(Xi-, = xt-))) is written as P(ys I do(x)_))). The identifiability of the total effect in FT-ADMGs is defined as follows.
Definition 3 (Identifiability of total effects in FT-ADMGs). Let X:-y and Y, be distinct vertices in FT-ADMG G = E). The total effect of Xt-y on Y, is identifiable in G if Pr(veldo(z==>)) is uniquely computable from any positive observational distribution consistent with g.
A total effect is uniquely computable if Pr(ye | do(x))))) can be expressed using do-free formula. Given fully specified FT-ADMG, there exists many tools to identify the total effect. For example the standard
backoor criterion (Pearl 1993b 1995) can be used to find a set of covariates B that is sufficient for adjustment: in such case the do-free formula of the total effect is written as Es Pr(ve Zt-y, .b) Pr(b). When such a set does not exist due to latent confounding, the standard front-door critcrion, as introduced by Pearl (1995) can sometimes euable the derivation of an alternative do-free formula.
However, in many real-world applications such as medicine or epidemiology, experts often cannot provide the FT-ADMG. Furthermore, discovering the true FT-ADMG from real observational data is challenging (Spirtes)
et al. 2000; Mogensen et al. 2018 Runge et al. 2019 Assaad et al., 2022) because 1) causal discovery aigorithms rely on untestable assumptions that are not always satisfied in real applications [Ait- et al, 2023) and 2) even when the assumptions hold, the output of such methods does not always correspond to the true graph but rather to a class of graphs that include the true graph GGerhardus & Runge 2020 Therefore,
be viewed as merely a snapshot of the larger graph. Under Assumption 2 sometimes possible extrapolate the entire graph from such a snapshot.
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experts typically rely on a partially specified representation of the FT-ADMG, known summary causal grapl M
Definition (Summary Causal Graph with possible latent confounding). Consider an FT-ADMG g = E). The summary causal graph (SCG) G = -0, E9) compotible with G is defined in the following way.
S (= = (Vig
V
Vi [1,
Ex1 : + Y
VX,Y 3t' S E [to, tmazl such that X + Y2 E E), VX,Y 3t' [to, tmael such that Xy e--- Yi [].
E =<xx ---- Y
where E = UE.2
SCG notations For an SCG G* directed path from X to Y and the edge Y + X form a directed cycle in go. Additionally, the specific case where a vertex has a directed edge to itself (self loop) is considered as a cycle too. Cycles(X. 9") denotes the set of all directed cycles containing X in G". A directed path between X and Y is a path between X and Y which starts by X + and does not contain any arrow on the path pointing strictly towards X. A back-door path between X and is a path between X and which starts by either X + or X c- or X =. If X > Y or X = Y, then X is porent of Y. The notions of ancestors and descendants are defined similarly as in the case of FT-ADMGs. A path in an SCG is blocked given a set W if it contains strict collider at W (i.c., -> W +, and not = W + or
=) such that W Desc(W, G*) 0 or if it contains strict non-collider al W (i.e., + W + or < W - or - W =, and not = W <- such that W E W and there exists a directed edge pointing from W to a vertex on the path that does not form a cycle with W. A path in an SCG is activated it is not blocked. In particular, a path is activated by an empty set if it does not contain any striet collider. The notions of interception is defined similarly as in the case of FT-ADMGs. Lastly, each vertex in an SCG is called a cluster or a macro vertex and it represents a time series.
Many FT-ADMGs might share the same compatible SCG. For example, Figure Ic presents the SCG com- patible with the two FT-ADMGs in Figures Ta and 1b Figure Ti presents the SCG compatible with the two FT-ADMGs in Figures Id and Ie and Figure Li presents the SCG compatible with the two FT-ADMGs in Figures 1g and Th For a given SCG G*, any FT-ADMG from which G can be derived is called as a candidate FTMA for G The set of all candidate FT-ADMGs for G is denoted by C(G*).
This paper focues on identifying the total effect when the only knowledge one has of the underbying DTDSCM consists in the SCG derived from the unknoum, true FT-ADMC In this setting, the identifiability of the total effect in SCGs is defined follows:
Definition& 5 (Identifiability of total effects in SCGs). Consider an SCG g Let Xi-r and Y, be distincl vertices in every candidate FT-ADMG in C(g"). The total effect of X,-y on Y, is identifiable in g if Pr(ye]d((x,-)) is uniquely computable from any positive observational distribution consistent with any FT- ADMG in C(G*).
Obviously, when the true FT-ADMG is unknown but the compatible SCG is accessible, it is possible to enumurate all candidates FT-ADMGs and then search for a do-free formula applicable for each of those FT-ADMGs. Within this approach, it is possible to identify the total effect if it is possible to find a set of micro vertices that satisfies the front-door criterion for each FT-ADMG in C(G*). However, enumerating all candidate FT-ADMGs is computationally expensive (Robinson 1977), even when considering the constraints given by an SCG. Therefore, this paper addresses the following technical problem:
"One key motivation for employing summary causal graphs stems from the current limitations of causal discovery methods, which often struggle in practical applications due to their reliance non-testable strong assumptions. Particularly in fields like medicine and epidemiology, researchers tend to prefer graphs built from prior knowledge rather than those inferred purely from data. However, fully specified graphs are very complicated to construct and validate manually and that is why it is important to work with (and ask experts to build) partially specified graphs such summary causal graphs. That being said, recent studies have demonstrated that inferring summary causal graphs from data more feasible than inferring FT-ADMGs from data Wahl al. 2024.. This supports the relevance of our work even if researchers choose utilize causal discovery, suggesting that our approach remains applicable when taking a data-driven approach get the summary causal graph.
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Problem 1. Consider on SCG G* and the total effect Pr(yoldo(z1--).. The atm is to find sufficient condi- lions for identifying Pr(va)do(xx-)) using an SCG with latent confounding without enumeraling all condidate FTT-ADMss in C(G*).
3 Unsuitability of the standard front-door criterion when applied to SCGs
This section clucidates why the standard front-door criterion does not straightffrwardly apply to SCGs. The standard front-door criterion consists of three conditions. Initially, the criterion was introduced for ADMGs (which means it can also be correctly applied to an FT-ADMG), but in order illustrate its unsuitability in the context of this paper, it is presented here with few modification (in blue) given Definition G
Definition& 6 (Standard front-door criterion naively applied to SCGs). Consider an SCG G*. A set of macro vertices W in 9 satisfy the front-door criterion relative to a pair of micro vertices (Xt-y: Yi) compatible with a pair of macro verlices (X,Y) in G if:
1. W intercepts all activated directed paths from X to Y; there is no activated back- door path from X to W; all back- door paths from W to Y are blocked by X;
To obtain the standard front-door criterion for FT-ADMGs, the terms in blue need to be modified as follows: since the variables of interest (X,-x: Yt) are already verticos in the given graph (FT-ADMG), there's no need to map these vertices to their compatible counterparts in the SCG. Therefore, the phrase 'compatible with a pair of macro vertices (X,Y) in gor should be removed. Next, replace all remaining instances of "SCGs" with "FT- FFT-ADMss' and "G"" with 'G'. Additionally, since FT-ADMGs do not include macro vertices, the term "macro" should be replaced with 'micro'. Finally, replace all remaining instances of "X" and 'Y" with 'Xr_," and 'Y,'. Pearl's insight is that if there exists set. W that intercepts all directed paths from X to Y and there is no hidden confounding that cannot be blocked by Xt-y, the total effect of X,-, on Y, can be identified. This is achieved by: (i) identifying the effect of Xt-y on W (which is identifiable because the unobserved confounders influence X,-, but not W); (ii) identifying the effect of W on Y, conditional on Xt-, (which is identifiable because the unobserved confounders affect Y, but not W); and (iii) multiplying the do-free free formulas Pr(xt_y w) and Pr(yt Zt-7, w) Pr(x)_n). Intuitively, in the context of an FT-ADMG (or an ADMG), the standard front-door criterion enables the identification of total effect that cannot be identified through adjustment alone, by decomposing it into two identifiable total effects. The previously proposed extension of the front-door criterion to time series settings Eichler & Didelez (2010)-referceh to here as the Granger front-door criterion equivalent to the one given in Definition F with one minor change which makes slightly more restrictive: the Granger front-door criterion forbid all back-door paths from X to W (activated and not activated) in Condition 2 In contrast to the criterion proposed in the next section, the Granger front-door criterion relies on two key assumptions: it implicitly assumes there is no latent confounding across different time points within a single lime series, and explicitly rules out instantaneous causal relationships. Under these assumptions i.e., when the SCG reflects the absence of latent confounding over time and no instantancous effects Granger front-door criterion allows identification of a total effect that is not identifiable via standard adjustment, by decomposing it into two identifiable total effects.
