Double Machine Learning Based Structure Identification from Temporal Data
Authors: Emmanouil Angelis
Francesco Quinzan
Ashkan Soleymani
Patrick Jaillet
Stefan Bauer
Venue: Accepted by TMLR ()

Abstract: Learning the causes of time-series data is a fundamental task in many applications, spanning from finance to earth sciences or bio-medical applications. Common approaches for this task are based on vector auto-regression, and they do not take into account unknown confounding between potential causes. However, in settings with many potential causes and noisy data, these approaches may be substantially biased. Furthermore, potential causes may be correlated in practical applications or even contain cycles. To address these challenges, we propose a new double machine learning based method for structure identification from temporal data (DR-SIT). We provide theoretical guarantees, showing that our method asymptotically recovers the true underlying causal structure. Our analysis extends to cases where the potential causes have cycles, and they may even be confounded. We further perform extensive experiments to showcase the superior performance of our method. Code: https://github.com/sdi1100041/TMLR_submission_DR_SIT

Published in Transactions on Machine Learning Research (09/2025)
Double Machine Learning Based Structure Identification from Temporal Data
Emmanouil Angelis"
Helmholtz AT, Helmholtz Center Munich Technincol University of Munich
Franccsco Quinzan'
Department of Engineering Science
University of Oxford
Ashkan Soleymani
oshkanso@@iil edu
Department of Electricol Engineering and Computer Science Massachusetts Institute Technology
Patrick Jaillet
joillet@mitt edu
Deportment of Electricol Engineering and Computer Science Massachusells Institute of Tecknology
Stefan Bauer
st. Subbuueruuutn de
Helmholts AI, Helmholtz Center Munich Technincal University of Munich
Reviewed OpenReview: https://openrreee net/ formm? id=4GHAoFVNaa
Abstract
Learning the causes of time-series data is a fundamental task in many applications, spanning from finance to carth sciences or bio-medical applications. Common approaches for this lask are based on vector auto-regression, and they do not take into account unknown confounding between potential causes. However, in settings with many potential causes and noisy data, these approaches may be substantially biased. Furthermore, potential causes may be correlated in practical applications or contain cycles. To address these challenges, we propose a new double machine learning based method for structure identification from temporal data (DR-SIT). We provide theoretical guarantees, showing that our method asymptotically recovers the true underlying causal structure. Our analysis extends to cases where the potential causes have cycles, and they may even be confounded. We further perform extensive experiments to showcase the superior performance of our method. Code: htt.::o411/AMAAAAammssiiiis.oo. submission, DR_SIT
1 Introduction
One of the primary objectives when working with time series data is to uncover the causal structures between different variables time. Learning these causal relations and their interactions is of critical
importance in many disciplines, e.g., healthcare (Anwar et al. 2014). climate studies (Stips et al.) 2016
Runge et al.] 2019a), epidemiology (Hernán et al. 2000 Robins et al. 2000), finance (Hiemstra & Jones 1994,, ecosystems (Sugihara et. al.) 2012), and many more. Interventional data is not often accessible in many of these applications. For instance, in healthcare scenarios, conducting trials on patients may raise ethical concerns, or in the realm earth and climate studies, randomized controlled trials are not feasible.
"Equal contribution.

---

Published in Transactions on Machine Learning Research (09/2025)
In general, understanding the underlying causal graph using only observational data is cumbersome task due to many reasons: i) observational data, as opposed to interventional data, capture correlation-lype relations instead of cause-effect ones. ii) unobserved confounders introduce biases and deceive the algorithms to falsely infer causal relations instcad of the true structure, c.g., the existence of a hidden common confounder iii) the number of possible underlying structures grows super exponentially with the number of variables creating major statistical and computation barriers iv) the identifiability problem, since multiple causal models can result in the same observational distribution, thus making it impossible to uniquely determine the true structure. To overcome these problems and determine the underlying structure, additional assumptions are imposed. Typical assumptions include faithfulness, linearity of relations, or even noise-free settings, which limit the types of causal relationships that can be discovered (Pearl 2000; Peters et al. 2017 Spirtes et al.) 2000 Glymour et. al. 2019). Almost all of these challenges extend to the problem of d'identilying the underlying causal structure from observational time-series datasets.
Subsequently, in many instances, the emphasis is placed on particular target variables of interest and their causal features. Causal features of a larget are defined as the set of variables that conditioned on them, the target variable is independent of the rest variables. Causal feature selection enables training models which are much simpler, more interpretable, and more robust (Aliferis ct al. 2010 Janzing et al.) 2020 However, learning the causal structures between variables and a specific target is still a demanding task, and many current approaches for causal feature selection face limitations by making unrealistic simplifying assumptions about the data-gener process or by lacking computational and/or statistical scalability (Yul ct al., 2021 2020). These challenges become particularly amplified in the context of time series data, where the number of variables grows linearly with the length of the data trajectories. As a result, to mitigate these problems, additional assumptions, c.g., stationarity no hidden confounders are included (Moraffah et al.)
2021 Bussmann et al. 2021 Runge 2018) and/or weaker notions of causality such as Granger causality
have been studied extensively [Granger 1988
1969 Marinazzo et al., 2011 Tank et al. 2018 Bussmann
ct al. 2021 Khanna & Tan
2019
Runge
2018
Hasan et al. 2023).
Existing causal feature selection from time-series data a algorithms often assume some level of faithfulness
or causal sufficiency (Runge et al. 2019b] Runge] 2018 2020). Oftentimes, they overlook the presence
of unknown confounding factors among potential causes MMoraffah et al., 2021). Moreover, most cannot
2010 which is relatively ubiquitous in many domains (Bollen 1989). Furthermore, many algorithms employ the popular vector auto-regression framework to model time-
adapt to cyclic settings (Entner & Hoyer
dependence structures (Bussmann et al. 2021 Lu et al.] 2016 Chen et al. 2009: Weichwald et al. 2020 Hyvärinen et al.] 2010). which again is restrictive. To overcome these problems, we propose an efficient algorithm for doubly robust structure identification from temporal data.
Our contribution.
1. We provide an efficient and masy-to-implement doubly robust structure identification from temporal data algorithm (DR-SIT) with theoretical guarantees enjoying vn-consistency.
2. We provide extensive technical discussion relating Granger causality to Pearl's framework for time series and show under which assumptions our approach can be used for feature selection or full causal discovery. As a consequence of this, ours is the first paper to propose a non-parametric Granger causality test that achieves the marammmraectric v7--rate We remark, however, that the same semi- marimearaettric rate was previously established for the related notion of Conditional Local Independence by Christgau et al. (2022).
3.
We provide theoretical insights showing that our amgorithm allows for general non-linear cyclic structures and hidden confounders among the covariates, while only requiring faithfulness within the parental subgraph of the target. In particular, faithfulness outside this local subgraph is not necessary in the context of local causal discovery, where the goal is to identify the direct causes of the target.
4. In extensive experiments we illustrate that our approach is significantly more robust, significantly faster, and more verformative than state-of-the-art baselines.
wweaker than Pearl's structural equation model
Pearl 2000
2

---

Published in Transactions on Machine Learning Research (09/2025)
2
Related Work
Causal Structure Learning for Timeseries A longstanding line of work intends to tailor the existing
causal structure learning and Markov blanket discovery for iidd. data to the temporal setting. To name few,
Entner & Hoyer (2010) adapted the Fast Cansal Inference algorithm (Spirtes et al. 2000) to time-series data. While the approsch shares the benefit of being able to deal with hidden confounders, it is not possible to account for cyclic structures. Runge et al. (2019b) introduced PCMCI, as an adjusted version of PC (Sirtes
et al. 2000 with an additional false positive control phase which is able to recover time-lagged causal dependencics. PCMCI+ modified the approach further to additionally be able to find contemporancous causal LPCMCI extends the scope by catering the case of hidden confounders (Gerhardus
edges [Runge 2020
& Runge 2020 Even though methods in this category are able to provide theoretical guarantees for learning the causal structure, DR-SIT has several advantages over them: i) For all of these methods, the faithfulness assumption is a key ingredient while DR-SIT does not need it. ii) These methods are based on conditional independence testing which is widely recognized to be a cumbersome statistical problem (Bergsma 2004 Kim et al.) 2022). [Shah & Peters (2020) have established that no conditional independence (CI) test can effectively control the Type-I error for all CI settings. Moreover in practice, conducting many conditional independence tests from lengthy time-series is burdensome. iii) Even having access to a perfect conditional independence test oracle, severe computational challenges exist (Chickering 1996 Chickering et al. 2004 (please refer to Appendix | for detailed comparison).
Another line of work relies heavily on the vector auto-regression framework. VarLiNGAM (Hyvärinen et al., 2010) generalizes LiNGAM (Shimizu et al. 2006) to time-scries and similar to that it assumes a lincar non-Gaussian acyclic model for the data. In the work of Huang et al. (2019), a time-varying linear causal model is assumed, allowing for causal discovery even in the presence of hidden confounders. More recently, deep neural networks are used to train vector auto-regression. Tank et al. (2018) adapted neural networks (named cMLP and cLSTM) for Granger causality by imposing group-sparsity regularizers. In similar fashion, Khanna & Tan (2019) used recurrent neural networks. In another work, Bussmann et al. 20021 designed neural additive vector auto-regression (NAVAR), neural network approach to model non- non-linearities. In contrast to previous works, they extract Granger-type causal relations by injecting the necessary sparsity directly into the architecture of the neural networks. This line of work is quite limited to ours as they consider confining structural assumptions over the underlying causal structural equations; details on these assumptions are discussed next to Axiom (A) in Section 3
Double Machine Learning The use of double robustness in causality problems has a long history mainly
concentrated on estimating average treatment effect Robins et al.] 1994 Funk et al.] 2011 Benkeser et al.
2017
Bang & Robins 2005 Sloczyński & Wooldridge 2018). Chernozhukov et al. (0018) introduced the
DML framework to achieve double robustness for structural parameters. Upon that, Soleymani et al. (2022) defined an orthogonalized score to infer the direct causes of partially linear models. Their approach is fast and allows for the assumption of complicated underlying structures but unfortunatcly, it is limitod to only linear direct causal effects. While this was later extended to the non-linear case (Quinzan et al. 2023). we propose a doubly robust approach for identifying causal structures from temporal data under the general assumptions discussed in Section 3
Conditional Local Independence. While our focus is on causal discovery for discrete time series, it is helpful to discuss the related notion of Conditional Local Independence (CLI). CLI is an asymmetric, continuous-time notion of (non-)influence: Y is conditionally locally independent of X (given other histories) when augmenting the filtration with the history of X does not change the predictable intensity (compensator) of Y. Foundational work introduced local-independence graphs and an appropriate separation criterion to represent such statements for point processes and related contimuous-iime models (Didelez) 2007 2008
Subsequent theory developed graphical representations that remain meaningful under marginalizatio lization (latents), via directed mixed graphs and p-separation, thereby characterizing the Markov equivalence classes identifiable from observations (Mogensen & Hansen 2020). Related learning procedures for partially observed stochastic dynamical systems were proposed in Mogensen et al. (2018a). Most relevant to testing, Christgau et al. (2022) provide a model-free (nonparametric) CL. testing framework for counting-process targets: they define
2T Leaarnng Bayesian Networks with conditional independence tests NP-Hard.
3