It turns out that directly applying the standard front-door criterion as introduced in Pearl 1995) to SCGs is not suitable. The main reason for this unsuitability is that not having a back-door path between two macro vertices in the SCG does not imply that there is no back-door path between two compatible micro vertice in any FT-ADMG in C(g*). Which means that even if Conditions SCG G*, this does not guarantee that Conditions 2 and 3 of the standard front-door criterion are met for any FT-ADMG within C(g*). Sometimes, this would imply that even if the standard front-door criterion is satisfied when applied to SCGs, the total effect of interest might be non identifiable. For illustratation consider the total effect P( | ( and the SCG in Figure Ii where W satisfies the standard front-door criterion with respect to X and Y. However, the total effect is not identifiable using W because in the FT- ADMG in Figure or or Figure [h (which is assumed to be unknown), there is a back-door path between Xt-1 and W, passing by X, which should not be blocked by X, (since X, is a descendant of X,-1 and ancestor of W.). Similarly, the Granger front-door criterion would also incorrectly conclude that the total effect is identifiable. Recall that the Granger front-door criterion assumes the absence of latent confounding across
2 and 3 of Definition 0 are satisfied in the
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different time points within single time series, and explicitly assumes no instantancous causal relationships and both of these assumptions are violated in the ADMGs shown in Figure Lg and Figure Th Therefore, it is not surprising that, when relying solely on the SCG and lacking information about the corresponding FT-ADMG, the Granger front-door criterion would mistakenly suggest identifiability of the total effect.
Let us also consider another similar example, with a similar SCG but where X - X is not in the SCG but there is a larger cycle containing X, like the SCG in Figure 3b and suppose that our aim is to identify the total effect Pr(se do(X,-1)) with Tmax 1. It is possible to construct FT-ADMG compatible with this SCG in which a back-door path exists: X(-1 + UL-
X, W,. This path cannot be blocked by conditioning on X₁ because the presence of a self loop on X implies that X, could be a descendant of XI-1- At the same time, blocking this path via U,-1 is also problematic, because it is possible to imagine an alternative FT-ADMG, also compatible with the given SCG, where the same path is instead a directed path: X,_ - U,- - X; - Wi, implying that U--1 is descendant of X,-1- In this case, conditioning on UL-1 would block a valid directed path, further complicating identification. However, in such cases, applying the standard front-door criterion or the Granger front-door criterion would incorrectly suggest that the total effect is identifiable when, in reality, it is not. Again, this is unsurprising, these tools were not designed for the specific sctting considered in this paper.
At an intuitive level, the failure of the standard front-door criterion and of the Granger front-door criterion in these cases arises due to the simultaneous presence of instantaneous relations between time series and either X +-+ X and a self loop on X in G or just simply larger cycles on X. The existence of X +-.-. X in g implies that there may be a back path between X1-y (for y > 0) and W,, starting with X,-y +- ---y+~> X,. Meanwhile, the presence of self loop on X means that this back-door path cannot be blocked using X since X₁ would be a descendant of Xr- As result, conditioning on X, would block directed path from X1- to Ws, leading to bias in the estimation of the total effect of X,-, to W,. It is important to note that this kind of back-door paths is not immediately apparent when examining only the SCG with classical tools built for DAGs or ADMGs such as the standard front-door criterion. The same intuition holds for the case when there is larger cycles containing X even in the absence of X +---+ X in the SCG.
In other cases, the unsuitability of the standard front-door criterion when applied to SCGs might imply that, even if the total effect of interest is identifiable, the corresponding do-frec formula could be more complex than the one associated with the standard front-door criterion (Pearl 1995) or the Granger front- door criterion (Eichler& & Didelez, 2010]. For example, consider the total effect P(ye xx-1) and the SCG in Figure Ic (no cycle containing X in the SCG but X --- X is in the SCG), where W satisfies the standard front-doo criterion with respect to X and Y. Now, consider one of the FT-ADMGs given in Figures La
and 1b compatible with this SCG. In this FT-ADMG, the micro vertices (i.e., (We-1, W.)) corresponding
to the macro vertex W, which intercepts all directed paths from X,-1 to Y,, do not satisfy the standard front-door criterion for (X,,1, since there exists back-door path X,-1 4-- X, + W,. However, since there is no cycle containing X in the SCG, this back-door path can be easily blocked by adjusting for Xt. Doing so results in a do-free formula similar to the one derived using the front-door critcrion, but with the additional step of adjusting for X,. These types of do-free formulas are derived by combining the back- door criterion with the front-door criterion, as discussed in (Pearl, 1995) and subsequently applied by other authors (Fulcher et al., 2019). However, it is important to note that the same do-free formula should not be used if the SCG in Figure If (there is cycle containing X in the SCG but X x X is not in the SCG) is considered. Moreover, distinguishing between these different scenarios is not feasible when relying solely the SCG, whether using classical tools developed for DAGs ADMGs, such as the standard front-door criterion, or tools tailored to SCGs that assume no latent confounding across time and no instantancous relations, such as the Granger front-door criterion. These types of unsuitability also exists when considering instantaneous total effects. For instance, take the SCG in Figure 2b and the total effect of X, on Y. In this case, there are no back-door paths between X and W within the SCG. However, it is possible to construct an FT-ADMG where a back-door path exists: X, < UU,+ U, +.... ... + U2-1000 - X - Wi-1000 + ... -> W (assuming lo < t 1000). This path must be blocked to ensure correct identification using the front-door formula. To block this and similar paths, one would need to either adjust all variables between Xt, and X,-1 (which is impractical) or simply adjust on U., U..l...., U, - However, the standard front-door criterion and its associated theorem cannot be applied to infer this requirement.
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4 The identifiability of total effects in SCGs with latent confounding
This section, presents the main results of the paper, with some of the corresponding proofs omitted and provided in the appendix. Let us start by giving the extension of the front-door criterion for SCGs.
Definition 7 (SCG-front-door criterion). Consider an SCG g" A set of macro vertices W in G satisfy the SCG-front-dow criterion relative to a pair of mácro vertices (Xex, Vi) compatible with a pair of macro vertices (X,Y) in 9 if:
W intercepts all activated directed paths from X to Y; 2. there is no activated bock-door path from X to W; 3. all back-door paths from W to Y are blocked by X;
4- one of the sollowing holds:
(a) Cycles(X.9))
10 or
(b) Cycles(X, (*))") = (X + }} and BZ Anc(X, such that X c.... Z in 9" or
(c) =0
Conditions 3 in Definition correspond to the three conditions in the standard front-doos criterion (Pearl
1995). For illustration of Definition Figure 2 provides several examples of SCGs, including the SCGs given in Figures T and Figures Di where Definition 7 is satisfied for W relative to (Xt-7: Yt). Notice that Desinition 7 remains isfie for W relative to (X,, Y2) if a cycle is added on X that does not involve any other vertex in the presented SCGs, as illustrated in Figure 3 which includes the SCG given in Figure Ic Note
that in these SCGs, W is not necessarily the only vertex satisfying Definition 7 for example, in Figure 2c U, also sat isfied the SCG-front-door criterion. In contrast, Figure | provides several examples of SCGs where Definition not satisfied for any vertex relative to (x)
In the following, lemmas that form the building blocks for the theorem introduced at the end of the scction is presented. The first lemma asserts that if a set of macro vertices intercepts all directed paths between X and Y in an SCG, then there exists a finite set of micro vertices that intercepts all directed paths between X;-y and Y, in any candidate FTMADMG. This mirrors the first Condition of the standard front-door criterion for ADMGs.
Lemma 4.1. Consider an SCG G If a set of macro verlices W intercepls all direcled paths from X to Y in G then ((bsstdoossss) intercepts all directed paths from Xt-y to Yt in any candidate FT-ADMG in c(g').