---

Published in Transactions on Machine Learning Research (09/2025)
Figure 1: Example of full-time graph (left), and the corresponding summary graph (right). Note that time series X and X causally influence the outcome Y with different lags. Note also that our framework allows for auto correlative lags, as well lagged causal effects from the target to any of the poential causes. The time series U is unobserved, and it acts confounder for X and X². Moreover, there is cycle between the confoundings variables X and X² of the outcome variable Y.
the Local Covariance Measure (LCM), estimate it with cross-fitted orthogonalized moments, and derive an (X)-Local Covariance Test ((X)LCT) with uniform control of size and power under modest estimation rates. It is important to distinguish, however, that CLI concerns instantaneous (intensity-level) influence in continuous time. By contrast, the Granger-causal target in this paper is a discrete-time predictive notion: whether past values of X improve prediction of beyond the past of and other covariates. The two notions answer related questions, but they are not interchangeable.
3 Framework
3.1 Problem Description
We are given given i.idd. realizations of joint time scries (Y, X) observed over time, where Y = (Yihez is
univariate larget lime series and X = [X),..., X hez is a multivariate lime series of potential causes. We assume that the target time series is specified by some of the potential causes by a deterministic function with posterior additive noise, and no instantaneous effects. We can formalize this model as
Axiom (A) Yr = (P) + ET for all time steps T E Z,
with Bx exogenous independent noise and pay (Y) S (Xt,.... Xy" Jıez is a subset of random variables of the multivariate time series X. Note that the independence of the additive noise is important for identifiability. In fact, if there are dependencies between the noise and the history, then one might run into identifiability problems. We refer the reader to Appendix B for countercxample. We are interested in identifying the time series X that directly affects the outcome Y. That is, we wish to identify time series X such that it holds X C (Y) for some time steps T.
We further use the following assumptions:
Axiom (B) there no causal effects backward in time. Specifically, X pY)(Y), for all 1,..., and for all time steps T Z with t > T;
Axiom (C) there are no instantaneous causal effect between Y and any of the potential causes X i.e., X3 & par(Y). for all i= I...., and for all time steps T E Z.
Note that according to Axiom (C) instantancous effects are allowed between potential causes X' and X', illustrated, for example, in Fig. Both Axiom (B) and Axiom (C) appear in previous related work (see, e.g.. Peters et al. (2013); Mastakouri et al. (2021) Löwe et al. (0022) ). Note that these axioms allow for cycles and hidden common confounders between the potential causes. Axiom (B) is a natural assumption
system called causal when the output of the system only depends on the past, not the future (Peters)
et al. 2017
Pearl 2000
Axiom (C) poses additional restrictions the class of models that we consider,
4