For an illustration of this lemma, consider any FT-ADMG depicted in Figure D or Figure 5 where in the corresponding SCG, all directed paths from X to Y are intercepted by W. Observe that any directed path from X,-1 to Y; must pass through micro vertex W.-x that corresponds to W. If X 7, then all active paths from X,-I to Y, passing through W,- are not directed paths since W,-x is temporally prior to t and y. Conversely, if A < O, all paths from X,-1 to Y, that pass through Wi- are blocked, as W would always act as a collider due to its temporal positioning after both t and 7-7-
The second lemma asserts that given Couditions 13 along with Condition 1a it is guaranteed that there exists finite set that blocks all back-door paths from X1-9 to any vertex in any set of micro vertices that intercepts all directed paths from X,_y to Y, in any candidate FT-ADMG. Furthermore, this finite set does not contain any descendants of X1-x. This mirrors the second Condition of the standard front-door criterion for ADMGs.
Lemma 4.2. Consider an SCG g = (S, and the pair of micro verlices (Xi-y; Yi) compal- able with the macro vertices (X,Y). Y). Suppose W is set of macro vertices that statisfies Con- ditions 1 and 3 of Definition 7 in G relative to the pair of micro vertices (Xi-,, Yi). If the set ((B(---)ossts
Cycles(X, = then for any W E {(We-stc)osss) ((BBa-e) ()ISE B E Ano(Wl, g*) Oi Desc(X, (o) ((X blocks all back-doon paths from X to W and does not contain any descendant of Xt-, in any condidate FT-ADMG in C(g*).
E Par(X, U
The intuition behind Lemma 4.2 is that conditioning on micro variables corresponding to the parents of X in the SCG blocks all back-door paths that start with X, < This still leaves some possible back-door paths
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Z
(a)
(b)
7
W
(d)
(e)
Z
(f)
(h)
Z
W
W
(i)
(j)
Figure 2: Ten SCGs satisfying Definition 7 for W relative to the pair of micro vertices (X Y)), Vy E (0.... 7max); Le., for the first half these SCGs, Conditions 1 Conditions 2 Conditions and Conditions 4a in Definition are satisfied and for the second half, Conditions D Conditions Ø Conditions and Conditions 4b in Definition 7 are satisfied. Each pair of red and blue vertices represents the total effect of interest and green verlices are those that intercept all directed paths from the cause to the effect of interest.
active, those starting with Xi-y e---,, which require additional conditioning sets to be blocked, namely some micro variables corresponding the ancestors of W. The set (X,-2, X,) satisfies the conditions of this lemma when considering the total effect of X,-1 on Y, and the SCG presented in Figure Ic Specifically, {X,-2, Xr} effectively blocks all back-door paths from X,-1 to (W,11, and does not. include any descendant of X,-1 in any FT-ADMG that is compatible with the SCG (e.g., Figure la Figure [b]. For another, more complete, visual explanation of why the selected set in Lemma 4.2 blocks all back-door paths, refer to the FT-ADMG in Figure 5a Tn the figure, the vertices corresponding to ((Ba--e)ostssmm B Par (or(X,, ())) are highlighted brown, the vertices corresponding are highlighted in pink, and the vertices corresponding to ((X==y4e)
(BB-n--)isés B Anc(w, (, 9*) n Deso(X, g*) highlighted in gray.
Notice that if Cycle(XX, G*) * 0, then there could exist an FT-ADMG in C(G*) where micro vertices related to X that temporally succeed X,-y can be descendants of X1-7. Additionally, these descendants can share a latent confounder with X,-y, and the path responsible of this latent confounding cannot be blocked. For example, in the FT-ADMG in Figure [g compatible with the SCG in Figure [i that contain a cycle of size on X, it clear that X, descendant of X,-1 and that the back-door path from X,-1 to W, starting with X,-1 c- X, cannot be blocked by any vertex. Thus, (W,11. Wi} cannot be used to intereept the relationship between X and Y, for identification using the front-door criterion. However, it turns out this
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Z
(b)
W
W
(d)
(e)
Figure 3: Five SCGs satisfying Definition for W relative only to the pair of miero vertices (Xt, Yt), i.e. Conditions I Conditions 2 Conditions 3 and Conditions Ec in Definition are satisfied. Each pair of red and blue vertices represents the total effect of interest and green vertices are those that intercept all directed paths from the cause to the effect of interest.
is not the case when either the only cycle on X is self loop and there is no latent confounding between X and its ancestors-whit implics that X X is not in 9 (as in the SCG in Figure or when y = 0, as stated in the following two lemmas.
Lemma 4.3. Consider an SCG G = (S,E") and the pair of micro vertices (Xi-y. Yi) compatible with the macro vertices (X,Y). Suppose W a set of macro vertices that statisfies Conditions 1 and of Definition 7 in G relative to the pair of micro vertices (Xt-,,
X} Z in G then for any Wr-x E (WVn-++)oossv/) the sel blocks all back-door paths from Xe-y to We- and does not contain any descendant of Xi-, in any candidate FT-ADMG in C(G*).
If Cycle(X, =
and /32 Ane(X, 9*) such that X
{(B--g-eoses tmax [B Par(X,9")) U (()----)ists
The intuition behind Lemma 4.3 is that, similarly to the previous lemma, conditioning micro variables (for t-y to t-7-7mma "max) corresponding to the parents of X in the SCG blocks all back-door paths that start with X1-, + U-- where X + U. This leaves only possible back-door paths starting with X,-, <- N,- which can be blocked by timepoints of X that preceeds { -- 7. The set {U,-2, X statisfies Lemma 4.3 for the total effect of X on when considering the SCG in Figure If Which means the set (U,-2, UU-1,X,-2) blocks all the back-door paths between X,-1 and W. in any FT-ADMG compatible with the SCG considered. To visually
why the selected set in Lemma 4.3 blocks all back-door paths, refer to the FT-ADMG in Figure 5b In the figure, the vertices corresponding to (Brr-e)oseStmal | Par(X,g*) are highlighted in brown and the vertices corresponding to ((x=y-e))sess )mmr)) are highlighted in pink.
Compared to Lemma 4.2 in Lemma 4.3 the set (X,-y+e)1 SES is not included the conditioning set for two main reasons: (1) in the context of Lemma 4.3 where self loops are allowed, the temporal variables this set may be descendants of Xt Xt-ry 7, and (2) in this scenario, no back-door paths start with e--+, For the same reason (2), the only subset considered from (B, & yymx B Anc(W, g*) Desc(X, g*) is (X,-,-e)1 SE ymax.
Remark that Lemma 4.3 would fail if larger cycles involving X were considered, meaning cycles on X beyond a simple self loop as discussed in Section 3
Let us now turn to a lemma, which permits even larger cycles and allows confounding between and its ancestors, but imposes the restriction that y must be zero.
Lemma 4.4. Consider an SCG ga (S,E*) and the pair of micro vertices (X,, Y,) com- patible with the macro vertices (X,Y). Suppose W set of macro vertices that stat- isfies Conditions and 3 of Desinition in go relative to the pair of micro vertices
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W
(a)
(b)
W
W
(d) SCG.
(e) SCG.
Figure 4: Five SCGs not satisfying Definition
7 for any vertex relative to the micro of vertices (Xz_yy Y)).
The SCG (a) and (d) does not satisfy Definition 7 because Condition 2 is not satisfied. The SCG in (b) and (e) does not satisfy Definition because Condition
not satisfied. The SCG in (c) does not satisfy Definition 7 because Condition and are not satisfied. Each pair of red and blue vertices represents the total effect of interest and green vertices are those that intercepts all directed paths from the cause to the effect of interest.
Then for any W, E We the set {(B:-))ostS E Ano(X, (o,g())))))) (eso(( (, U (BB-e))sttrma |B (Anc(X, 9*) ( Dese(X, }r blocks all back-door paths from X, to W2 and does not contain any descendant of X, in any condidate FT-ADMG in C(G*),
(X,; V1).
The intuition behind Lemma 4.4 is that conditioning on micro variables corresponding to the parents of X in the SCG cannot block all back-door paths because a parent can also be a descendant of X (since cycles are allowed in this case), but the set micro variables (for t to t Tmaz) corresponding to the ancestors of X that are not descendants of X in the SCG and the set micro variable (for ( to L -mex) (a subset of the set in purple) corresponding to the ancestors of X that descendants of X in the SCG blocks all back-door paths that start with -> This still leaves some possible back-door paths active, those starting with e--,, which require additional conditioning sets to be blocked, namely some of the ancestors of W. The set (x,-1, W statisfies Lemma 4.4 for the total effect of X, on Y, when considering any of the SCGs given in Figures Ic 1f and li Which means the set {X} blocks all the back-door paths between the X, and Wt in any FT-ADMG G compatible with the SCG considered. To visually see, on a more complete example, why the selected set in Lemma 4.4 blocks all back-door paths, refer to the FT-ADMG in Figure 5c In the figure, the vertices corresponding to {BBg-e)useS |B E Anc(X, (*)\Dec(XX, (
9)} are highlighted in brown and the g*) U Ane(W, G*)) n Desc(X, 9)} are highlighted in vertices corresponding to ((B, B (AAn(X,,60 pink.