---

Published in Transactions on Machine Learning Research (09/2025)
since instantaneous effects may be relevant in some cases and applications (Lippe et al. 2022a
However,
without this assumption, it is impossible to learn causes from observalional dala. The necessity of Axiom (C)
for causal discovery is a well-known fact (Peters et al. 2013
3.2 Generality over Previous Work
Our framework retains some degree of generality over previous related work. In fact, Axiom (A) Axiom (C)
allow for hidden confounding and cycles between the potential causes (see Figure 1 providing more general framework than the full autoregressive model studied, e.g., by Hyvärinen et al.
2010
Peters et al. (2013)
Löwe ct al. (022)] Wu ct al. (2020). Furthermore, in contrast to several previous works
[Kemakhen et al,,
we do not assume that the variables Yr, "T........ XT are independent conditioned on the observed variables at previous time steps. Importantly, we also do not assume causal faithfulness in the full time graph, or weaker notions such as causal minimality[ This is a major improvement over some previous works, e.g., Mastakouri et al. (2021); Gong al. (2022), since there is no reason to assume that faithfulness or causal minimality hold in pract ice. Theorem 1 by Gong et al. (2022) provides an identifiability result for a model with history-dependent noise and instantaneous effects. This result, however, requires causal minimality.
2020 Gresele el al. 2021 Lachapelle el al. 2022 Lippe el al. 2022b Yao el al.
2021
Furthermore, as discussed in Section 2 vector auto-regression methods enforce heavy structural assumptions on the underlying causal structural equations. NAVAR (Bussmann et al. 2021 assumes that each variable is influenced by its causal parents exclusively in an additive way and higher-ordes interactions among them are precluded. In mathematical terms, they follow Yr- 3 + Ex ExEes, fx (X))-11-11) + Er, where / is a bias term, & is the time lag, and E, is an independent noise variable. VarLiNGAM (Hyvärinen et al. 2010 imposes considerably stricter constraints on the structural equations' functional form than our framework, restricting fx be a linear transformation of X As a result, it cannot capture even simple noulinear relationships e.g., Y, X x x2_, or Y, = log(X,_1 + x2- which are well within the scope of Axiom A] In contrast, our approach flexibly models arbitrary interactions between covariates and the timestep T, marking a significant advancement over prior methods.
3.3 Causal Structure
We are interested in direct causal effects, which are defined by distribution changes due to interventions on the DGP. An intervention amounts to actively manipulating the generative process of polential cause X at some time step t, without altering the remaining components of the DGP. Then, time series X' has direct effect on Y if performing an intervention on some temporal variable X will alter the distribution of Yr, for some time steps f, T.
We consider interventions by which a random variable X/ is set to a constant X/ + x. We denote with Yr | do(X; = z) the outcome time series at time step T, after performing an intervention as described above. We can likewise perform multiple joint interventions, by setting a group of random variables I different time steps, to pre-determined constants specified by an array i. We the symbol Yr do( i) to denote the resulting post-interventional outcome, and we denote with P (Yr y do i)) the probability of the event {Yr do(I = i) = -
Using this notation, a time series X has a direct cansal effect on the outcome Y, if performing different interventions on the variables X', while keeping the remaining variables fixed, will alter the probability distribution of the outcome Y. Formally, define the sets of random variables I I = (X...... X?, Y.J.eT,
An example of cases where time series exhibit instantaneous causal effects is given by dynamical systems (Mogensn et al. 2018b Rubenstein et al. 2016). In dynamical systems, a variable may instantaneously affect another variable of the model. Instantancous effects have boen studied in previous related work (Gong et al. 2022 but due to identifiability issues, they rely on stronger assumptions such as faithfulness.
'In general, Peters et al. (2013) show that causal discovery is impossible with instantaneous effects. Please refer to Appendix C for example of this iontiiiinti.ailityy However, Peters al. (2013) also provide special case in which the causal structure is identifiable with instantancous effects. This special occurs when the random variables of the model are jointly Gaussian, and the instantancous effects are linear. Our framework extends to this special case.
5 Recall that distribution faithful to causal diagram if no conditional independence relations are present, other than the ones entailed by the Markov property.
5

---

Published in Transactions on Machine Learning Research (09/2025)
which consists of all the information before time step T. Similarly, define the random variable X <T {Xiher,
consisting of all the information of time series X before time step T. Define the variable I 127 == I X which consists of all the variables in I<T except for X-eT. Then, a time series X' has direct effect on the outcome Y if it holds
P YY = y do x I = =P)) y do Xxer x", x=,12 1 =1)
(1)
We say that a time series X' causes Y, if there is a direct effect between X' and Y as in Eq. for any time step T.
Following, e.g,, Mastakouri et al. (2021). we define the full time graph 9 as a directed graph whose edges represent all direct causal effects among the variables at all time steps. Given the outcome Yr at a given time step, we refer to the parent nodes in the full time graph as its causal parents. We further define the summary graph whose nodes are X' and Y. and with directed edges representing causal effects between the time series. We refer the reader to Figure [ for a visualization of these graphs. Note that the causes of Y correspond to the parent nodes of Y in the summary graph. In this work, we always assume that the Markov property holds (sec, e.g., Peters et al. (017
3.4 Granger Causality
Granger causality (Granger 1988 is one of the most commonly used approaches to infer causal relations from observational time-series data. Its central assumption is that "cause-effect relationships cannot work against time". Informally, if the prediction of the future of a target time-series Y can be improved by knowing past Y. Formally, we say that X Granger causes
elements of another time-series X', then X "Granger causes"
Y if it holds
Yr = y X
TT = x, I' = + Yr Y (
(2)
for a non-zero probability event (X' = x, v <T = i}, where I er stands for the set (x) er x² 22r..................... and I'V represents the set IeT {x'r).
m
Granger causality is commonly used to identify causes. Assuming stationary, multivariate Granger causality analysis usually fits vector autoregressive model to given dataset. This model can be then used to determine the causes of target Y. However, it is important to note that Granger causality does not imply true causality in general. This limitation was acknowledged by Granger himself in Granger (1988).
4 Methodology
4.1 Double Machine Learning (DML)
DML is a general framework for parameter estimation, which uses debiasing techniques to achieve vn- consistency (see, e.g., [Rotnitzky et al. [2020) Chernozhukov et al. (2022)). ). In DML, consider the problem of estimating a parameter 6g as a solution of an equation of the form E [[(o, mo)l = 0. The score function L depends on two terms, the true parameter A that we wish to estimate, and a nuisance parameter no- We do not directly care about the correctness of our estimate of 7hs as long as we get a good estimator of 0g. The muisance parameter To may induce an unwanted bias in the estimation process, resulting in slow convergence. To overcome this problem, we use score functions that fulfill the Mixed Bias Property (MBP) [Rotnitzky et al. 2020); and learn the desired parameter On using DML.
The MBP is a properly that ensures that small changes of the nuisance parameter do not significantly affect the score function computed around the true parameters (6o. m7) (see Definition 1 by Rotnitzky et al.
Recall that the distribution of the DGP fulfills the Markov property if each variable in the graph G is conditionally independent of its non-non-dessendduts, given its causal parents.
This formulation assumes discrete target variable Yr for notational simplicity. The definition extends naturally to continuous random variables by comparing conditional distributions using conditional densities where they exist. We refer the reader, e-g., to (Shojaie & Fox (002) for comprehensive discussion.