Notice that, compared to Lemma 4.2 in Lemmas
4.3 and 4.4 two key replacements are made regarding the selected conditioning sets: { (BB-c)ooes mmx B E Par X, } is replaced with ((B_ee)osesymxx] B e Ana(X,g*) (", (ee))eee((X,,)) (X,)) and {(B e) Disesmmas B E Anc(W, g*) n Dese(X,g*)) is replaced with {[B, ((B==))sessomxll)
(Anc(X, 9*) U Amo(W, 9*)) n Deso(X, 9)} to account for potential cycles involving X. For example, in the FT-ADMG shown in Figure 5c conditioning only on the parents, specifically U, and U,- (without conditioning on 2, and 7,,1), would be sufficient to block all back-door paths. However, there cxists another FT-ADMG compatible with the SCG in Figure 3b which is almost identical to the FT- ADMG in Figure 5c except that now VI E [lo, (mar]] Xx + U, (which means U. A X.). In this FT-ADMG, U, becomes a collider between X, and Z,, so conditioning on it would activate a new path that cannot be blocked unless the ancestors of X, are also included in the conditioning set.
The last lemma asserts that given Conditions 13 it is guaranteed that there exists finite set that blocks all back-door paths from some micro vertices (that intercept all directed paths from X,-, to Y, in any candidate FT-ADMG) to Y, Furthermore, this finite set does not contain any descendants of the micro vertices that intercept all directed paths. This mirrors the third Condition of the standard front-door criterion for ADMGs.
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Lemma 4.5. Consider an SCG 9 (S) and the pair of mácro vertices (X1- Yi) compatible with the macro vertices (X, Y). Suppose W is set of macro verlices that slatisfies Condilions 7 and of Desini- tion ? in 9 relative to the pair of micro vertices (Xt-,,Y.), Then, for any Wr- E to y the set (Bi-g-e) -------sssmss E Ano(W,g)) (e) blocks all back-door paths from Wr-x to Y not passing by ((w)-otclosss)) N( and il does not contain any descendent of W
mmax [ E Ano(W, (ao(w.g))) n Desc(W. g')
The intuition behind Lemma 1.5 is much simpler than the intuition behind the previous lemmas since according to the conditions of Definition 7 there cannot be any back-door paths from W to Y, starting with +-- and cycles on W would not create any issues since they either implies active back-door paths that pass by W such that X' > 7 which can be casily blocked set of micro variables, conceptually very similar
to the one used in Lemma 4.3 and 4.4 Or they pass by Wr-x such that W ((W/ ) (WW+>>) >). satisfies the conditions of this lemma when analyzing the total effect of Specifically, X X1-1, X,} effectively blocks all back-door paths from W. (respectively, Wi-1) to Y, that do not pass through W (respectively, Wt). Additionally, this set does not contain any descendant W or W,-1 in any FT-ADMG that is compatible with any of these SCGs (e.g., Figures La visual understanding, using a more complete example, why the selected set Lemma
The set 2x 1, X,}
X,-1 on Y, in any of the SCGs presented in Figures
Ic Lf and Ii
lb
Id
le
L or 1h
To gain clearer
4.5 blocks all back- door paths not passing by ((Weyte)osess) M(Wia)) refer to the FT-ADMs in Figure 5d In these FT- ADMGs, the vertices corresponding to {(B(---t) -SS-- mmms B E Anc(W, (P))DessWW, }o)} are highlighted BB E Ane(W, meow,,9 9*) n Dese(W, (o)} are highlighted
in orange, the vertices corresponding to in purple, and the vertices corresponding to ((We-yte)osess) are highlighted in green. The vertex X,-y is highlighted in red and in orange since it is the main cause of interest and at the same time it is in the set (BB-x-t)--s(En |B E Anc(W, Desc(W.G"))
In the following, sound theorem is introduced for identifying total effects using SCG, even in the presence of cycles and latent confounders.
Theorem 1. If set of vertices W satisfies the SCG-front-door criterion (t.e., Definition relative to Yi) and if Pr(s)= > 0 then Pr(y do(x)_n)) is identifiable and is given by the do-free formula:
Pr(y do(xx=z))
ΣΣ
Pr(s zr-;, Pr(((")
(1)
F
E'
Σ
Pr(ye 1f,b,, ,x(- (,,b,, Pr((b', x.--)
x
by
where F = ((W: ((WL-tt)0ssssh
B" *==((B =B= B (e)eXXX (X,9))
U ((B 111ss BB (Anc(X, G*) U Anc(W, G*)) n Desc(X, G)} U such that B ((X if Condition
4a is satsified, and I = 0 otherwise;
and
B
B Ane(W, G') 9*))}
U U(BB-n-()s 11sts Tmax [ E (Anc(W. ("
Proof. By Lemma 4.1 since ' intercepts all directed paths from X to Y in the SCG, then (Woes+e) i.c., F, intercepts all directed paths from X to Ye in any candidate FT-ADMG in C(g*). Given this result, by the law of total probability, the total effect Pr(m do(xx-z)) can be computed in two steps: first, by computing Pr(l] do(xx_))) and Pr(y: do(f)), and subsequently multiplying the two quantities together while summing over f.
Notice that the set used Lemma 4.2 and the set used in Lemma 4.3 is a subset of B and at the same time 8 cannot contain any descendant of X,-, in the context of Lemma 4.2 and in the context of Lemma 4.3
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We
W.
(a) A candidate FT-ADMG of the SCG in Figure 2b showing the set introduced in Lemma 4.2 to block all back-door paths from X to W and W..
(b) A candidate FT-ADMG of the SCG in Figure 2g showing the set introduced Lemma 4.3 to block all back-door paths from XI-1 W and W..
Ue
W3
(c) A candidate FT-ADMG the SCG in Figure 3b showing the set introduced in Lemmma 4.4 to block back-door paths from X, to Y..
(d) A candidate FT-ADMG of the SCG Figure 2 showing the set introduced in Lemma 4.5 to block all back-door paths from W. and W. to Y..
Figure 5: Four FT-ADMGs that correspond respectively to the SCGs Figures 2b 2g and 2c In each
graph, the red and blue vertices represent the total effect of interest, while the green vertices are those that intercept all directed paths from the cause to the effect of interest. In FT-ADMGs (a), (b), and (c), all back-door paths from the red vertex (in bold) to the green vertices (in bold) are blocked by the brown, pink, and gray vertices. In FT-ADMG (d), all back-door paths from each green vertex (in bold)to the bluc vertex (in bold) are blocked by the orange, purple, and other green vertices that are temporally prior to the selected green vertex.
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Therefore, by Lemma 4.2
4.3
and 4.4 B statisfies the standard back-door criterion Peaal 1995 relative
to (Xt-,, F) which means that by Pearl (1955] Theorem 1), Pr([T do(xx-1)) = Σ.. Pr([ zt-7, b²) Pr((b").
By Lemma 4.5 B U (X-=x} blocks all back-door paths from Wi-x to Y for all 0 S X S 7, except those passing by (W- +eloses (W--> and it does not contain any descendant of W Therefore, since it is
intervened on F [[W )oses,,, Bf U (Xt=)) satisfies the back-door criterion relative to (F,Y)), which Theorem 1), Pr(ye do(()) = EE any_. Pr(y [f, 1b , Pr(b/)
means that by Pearl 1995
For simplicity, Theorem 1 is stated under a strict positivity condition on the distribution, which should ide-
ally be relaxed, as discussed in Hwang et al. (2024). Furthermore, the sets used in Equation of Theorem [
are applicable whether Condition 4a or Condition 4b or Condition 4c of Definition is statisfied. Never-
theless, by Temma 4.2 the set. IB in Theorem can be reduced to (BB-g-()osts S
Tmaa B E Par(XXg9)} U
Trowe can be reduced to (BB=-g-))SSSSmum |B E Par(X, ( ((X))U((--))))tst)
((B---e)sses Trows
Anc(W, 9o) n Desc(X, 9 (((
when Condition 4a is satisfied and it
when Condition
4b is satisfied.
Definition 7 reveals a broad class of SCGs that enable the identification of the total effect using the do-free formula provided in Theorem 1 This significantly enhances the applicability of causal inference in real- world scenarios where the fully specified causal graph is unknown, but its SCG available. For example, consider a conceptual study aimed at evaluating the effect of sedation levels administered today on blood pressure regulation tomorrow. Mathematically,, this corresponds to estimating Pec[Bloodreesureeoomore bl Sedativetoday = s) for given patient in critical care. Suppose daily patient monitoring records sedation levels, arterial pressure, and heart rate variability over time. Let us assume that the SCG corresponding to these time series is the one depicted in Figure [d] where X represents sedation levels, Y corresponds to arterial pressure, and W denotes heart rate variability. In this setup, the effect of sedation on blood pressure is fully mediated by heart rate variability. Note that the SCG encodes the assumption of the presence of a latent confounder (physiological factor) between sedation needs and blood pressure regulation, and latent confounding between different time points of X and latent confounding between different timepoints of Y. In this scenario, the total effect of interest is not identifiable by adjustment alone, the back-door criterion is not applicable. The standard front-door criterion and the Granger front-door criterion are not applicable for the reasons given in Section g However, the SCG-front-door criterion and the corresponding theorem provide the appropriate tools to correctly quantify the total effect of sedation levels administered today on blood pressure regulation tomorrow from non- experimmmie data, assuming all underlying assumptions hold. In this case, the appropriate do-free formula is given by:
Pr(se | xT-1)= = Σ
Σ
Pr(we, w | It-1, 2x-2
als, 1 Zt- 2,0
Σ
Wt, W, wtwp://w.......oterrrreeddd..ee.
5
Towards non identifiability results
This section explains why the total effect Pr(ye do(x)-))) is not identifiable in any of the SCGs shown in Figure
Consider any SCG in Figure Pr(se do(xx))))) not identifiable by the back-door criterion Pearl 1995
because the back-door path X X +-..+ Y, cannot be blocked by any observed micro vertex. In the fol following, it is demonstrated that for each SCG, the total effect cannot be identified by decomposing it into other total effects and then multiplying them together.
Consider the SCG g in Figure 4a Let us show that x = W in G*, then there exists at least one X where 0 < À < "max such that Pr( Pr(we=) do(2x-z)) is not identifiable. For 7 = X, there exists an FT-ADMG 9, E C(g*) such that X, E Par(W, 91) and another FT-ADMG 92 E C(g²) where W, E Par (X,, 92). Thus, the total effect is not identifiable. The same logic can be applied to the SCGs in Figure and similar logic for Figure 4b to show that Pr(ye do(w:)) not identifiable.
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Consider the SCG 9 in Figure 4d Let demonstrate that if X - W + X exists in g then there exists at least one X where 0 S À < "mmxx such that Pr(uz-> | do(xt-y)) is not identifiable. Let's take ? = X. Since there is a back-door path x* = (X) there exists an FT-ADMG 91 with the back-door path 11 = Xx t U, t W,, which would need to be blocked by conditioning on Ug. Howover, there is also a directed path x = (X, W, 2), which corresponds to an FT-ADMG 92 with the directed path #2 = Xx + W + U,, where conditioning on U, would bias the estimation of the total effect. This ambiguity around U, implies that the total effect is not identifiable. A similar argument can be made for the SCG in Figure 4e to show that Pr(y) do(w))) is not identifiable.
6 Conclusion
This paper focuses on identifying total effects from summary causal graphs with latent confounding. Def- inition 7 establishes graphical conditions that are sufficient, under any underlying positive probability dis- tribution, for the identifiability of the total effect. Additionally, in cases where identifiability is established, Theorem 1 provides a do-free formula for estimating the total effect. These results contribute to the ongoing effort to understand and estimate total effects from observational data using summary causal graphs. The main limitation of our result is that, while it is sound, it is not complete for identifying total effects using summary causal graphs. Consequently, future work should aim to develop a complete identifiability result that accounts for latent confounding and cycles, along with corresponding do-free formula that does not require any information about the distribution.
Acknowledgments
thank Fabrice Carrat, Nathanael Lapidus, and Benjamin Glemain for several discussions about applications of this work in epidemiology. This work was supported by the CIPHOD project (ANN-23-CPPJJ-221--0))
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A Appendix
Lemma 4.1. Consider an SCG G*, If set of macro vertices W intercepts all directed paths from X to Y in G then ((we))))ssts>) intereepts all directed paths from Xi-y to Yt in any condidate FT-ADMG in c(g*).
Proof. Let G be candidate FT-ADMG in C(G*). Consider directed path from Xi-y to Yi in G. Suppose Then, π must include at least one vertex from a does not contain any vertex from {(W) ((we-x+))sses))) as all paths from X to Y are intercepted by W in G Consider the case where there exists W in π such that & <<< O. In this case, notice that Wt- e is temporally prior to X,-, which means that by Assumption | ı cannot be a directed path because there exists no directed path from X,-y to Wi-++e. Consider the case where there cxists Wi-zé π such that E > y. In this case, notice that Y, is temporally prior to Wi-++e which means that by Assumption [ camnot be a directed path because there exists no directed path from We-++e to Y,. Therefore, it must be the case that π includes at least one vertex from (W (We=g+e)oses> Since G and i. are arbitrary, this conclusion applies to all directed paths between from X1-, to Yi in any candidate FT-ADMG in C(G*).
Property 1. Consider an SCG 9 If there is no backdoor palh from X to W in G, then each active backdoor paths from X1-y to Wr-x in any FT-ADME in C(G") has to pass by another vertez Xe-x with X # 7,
Proof. Consider an SCG G and a given candidate FT-ADMG 9 in C(g*). First, note that a backdoor path from X,-y to Y, in G cannot consist solely of edges pointing toward X,-y because if this were the case, there would be a backdoor from X to W path in G This would contradict the assumption that there is no backdoor path from X to W. Therefore, the only possible backdoor path, is a path containing at least one ancestor of Xt-y and W.-x- Consider an active backdoor path π in G: Xt-y +- Un,.... Uv",.. + Spm....,, + .... Zemm.... ... Zinon + Wr-x. Let's assune that there exists no X,-x such that X + y that belongs to π. Since the path π does not pass through any instance of X other than X,-y, the path that is compatible with π in G is the following path : X + U.... S....... Q- ..... Z - W where each subpath of micro vertices with the left and right endpoints shares the same macro vertex are replaced by this macro vertex. The path we've identified corresponds directly to a backdoor path in G This contradiction implies
that the assumption we took must be false. Moreover, if we assume that X2-x belongs to the path then if we replace Q by X then the compatible path in G would be X + Z + W which is not a backdoor path.
Lemma 4.2. Consider an SCG 9 (S,E*) and the pair of micro vertices (Xt-y, Y.) compat- ible with the macro vertices (X,Y). Suppose W is set of macro vertices that statisfies Con- ditions 1 2 and 3 of Definition 7 in Ge relative to the pair of micro vertices (Xt-,, Vi). If
Cycles( G = 4 then for any Wi-x E ((WW {(WW-tteDsssshh the set ((Br---Doosts [ E Par(X, {(B, -IsSSs |B E Ane(W,g") Dese(X, [(X
blocks all back-door poths from Xe-ry to W,- and does not contain any descendant of Xi_, in any candidate FT-ADMG in C(g*).