---

Published in Transactions on Machine Learning Research (09/2025)
In this work, we construct score with the MBP, following Chernozhukov et al. (2022) Rotnitzky et al. (020). For fixed time step T, let. X be random variable in the set of random variables V. 9 any real-va vall-vllud function of X such that E [x)] < co. We consider parameters of the form 0g = E [m (g where g)) is a lincar moment functional in g. The celebrated Riesz Representation Theorem ensures that, under certain conditions, there exists a function og of X such that [m[V; 9)] = E [oo ( (go)))g(X) ()). The function ogg called the Riesz Representer (RR). Chernozhukov et al. (2021) shows that the Riesz representer can be estimated from samples. Using the RR, we can derive score function for the parameter Oo with gu X) E
22200
[Y|]]] X] that fulfills the MBP. This function is defined as
( n) = (VV g) + +((X) (Y -Y- g(()) -.
(3)
Here, 77 = (a. g) a nuisance parameter consisting of a pair of square-integrable functions. As shown by Cher
nozhukov et al. (2022). , the score function Eq. (3) yields E (p((0, n) = - [(a(X) (o - -o
which gives the MBP in Definition 1 of Rotnitzky et al. (2020)
For score functions that fulfill the MBP, we can use DML to partly remove the bias induced by the nuisance parameter. The DML is defined as follows D
Definition 4.1 (DML, following Definition 3.2 by observations, split the dataset D into & random disjoint subsets Dj of the same size. Consider a score function (00,,) n)) that fulfills the MBP as in (3). Construct estimators f (åj, 0j) for the nuisance parameter of the score using datascts D Dy. Then, construct an estimator â of the paraemeter @ as the solution to the following equation:
Chernozhukov et al. 2018)). Given dalaset D of n
Boy (0) =0,
where Bo,1 ] is the empirical expected value over Dj-
4.2 Granger Causality Implies True Causation under Axiom Axiom (C)
As discussed Section 3.4 Granger causality does not imply true causality general. In our case, however, Granger cansality corresponds to true causation, as stated in the following result.
Theorem 4.2. Consider a causal model as in Aziom (A) Ariom (C), Then, it holds Xj E pay (Y) for some T,T E & is and only if (iff-) (2) holds. That is, time series X' has a direct causal effect on Y iff. x'
Granger causes Y.
Proof of this result given in Appendix D Importantly, Theorem 4.2 does not require causal faithfulness. We remark that Peters et al. 20133) shows that Granger causality implies true causation for fully autoregressive models (see also Löwe et al. [2022) ). This result is based on the identifiability of additive noise models, in observed (Peters et al. 2011). Our framework, however, is more general than Peters et al. (2013): : Löwe et al. (0022). since it allows for confounding among the covariates (sce Figure
which all relevant variables
In the special case of a fully autoregressive model, Theorem 4.2 is equivalent to previous results (Peters el al.
2013
Löwe et al. 2022
4.3
Testing Granger Causality with DML
Our approach to identifying potential causes consists of performing a statistical test to determine if Eq. 2 holds. Due to Theorem 4.2 a straightforward approach would then consist of using a conditional independence test, to select or discard a time series X' as a cause the outcome Y. However, conditional independence testing is challenging in high-dimensional settings. Furthermore, kernel-based conditional independence tests (Fukumizu al. 2007 Zhang et al. 2011 Park & Muandet 2020 Shong & Sriperumbudur 20200 are computationally expensive. Instead, we provide a new statistical test based on DML. Our approach is
WW remark that DML requires weaker assumption on the function than the MBP, namely the Neyman Orthogonality
Chernozhukov et 2018). For simplicity, we give description of DML only in terms the MRP for linear moment funelionas.. However, Definition be generalized.
Condition (Nymann & Scott
1965
7

---

Published in Transactions on Machine Learning Research (09/2025)
Algorithm The DR-SIT
1: split the datasct D into k random disjoint subsets Dj of the same size;
for each dataset partition j do estimate f) = (a)50)) on dataset D \ Dj, with Ø9 an estimate for g and ag an estimate of the RR og
of mo g as in Theorem 4.3
end for 0 - Eー ED, [mo( (Mno(V:))) V:0g) + ()) (Y (](x))]
for each potential cause X' do
for each dataset partition do estimate nj (a).0) 03) on dataset D Dj, with Øf estimate for gò, and 6j estimate of the RR ag of m: (Vg'") as in Theorem 4.3 end for
0' - Σド= ED, (v:(V:]) + (X) (Y -V,((()))
10:
11:
perform paired Student's f-test to determine if 6 ~ 60, and select time scries X as cause if the null-hypotheses is rejected;
12: end for
13:
return the selected time series;
based on the observation that under Axiom Axiom (C) the condition in Eq. can be written in terms of simple linear moment functionals. The following theorem holds.
Theorem 4.3. Consider the notation in Eq. g and fiz time step T. For any squore-integrable random variable gd (X'sr, I consider the moment functional mo(V:g") = YT g.. Similarly, for any squore- integrable random variable (T(T)T), consider the moment functional mc(Vig') == YT g². Assuming Aniom
AA) Aziom (C) X Granger causes iff. it holds
E [mo V: :999] - E [m (V;:VV: ([Vb)))) :b)] # 0,
(4)
with (0 B[YT| X <T x, 1' i], and (b) = E| 1h = i].
The proof deferred Appendix Intuitively, the parameter = EYrr]] quantifics the cross- correlation between the target. variable Yr and its conditional mean B[Yr| given the set of variables T. Tn general terms, variations in the parameter 0 due to changes in the set 1 suggest the presence of causal relationship between alterations in the set / and the target variable Yr.
By this theorem, we can identify the causal parents of y by testing Eq. 4 This boils down to learning parameters 88 = E (Vo(V; [V f = E (V:6))) and then performing a sample test to verify holds 0 03 * O. Since both mo V; (0) and m;(V; (V:6)) are linear moment functionals, we can use DML as Definition 4.1 to estimate 88 and 8% Under mild convergence conditions on the muisance parameters for mo V: ) and (V): DML ensures fast convergence in distribution to the true parameters.
5
The Algorithm
5.1
Overview
Our algorithm learns causal relationships between time series, by testing Granger causality using DML, outlined in Section 4.3 We refer to our approach as the DR-SIT (Structure Identification from Temporal Data). Our method is presented in Algorithm 1 This algorithm essentially performs the following steps: (1) Select potential cause X to test if X Granger causes Y. Split the dataset D into disjoint sets D
with k > 2.
(2) Estimate 7 = (49.0) 09) on dataset D D,, with 0f an estimate for g8. and 69 an estimate of the RR og of mo(V: (V:g)) in Theorem 4.3. Similarly, provide estimate n = ( dataset D Dj for the
8

---

sequencc-proccesinng layers that the input values for features X should be skipped. We then compute of using the resulting surrogate models z and āt. Please refer to Appeudix F on an intuition behind pero-masking, and why pero-masking may not hurt the estimations. Using zero-man masking dramatically improves run time, Published in Transactions on Machine Learning Research (09/2025)
pair no -90. ag); with dà in Theorem
4.3 and ob the RR aj of mV:V''"). This step corresponds to Line 3 and Line 8 of Algorithm [
(3) Provide an estimate 0 x Elmo Vigg);; by solving the equation k. - Σド
Eo, [00 f,)] = 0 with a (V: 00)) + (X) (Y x((X) - 8. This step corresponds to Line 5 of score of the form (p)( mo (V: mo(V
Algorithm Provide an estimate 0' = E/m, (V:g") in a similar fashion, as in Line 10 of Algorithm
(4) Use a paired Student's 1-test. to determine E [00 - 0*] is approximately zero. Select X" as & cause of
Y the mull hypothesis is rejected. This step corresponds to Line 11 and 8 of Algorithm
5.2 Strong Consistency Guarantees
In this paragraph, we provide an explanation for Step (4) of our algorithm. Under mild structural conditions on of and af. a² Chernozhukov et. al. (2022 2018 /n-consistency. Hence, it holds
Rotnitzky el al. (2020), the quantity 0 - 0¹ has
vn ( 0¹)
4 N (0,02)
(5)
if and only if E [mo V; 99)] - E |m, V: (V: = 0. Then, by Theorem 4.3 Eq. 5 holds iff. X' does not Granger causes Y. The notation in (5) means that the difference 0 0' converges in distribution to a zero-mean Gaussian for an increasing number of samples. That is, for all e 0 and Ç > there exists 8 > 0, such that given n 8 samples it holds TP (100 64 > S - +
S. if and only if E (V E (V: st)l 0. Here, & is the CDF of the standard Normal distribution. By this inequality, we can use a paired Student's t-test to determine if X' Granger causes Y, as in Line 11 of Algorithm !
We refer the reader Chernozhukor et al. (2022) 2018); Rotnitzky al. (020 for a detailed survey strong consistency for DML and its relationship with the MBP.
5.3 Complexity of Algorithm 1
Much of the run time of our algorithm consists of performing regression to learn n7 and n/ on dataset Dj. Denote with the time complexity of performing such a regression. For a given k-partition of the dataset and fix potential cause X can upper-bound the time complexity of our algorithm since a regression to learn n) is performed for each potential cause, i.e., times, the complexity of the algorithm is (dkmm). Here, d depends on the specific techniques used for the regression. Non-parametrie regression can be performed efficiently in the problem size. We analyse the computational complexity in Appendix
O(dk). Furthermore,
D further and show runtime plots in 6.4 We further discuss the run time and extension for full causal discovery in Appendix [..
We further improve efficiency in practice as follows: Instead of computing mi = (g6, (g))aa) directly, we apply a zero-masking layer to the NNs used to estimate of and ag for the features X This masking layer tells the
since it allows to perform only two regressions through the entire run time of Algorithm !
5.4 Practical implementation at a glance
When deploying DR-SIT, we follow the workflow below.
Single model for both nuisance parameters. In our setting the Riesz representer coincides with the regression muisance, because E[m(V: V: = Ex E|| X| = a(X) = E[y X (X)). Hence we train one regressor and reuse it for both quantities; training two identical copies yields indistinguishable results.
Choice and tuning of the regressor. DR-SIT agnostic the model class. A small validation split is reserved select the family (kernel, MLP, ...) and its hyper-parameters. For the synthetic datasets (Sec.
6.1 abundant. samples justify fixed two-layer MLP throughout. For DREAM3 (Sec. 6.2). tune
9