Proof. By definition of Tunxx, the set 1-B----0ssssm Tmox B E Par(X,99 g)) contains all observed parents of X1-, in any FT-ADMG in C(G*). Given that Cycles(X, = 0, if BE Par(X,9") then B & Desc(X, 9*), implying that {(BI---eossss Ymaae |B E Par(X, cannot include any descendant of X Therefore, this set blocks all backdoor paths from X,_, to W that begin with an undashed left arrow, i.e., 'X,_ +" without activating a new path between X,-y and W Again, since (y,les(X,99)) = 0, the only backdoor paths from Xt_, to Wi_x that is not blocked or becomes activated by (BBr-n-)osessymal B Par(X, g))} are those that are those starting with "X,_y ---,.. By Property [ all backdoor paths from X1-, to Wi-x in any FT-ADMG G in C(g*) has to pass by another vertex X,-x with X' * 7. Let us consider three cases:
Case 1 If X' < O then obviously, by Assumplion [ the backdoor path is necessarily blocked by the emply set since Xx-x cannot be an ancestor of Wi-x nor an ancestor of X,-y. Case 2 If X' > 7, then by Condition D {(X,-y-))est) can blocks the backdoor path, however this set is unbounded and therefore unpratical. So instead, we can replace by ((B---e)ssEs JB e Ane(W, go) n Dese(X, (*)) (remark that (X)=n-)))30s-mma) is included in that set) to blocks the
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backdoor path. Furthermore, by Assumption cannot contain descendants of X; Case 3 If0 < X < 7, then by Condition 2 (Xx++t))sess) blocks the backdoor path. Furthermore, since
Yamax [B E Ane(W, n Desc(XX, (*))
Cycles(X,g9) = 0, {(x
cannot contain descendants of Xi-y-
Lemma 4.3. Consider an SCG G (S, and the pair of micro verlices (Xi-;, Y2) compatäble with the macro vertices (X2Y). Suppose W is a set of macro vertices that statisfies Conditions 7 2 and g of Definition 7 in G relative to the pair of micro vertices (X--,,Yi), If Cycle(X, = {X }
and /3Z E Anc(X, g") such that X k-, Z in G then for any Wi-x E ((We---e)sstsyh the set (BB--g-))ssSSumm BB E Par(XXg9) U ((x--g-))stsrum and does not contain any descendant of Xi-7 in any candidate FT-ADMG in C(g*).
blocks all back-door paths from X Xe-y to W
Proof. Condition 2 implies that there exists no active backdoor path x² = (V1 X ,...,, V W) from X cycles that contain X
to W in 9 such that (V?...., V=-1) C Desc(X, 9") which means that there and another vertex on path from X to W. Furthermore, by Property [ all backdoor paths from Xt-y to W in any FTT-ADMG G in C(g*) has to pass by another vertex X with X +< y. Let us consider three cases:
Case 1 If X' < O, then obviously, by Assumption D the backdoor path is necessarily blocked by the emply set since Xt-x cannot be an ancestor of Wi-x ancestor of X1-7.
Case 2 If X 2 y and if the second micro vertex U--X" on the backdoor path we are considering is com- patible with the macro vertex U & Cycles(X, 9t) then the backdoor path can be blocked by (BBe--e)ostt
Ymax ymar
B Par(XX
Case 3 If 2 y and if the second micro vertex U- on the backdoor path we are considering is compatible with the macro vertex U E Cycles(X, then, since we consider that Cycle(X, {X + X}, U = X and X > 7. Therefore, the backdoor path starts with Xe- XX-x which means it is obviously blocked by ((xe-y-))ssss
In addition, by construction and by Assumption I ((Br-c)osestmal |B (O)(XXX))))((XX cannot contain any descendant of X1-y- Which means that ((Br-c)os(STma B E Par (x)))) u ((x=yy)))sssmmal blocks all backdoor paths from X._, to WaLx starting with an undashed right ar- row, i., 'X,,, t'. Since we consider that BZ E Anc(X, such that X c Z,, no additional back-door paths can exist from Xe- to W
Lemma 4.4. Consider an SCG G = ES,E*) and the pair of micro vertices (X,, Yt) com- patible with the macro vertices (X,Y). Suppose W is a set of macro vertices that stat- isfies Conditions 7 (X,, Y)). Then for any We (Be-e) |B E (Anc(XXg*) U Anc(W, Desc(X, blocks all back-door paths from X, to W, and does not contain any descendant of Xt in any condidale FT-ADMG in C(G*).
E and 3 of Desinition 7 in G relative to the pair of miero verlices We, the set ((Brs))sessmamnl) |B E Anc(X, Desss"( U
Proof. Condition 2 implies that there exists no active backdoor path n² = (V1 X...., <"= W) from X to W in 9 such that (V²,..., v=-1) C Desc(X, 9*) which means that there are no cycles that contain and another vertex on path from X to W. Furthermore, by Property [ all backdoor paths from X, to W
in any FT-ADMG in C(G*) has to pass by another vertex X,-x with X' + 0. Let us consider three cases:
Case 1 If X < O, then obviously, by Assumption [ the backdoor path is necessarily blocked by the empty set since X,-x cannot be an ancestor of W, nor an ancestor of X,.
Case 2 If X > 0 and if the second micro vertex U,~ on the backdoor path we are considering is com- patible with the macro vertex U & Cycles(X.9) then the backdoor path can be blocked by (Bos-) B E Anc(X, Anc((X De)))ees(((XX))
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Case 3 If X > 0 and if the second micro vertex UL-A on the backdoor path we are considering is com- patible with the macro vertex U E Cycles(X, 99) then the backdoor path can be blocked by Ann((,
Yamax E Anc(X, 9*))Deso(XX 9)} U ((Br BB E Ano(X, 9*) (P) n Desc(X,)))
In addition, by construction and by Assumption ! ((Bee))sssr Tmax 1B A ( Desc(X, 9)} U
{ (Bi-)))seSSmm) Trase [B E Anc(X, 9*) Desc( X, G)} cannot contain any descendant of X,, Which means that { (BB--)oSeSTuam |B E (nn(X Desc(XX 9)} (BB--)) |B E Ano( g") n Desc(X (X, 9)) blocks all backdoor paths from X, to W2 starting with an undashed right arrow, i.e, 'X, +.. This means that the only backdoor paths from X, to W, that are not blocked are the ones statring with "X, --+'. Let us consider two cases:
Case 1 If < O, then obviously, by Assumption D the backdoor path is necessarily blocked by the empty set since X cannot be an ancestor of W, nor an ancestor of Xx.
Case 2 If X' > 0, then by Condition 2 {(Xi-x-e)esi) can blocks the backdoor path, however this set is unbounded and therefore unpratical. So instcad, we can re- place it by {(BL-c)istS |B E Ann(ww (oo, W, nn Desc(X,)), Furthermore, by Assumption [ ((Beee)ssen BB E Anc(W,g*) Dese(X, 9)) cannot contain any descendant of X, in any can- didate FT-ADMG in C(g*).
Property& 2. Consider an SCG g Given Condition if all backdoor path from W to Y in G are blocked by X, then each active backdoor paths from Wr-> to Yt in any FT-ADMG in C(G*) has to pass either by a verter Xx-x or a verter WT-x" with X" +X x.
Proof. Consider an SCG G and given candidate FT-ADMG G in C(G*). The proof that there might exists an active backdoor path from Wi-x to Y, that does not pass through any Xi-x must pass through vertex W follows a similar reasoning to that used in the proof of Property [] Let's assume there is an active backdoor path between Wi-x and Y, that does not pass through Wi-x". If the path between W,-> and Y, is active and does not pass through W,w,, then this path must be relying on connections in the SCG. Consider such an active backdoor path in G: W < U...... UUr..... Spm... ...........e -.... Zeom, .... Zprn Y,. The compatible of the path considered in G is the following path W - U.....+ S... Q where each subpath of micro vertices with the left and right endpoints shares the same macro vertex are replaced by this macro vertex. The identified path corresponds to backdoor path in ga. If X is not on this path, it would lead to a contradiction, as in that case, not all backdoor paths would be blocked by X. This implies that our initial assumption that there exists a backdoor path between W and Y, that does not pass through W must be false. Consequently, under our assumption, X must be on the path in 9", and Xx-x must be on the corresponding path in g.
,2 + Y
Now, under similar scenario, let us assume that the backdoor path between Wi-x and Y, does pass through W For instance, suppose W = Qu!: this path. In this case, the compatible path in G would be W + Z + Y, which is not a backdoor path in g".
Lemma 4.5. Consider an SCG g ES,E*) and the pair of micro vertices (Xi_y, . Y)) compatible with the
macro verlices (X,Y). Suppose W is set of macro verlices that statisfies Condilions 7 2 and of Desini- tion 7 in G relative to the pair of micro vertices (Xi-,, Y.). Then, for any W {(W (Wi-g+00ssss set
the [ E Anc(W, 9*) Desc(W, G')
B E AAnc(W, g") Desc(W, (B))) blocks all back-door paths from W,-x to Ye not passing by ((W1)+ (WW->> and it does not contain any descendent of W
Proof. If there is a cycle that includes W and any other vertex not in 'W' on directed path from X to Y that is intercepted by W and that is closer than W to Y on the path, then there exists an active backdoor path from W to Y that does not pass through X (which means cannot be blocked by X). By Conditions 2 and 3 this cannot be true. Therefore, it must be the case that all cycles containing W do not include any other vertex on a directed path from X to Y that are neither W nor closer to Y on the path. By Property 2 all backdoor path from Wi-x to Y, in any FT-ADMG in C(g*) has to pass by either by X or by W with X" * X.