---

Published in Transactions on Machine Learning Research (09/2025)
Table 1: AUROC score for the DR-SIT and common algorithms for Granger causality. Models with * sign use deep neural networks. When reported, the error bounds represent standard deviation of the AUROC score over experiment repetitions
Method
E.Coli
E.Coli 2
Yeast
Yeast
Yeast 3
cMIP*
0.644
0.585
eLSTM
0.355
TCDF*
0.614
0.647
0.581
0.556
0.357
SRU*
0.657
0.666
0.617
0.575
0.55
6SRU*
0.66
0.629
0.627
0.557
0.55
DYNO.
0.590
0.547
0.527
0.526
0.510
PCMCI*
0,530
0.519
0.530
0.510
0.512
Rhino" Reprod.
0.671 1:0.014
0.640
#0.022
0.656
+0.011
0.565
10.012
0.549
#0.004
Rhino+g" Reprod.
0.665 +00 023
0.646 5.2
0.649 =0.011
0.582 #0.011
0.571 #0.010
DR-SIT (aurs)
0.704:10.005
0.680.10.004
0.653:10.001
0.585110.003
0.544110.003
the first sub-task (E. Coli 1) and adopt KernelRidge with third-degree polynomial kernel (alpha 1, coef0 1) for all remaining tasks.
Lag selection. Lag is treated as byper-parameter. For the synthetic datasets 66.1). ) we use the ground- truth lag employed in data generation. For DREAM3 (.2 we fix lag = 2 for every method, following prior work. However note that stationarity is not required by our theory; if it fails, the Granger-causality condition must simply be checked at all time points rather than single T. Cross-fitting scheme. We employ 5-fold cross-fitting, splitting trajectories uniformly at random into inference.
equal-sized folds an essential step for valid haabble-acchine-eer
6
Experiments
In this section we provide an overview of the main experimental results. We provide additional extensive experiments in Appendix J
6.1 Synthetic Experiments
Dataset generation. We first set the number of potential causes m, and we fix the lag A and the number of time steps for the dataset T. We create a covariate adjacency 3-dimensional tensor Σ of dimensions A x m x m. This tensor has 0-1 coofficients, where Σk.i,j = X; has a casual effect on X for all time steps t. Similarly, we create an adjacency matrix Σ for the outcome Y. y is binary A x m array, such that 2 = if x/_, has a causal effect on Y, for all time steps t. The entries of Σ and y follow the Bernouli distribution with parameter p = 0.5. Note that the resulting casual structure fulfills Axiom
[B) Axiom (C)
We then create m transformations that are used produce the potential causes X ,..., X" Each one of these transformations is modeled by an MLP with 1 hidden layer and 200 hidden units. We use TANh nonlinearities (included also in the output layer) in order to control the scale of the values. The final output value is further scaled so that all transforms generate values in the range [-10, 10j. The input layer of each MLP is coming from the corresponding causal parents of the corresponding time series, as calculated from Σ.
In order to generate the potential causes X we use Σ and the MT.P transforms. The first value of each time scries is randomly generated from uniform distribution in [-10, 10). Then, each X' is produced by applying the appropriate transform to its causal parents, as determined by Σ, and a zero-mean unit variance Gaussian noise is added. We generate the target time series in similar fashion. Each variable Y, is produced by applying the MIP transform to its causal parents, determined by the target adjacency matrix ΣY. We then add zero-mean Gaussian noise. The scale of the posterior additive noise for the outcome is referred to as the noise-lo-signal ralio (NTS).
Description of the experiments. We are given a datasct as described above with m potential causes and fixed NTS for the generation of the outcome Y. We determine which series X! X!.... Y'..... X" are the causal parents of the outcome, using Algorithm For given choice of m and NTS, we repeat the runs five times.
10