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Let's consider of these backdoor paths that pass by vertex Xe-x but does not pass by any tex Wz-x" such that X" + X. By Condition 3 -BS----)-SSShmmm BB E Anc(W, DescD W. 9*)3 block this path, however it can activate path of the form W(-1 + ....I-xx' --, Y, where X > 7 + max Since I y and X > y + Ymax then the parents of W which are temporally prior to B Anc(W, 9o) n Dese(W, 9*)) Fur-
can
block this path. Notice that these parents are (BB-g-()ssss thermore, by construction and by Assumption 1 ( |B E Ane(W, g*) Desc W; g*) U ((B=-)-))sttrmmll Anc(W, 9*) Desc(W, 9 }} does not contain any descendant of W
Now let's consider one of the remaining backdoor paths that pass by vertex Wi-x", Let consider two cases:
Case 1 If X" -BBi---lises Timase B Anc(W, 9t) n (es
7, then obviously the backdoor path
can be blocked by
Case 2 X" & y then the backdoor path pass by /WW which means it does not need lo be blocked.
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Reviews:
[{"review_id": "parbAIMNzl", "reviewer": "Reviewer_g9tu", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**  \n- The paper offers a rich literature review that effectively summarizes the historical development of theories related to causal inference, particularly in the context of partially specified graphs.  \n- It provides a solid proof sketch for its main results, and all technical derivations appear sound.  \n- The work is technically self-contained, making it accessible to readers who may not be deeply familiar with every aspect of the underlying literature.\n\n**Weaknesses:**  \n- The paper does not provide any explicit working example that shows an SCG derived from an FT-ADMG in which the SCG front-door criterion is satisfied. This absence makes it difficult to assess the practical relevance and applicability of the theoretical results.  \n- The definitions and notations are sometimes unclear. For example, in Definition 4 the set $S := \\{{Y \\mid \\forall Y_t \\in V \\}}$ is ambiguous, leaving it uncertain whether $Y$ represents a cluster of temporal observations. Similarly, the notion of \"macro vertices\" in Definition 6 is not explicitly defined.  \n- The motivational aspects of the paper are weak. Although the paper reviews the literature extensively, it lacks a real-world or empirically driven working example that demonstrates the utility of the proposed approach.  \n- The presentation could be improved for accessibility; for instance, the color choices in the figures (e.g., brown, purple, gray) might not be easily distinguishable for color-blind readers. \n- To my understanding, the SCG front-door criterion essentially adds the constraint that there must be no cycle involving \\(X\\) (or equivalently, \\(Cycle(X, G_s) = \\emptyset\\))—a condition needed to avoid back-door paths from \\(X_t\\) to \\(X_{t'}\\). If this is indeed the case, the contribution appears somewhat incremental rather than truly novel. I recommend that the paper be reorganized to more clearly highlight a concrete example where the standard front-door criterion fails, but the SCG-specified front-door criterion succeeds. A fully worked-through example that penetrates the entire paper would greatly enhance both the clarity and impact of the contribution.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper provides a front-door criterion specifically tailored for Summary Causal Graphs (SCGs). SCGs serve as a useful abstraction when there exists causal dependencies between temporal variables. In SCG, those temporal variables are considred as a single vertex—thereby abstracting away detailed temporal lags. The paper demonstrates that the standard Pearl's front-door criterion, which is applicable to fully specified causal graphs (such as FT-ADMGs), is not directly applicable when working with SCGs. To address this limitation, the authors develop an SCG-specified front-door criterion that includes additional conditions (e.g., no cycles in treatment-temporal variables or no time lag in the treatment).", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "1. **Working Example:**  \n   Add a fully worked-through example that starts with an FT-ADMG and shows how it is projected into an SCG. Use this example to illustrate a scenario where the standard front-door criterion fails—because latent backdoor paths or cycles exist—but the SCG front-door criterion (with its additional no-cycle constraint on \\(X\\)) succeeds. This example should be central to the paper and help readers see the practical impact of the theoretical contribution.\n\n2. **Clarify Definitions and Notation:**  \n   - Revise Definition 4 to clearly explain that \\(S := \\{Y \\mid \\forall Y_t \\in V\\}\\) means that \\(S\\) is a set of clustered variables, where each element \\(Y\\) represents a collection of temporal observations \\(\\{Y_t\\}\\).  \n   - Provide an explicit definition of “macro vertices” in Definition 6 and explain how they differ from the micro (time-indexed) vertices.\n\n3. **Strengthen the Motivation by adding Real-world Examples:**  \n   Include a real-world inspired case study that shows why working with SCGs is useful—especially when the detailed temporal information is hard to specify. Clearly explain the limitations of applying Pearl's standard front-door criterion in dynamic settings and why the added constraint in the SCG front-door criterion is necessary.\n\n4. **Improve Accessibility:**  \n   Update the text by not using colors like purple or brown so that color choices are easier to distinguish. Consider using colors that are friendly to color-blind readers or adding patterns/labels to ensure the figures are accessible.\n\n5. **Address Novelty Concerns:**  \n   Acknowledge that the additional constraint (e.g., requiring \\(Cycle(X, G_s)=\\emptyset\\) or \\(\\gamma=0\\)) might seem incremental. Explain in more detail how this condition is essential for identifying the total effect in SCGs and why it matters in practice. You may highlight the importance or significant through designing examples.", "broader_impact_concerns": "This paper doesn't have any ethical implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "VKNLe7T4PZ", "reviewer": "Reviewer_QeaU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "A fundamental weakness of the work is that a version of the front-door identifiability criterion has already been established in a very similar class of time series summary graphs by Eichler and Didelez (2010). They establish both back-door and front-door identification criteria. (The front-door result is their Theorem 4.12. A version of the result was stated without proof in an earlier conference paper, Eichler and Didelez (2007), which is cited by the authors of this manuscript, but this result is not mentioned and the 2010 paper where all details are spelled out is not cited at all.) I say that this result is \"similar\" but not \"exactly the same\" because Eichler and Didelez use a slightly different graphical set-up; they are also using mixed summary graphs (also allowing for cycles, only using dashed instead bidirected edges to represent unmeasured confounding) though the relationship between the Eichler and Didelez summary graph and the underlying \"fine-grained\" time series graphs is a bit different than the one assumed here. Eichler and Didelez assume no \"contemporaneous\" causal connections within a time-slice. The present authors allow contemporaneous connections, but assume these are acyclic. Besides this, I cannot really see any important difference between the setting here and the setting of Eichler and Didelez.* I could be wrong: maybe there is something that I am missing. But if that is correct then this manuscript does not seem to be adding much to the existing literature. The role of contemporaneous relationships is barely discussed in the manuscript. Perhaps if the authors made some theoretical argument as to what difference contemporaneous (acyclic?) connections make to the (non)-identifiability of causal effects across lags,** then at least the added contribution could be evaluated.\n\n*I also think the Eichler & Didelez treatment is more rigorous, since they take into account some important statistical subtleties about the underlying time series processes that the authors here gloss over or do not address (see below).\n\n**Obviously, if the effect of interest is purely contemporaneous, i.e., of X_t on Y_t, then there is nothing new being added, because the front-door criterion in that case is just exactly the same as Pearl's. It is only lagged effects that are of interest here: X_{t-h} on Y_t for some h.\n\nOther weaknesses:\n\n- It seems that the only real difference between the result proposed here and existing results is the last condition in the modified front-door criterion, which requires Cycles(X, G^s) = \\emptyset. This is not really discussed in any detail or explained, and the consequences of this are not very clear to me. First, it is unclear if the authors include \"self-cycles\" in this definition. Many of the example graphs have X_{t-1}-->X_t edges and thus X-->X cycle in the summary graph. In the caption to Figure 1(g), the authors say their criterion doesn't hold but the only cycle is a self-cycle, so I guess X-->X means Cycles(X, G^s) = X and that is what precludes identifiability here. But then the criterion basically does not apply to any graph where the exposure variable causes itself over time, which is arguably most realistic dynamical systems. It is also well-known in this literature that X-->X and X<-->X are not distinguishable from each other empirically. (If there are such self-cycles, then the criterion only works for contemporaneous not lagged effects, which is just Pearl's criterion as mentioned at ** above. The examples in Figure 3 are all graphs where the effect of interest is X_t on Y_t and so it is just Pearl's front-door.) If I am misunderstanding something about the criterion and self-cycles are allowed, please correct me. But if I am correct in my understanding, then the result as stated is far too restrictive to be useful. And notably the Eichler and Didelez result does not make this restriction, or at least not in the same way.\n\n- The definition of the underlying dynamical model leaves out many important details. Definition 1 does not make clear if the (full) process is of finite Markov order, or if infinite order processes are allowed. Even with finite Markov order in the full process, there may be no finite order process over the observed marginal because of \"infinite lag\" association induced by confounding. Assumption 2 says that the functional relationships to not depend on time, but this is not sufficient to imply stationarity of the joint distribution over all finite collections of random variables. There are additional conditions required such that the induced distributions behave well. For example, see conditions (T1-T3) and (I1-I4) in Eichler and Didelez (2010).  \n\n- Pg. 1: Describing \"experimentation\" as simply \"free from bias\" is not accurate. There is a huge literature on possible sources of bias in randomized experiments. \n- Pg 2: Pearl's front-door criterion does not assume a DAG over the observed variables, it assumes an ADMG (or what he calls a semi-markovian model).\n- Pg 5: the discussion of learning time series graphs from observational data is much too quick and dismissive. What does \"not satisfactory\" mean? What are the \"non-testable strong assumptions\"? I agree that causal discovery from time series is challenging, but it is not obvious that the challenges are due primarily to some \"strong assumptions\" or other reasons. The brief sentences here and footnote do not fairly reflect the large literature on causal discovery from time series.\n- First paragraph on pg 7 is not written very clearly. What does \"there is a backdoor path between Xt−1 and Wt passing by Xt which should not be blocked by Xt\" mean? (\"should not be\"?)\n- I only skimmed the technical Lemmas briefly and they seem mostly sound, but some parts of the proofs are written in a confusing way that makes them hard to follow. For example, in the proof for Property 1: \"Since the path does not pass through another instance of X, the compatible of the path considered in Gs is the following path...\" The proofs could be cleaned up.\n- There are many other points in the text with typos or unclear writing, which would benefit from some careful editing.\n- There is other existing literature on causal identification theory for dynamical systems that should be cited. For example, in addition to the work of Eichler and Didelez, there are papers by (e.g.) Niels R Hansen and Søren W Mogensen et al on causal representations of stochastic processes, there are papers by Joris Mooij, Jonas Peters, et al on identification in cyclic graphs, related papers by Jakob Runge et al on identification using certain other time series representations, etc. \n\nRefs:\nEichler, M., & Didelez, V. (2010). On Granger causality and the effect of interventions in time series. Lifetime data analysis, 16, 3-32.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The topic of the paper is identifiability of causal effects in graphical models that represent relationships among time series variables. Some literature has representing causal relationships among time series in a fine-grained way, with a separate set of variables for each time point, and other literature has instead used a coarse-grained summary graph representation, which abstracts away from some details of the dynamics. This work takes the latter approach, and investigates the conditions under which a causal effect may be identified in the presence of unmeasured confounding, focusing on the front-door strategy. The contribution is entirely theoretical: there is basically one main result which is a set of sufficient (but not necessary or complete) graphical conditions under which a causal effect is identifiable from the summary graph.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "For the paper to be of interest to the TMLR readership, the result would need to be clearly distinguished from existing results and shown convincingly to somehow generalize or improve on existing identification theory. The other issues raised above would also need to be addressed.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "1dD6gYErHB", "reviewer": "Reviewer_k1g5", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strength\n\n**Contribution.** The paper proposes a valuable contribution, in particular\n\n1. An example demonstrating the deficiencies of the standard front-door criterion applied to SCG for identifiability of causal effects on micro (temporal) variables\n2. Sufficient graphical condition on the SCG with latent confoundings for the identifiability of total causal effect on micro variables\n3. A do-free formula for the computation of identifiable causal effects.\n\nThese contributions are surely of interest to the causality community. \n\n**Presentation.** The paper is well-structured and, in general, clearly written. In particular, I praise the clarity of Sections 2 and 3, which introduce all the required background and the limitations of the *standard* front-door criterion naively applied to SCG. The use of graphs as a visual aid for understanding definitions and theoretical results is very helpful.\n\n## Weaknesses\n\n- The lack of intuition on some important definitions or results of the paper makes understanding Section 4 challenging. In particular\n    1. It’s clear by example how the standard front-door criterion on SCG is insufficient for the identifiability of causal effect on micro variables. I don’t understand, at an intuitive level, why removing all cycles — as per condition 4a in Def. 7 — would be the right choice. Could the authors clarify this point?\n    2. Lemma 4.2, 4.3, 4.4. lacks an intuitive explanation of why the selected variables block the intended backdoor paths. For example, about Lemma 4.2, without reading the proof I couldn’t understand why the proposed set of variables is the right choice to block the backdoor paths between $X_{t-\\gamma}$ and $W_{t-\\lambda}$. On the other hand, I think the proof does a good job of providing this intuition: conditioning on the parents of $X$ in the SCG blocks all backdoor paths that start with $X_{t-\\gamma} \\leftarrow$. This still leaves some possible backdoor paths open, those starting with $X_{t-\\gamma} \\leftrightarrow$, which require additional conditioning sets to be blocked. An explanation of this kind helps the reader break the overall problem - understanding why the big conditioning set is the right one - into simple subproblems - i.e. what *type* of backdoor paths each of the individual subsets is contributing to block.\n    \n    A proof sketch in the main text (or a short discussion after the Lemma) might be the right way to provide this intuition. \n    \n    In general, the paper is technically very dense, which is why I believe additional guidance in the reading of Section 4 is beneficial. \n    \n- Could the authors clarify why Theorem 1 is not complete, i.e. which conditions of Definition 7 are not necessary for the identifiability?\n\n## Questions\n\nI use this place to ask for generic clarifying questions\n\n- To make sure I understand all relevant graphical notions for SCG:\n    1. Is $X \\rightarrow X$ a cycle? I’d say no, as there is no direct path, as required by the definition\n    2. Is $X \\rightarrow Y \\rightarrow X$ a cycle? Again, I’d say no, as there is no direct path, as required by the definition\n- Fig 4a: why is this not satisfying Definition 7? (Perhaps, this is connected to my doubts above on cycles.) To allow the reader to check understanding, it would be good to have in the figure caption which conditions of Def. 7 are violated by Fig. 4 example.\n- Def. 5: what does *uniquely* mean in the definition?\n\n## Miscellanea\n\n- Typos: illusrtation, necessartly (page 8); necesserly (page 15), ..\n- Often $\\mathcal G$ in place of $\\mathcal G^s$, e.g. in Lemma 4.3, Property 1\n- In Lemma 4.2: $Cycles (X, \\mathcal G^s)$.\n- After Def. 6, page 6: “macro” (in purple) → “macro” (in blue) - I think.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This is a theoretical paper, that establishes a sufficient graphical condition on summary causal graph (SCG) with latent confoundings for the identifiability of total causal effect on the full-time ADMG (FT-ADMG) - intuitively, formulating a version of the front-door criterion for SCG. For cases when this condition is satisfied, it provides a do-free formula for the computation of the identifiable causal effect.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The requested changes all reflect the points in the weaknesses section in terms of clarity:\n\n1. An intuitive summary on the conditioning variables in Lemmas 4.2-4.4 / a proof sketch in the main text that carries such intuition.\n2. An intuition on why, for $\\gamma \\neq 0$, removing cycles through $X$ is the right choice for defining the front-door criterion of Def. 7.\n3. Explanatory captions on Fig. 4, i.e. how the graphs violate Definition 7.", "broader_impact_concerns": "I don't see a clear broader impact or risk of harm from this paper, especially given its theoretical nature", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers disagreed on the assessment. I suggest that the paper be accepted with minor revisions, on the conditions:
* an additional paragraph is added highlighting the differences with Eichler & Didelez (2010), including at least one example where contemporaneous effects yield different results
* A general improvement in the clarity, especially around the point of one vs multiple observations per time series.
* Addition of a clear practical example showing an SCG derived from an FT-ADMG where the standard front-door fails but the SCG version succeeds (weakness 1 and change 1 from g9tu -- the reviewer stated in the current revision this was not yet implemented)
Decision: Accept (poster)