---

Published in Transactions on Machine Learning Research (09/2023)
AURDD NJ Jeat Noice level = 0.0
AURDC N, feut Nolce level = 0.1
AUROCC N, (_feot Noise level - 0.25
90
0.90
02:S1T
D1ST
20
3.90
B. foat
N fost
Figure 2: AUROC metric for DR-SIT, RHINO and PCMCI for various noise levels on the synthetic dataset.
0.70
0.66
0.60
0.60
0.65
RHINO RHINO
0.65
0.55
0.60
RHINO
0.60
0.64
0.62
0.58
RHINO
0.58
0.54
0.56
RHINO
0.55
0.50
nHINO+g
0.56
ReiNO+g
0.52
RHINO
DR-SIT
0.54
0.50
0.45
0.50
10
20
30
40
10
20
30
40
10
20
30
40
10
20
30
40
(a) Task: Coli 1
(b) Task: Ecoli 2
(c) Task: Yeast 1
(d) Task: Yeast
Figure 3: This figure demonstrates the consistent performance of DR-SIT w.r.t number of training obscrvations (i.e trajectories) compared to state-of-the-art methods Rhino and Rhino+g. For the same plots in all 5 tasks depicted in greater resolution, see Fig. in Appendix J.2
This experiment repeated for an increasing number of potential causes, and increasing noisc-to-signal ratio, to evaluate the performance of Algorithm on challenging datasets.
In this set of experiments, model. We found that this simple approach, combined with zero-masking, dramatically reduces the false positives of the Student's I-test in Line T1 of Algorithm [
learn vi} in Line 3 of Algorithm [ using for the regression task an MLP
Results In Figure 2) (also shown greater detail in Appendix Figure 4 we plot the AUROC performance of DR-SIT against RHINO and PCMCI on the synthetic dataset, confirming the competitive performance of DR-SIT. In order to calculate the AUROC for DR-STT, we sort our predictions (for existence of edge) on the standard deviation of Z Yr 0) + 6 (Yr 9)) Yr og. -a.- -Yr ,g), which simply the difference of the doubly robust statistics for 6¹ and 8 for a datapoint in partition Dj. Moreover, Appendix J.1 we show the performance of our method with respect the accuracy, F1 and CSI scores (see Tablse We see that the DR-SIT is stable for increasingly higher posterior noise and scales reasonably w.r.t the number of observations.
6.2
Semi-Synthetic Experiments
The Dream3 dataset. Following previous related work (Tank et al. 2018 Khanna & Tan 2019 Nauta et al, Bussmann et al.) 2021 Gong et al. 2022 we evaluate performance with the Dream3 benchmark (Prill et al. 2010 Marbach et al. (209)) The Dream3 benchmark is collection of gene expression level measurements across five different networks, where cach network comprises 100 genes. The data Dream3 are organized as time series. Specifically, for each of the five networks 46 trajectories are given; each trajectory details the progression of the 100 genes over 21 time steps after an initial perturbation in the gene values.
2019
Description of the experiments. Our goal is to infer the causal structure of cach network. This gives us a total of five tasks, i.e., E.Coli 1, E.Coli 2, Yeast 1, Yeast 2, and Yeast. 3. We run our algorithm on this dataset. and we use the area under the ROC curve (AUROC) as the performance metric. Following (Gong et al.) 2022), we compare against the following basclines: cMLP (Tank et al. 2018), cLSTM (Tank al. 2018). TCDF (Nauta al. 2019). SRU [Khanna & Tan 2019), eSRU (Khanna Tan 2019). Dynotears (Pamfil
et al. 2020 Rhino (Gong al. 2022), and Rhino [Gong et. al. 2022).
11

---

Published in Transactions on Machine Learning Research (09/2025)
Table 2: Run time and Hardware used for our method (DR-SIT) and the state-of-the-art baseline Rhino.
Category Method
Rtino
DR-SIT (ours)
Runtime
18 mins = 30 sec
37.12 sec 1.6 sec
Hardware
NVIDIA A100 GPU AMD EPYC 7402 24-Core CPU
11th Gen Core i5- 15-1140F CPU
Results. Shown in Table 1 The results for cMLP, cLSTM, TCDF, SRU and SRU are taken directly from Khanna & Tan (2019); Gong et al. (2022), where error bounds are not reported. Regarding Rhino+g and Rhino, we partially reproduce the experiments by GGong et al. (2022), using their source codc. Our implementation of Rhino+g and Rhino differs from Gong et al. (2022) only in the choice for the hyper- parameters, which is the same on all five tasks. We specifically use the following pper-paraaeters for Rhino: Node Embedd. = 16, Instantancous eff. Falso, Node Embedd. (flow) 16, lag 2, X = 19, Auglag 30. And use the following for Rhinotg: Node Embedd. = 16, Instantancous eff. Falsc, lag 2, X, = 15, Auglag- = 60. This is the setting that is reported for the Ecolil subtask and found in their corresponding code implementation. We opted for this approach because in the experiment by Gong et al. (2022), it seemms that Rhino overfitted to the dataset. We learn 7/J as in Line of Algorithm 1 using a simple kernel ridge regression model with polynomial kernels of degree three combined with zero masking.
We observe that DR-SIT outperforms all the other benchmarks on three tasks (E.Coli 1, E.Coli 2, Yeast 2). Furthermore, DR-SIT obtains comparable performance on the remaining lasks (Yeast, Yeast 3). We also would like to emphasize that our estimator is much simpler than deep nets such as Rhino Rhino+g. As such, it has lower sample complexity and lower run time than the other algorithms.
6.3 Low sample regime
One appealing property of DR-SIT is the strong consistency of the estimators (see Section 5 Due to this property, the estimators @" and 6 exhibit fast convergence to the true parameters. Hence, algorithm enjoys competitive results in low sample complexity settings, than richer models such Rhino (Gong et al. 2022). We illustrate a compelling example of this, by comparing Rhino and Rhino+g against DR-SIT. In this example, DR-SIT learns nuisance parameters (Line 3 and Line 8 of Algorithm [ using a simple kernel ridge regression estimator with polynomial kernels of degree three. In Fig. 3 we explicitly compare the performance of DR-SIT, Rhino, and Rhinotg as the number of trajectories used in training varies and demonstrate significantly improved performance especially in low sample settings.
6.4 Run Time and Hardware
We report on the average runtime of DR-SIT and the competitive rival Rhino for experiment Table I across all five tasks ColColi 1, F. Coloii 2, Yeast 1, Yeast 2 and Yeast 3) in Table 2 Despite having access only to single CPU in contrast to the GPU-equipped execution of Rhino our method is almost 20x faster. This is because of the fact that DR-SIT algorithm will provide reasonable results even when employing simple fast efficient estimators (in this case a kernel regression). For visual presentation of the AUROC performance progression vs runtime for various combinations of tasks and training sample sizes, see Figs. to 10 in appendix.
7 Discussion
In this work, propose efficient algorithm for doubly robust structure identification from temporal data. We further provide asymptotical guarantees that our method is able to discover the direct causes even when there are cycles or hidden confounding and that our algorithm has /ūl-consistenyy We extensively discuss the relations of the approach between the popular frameworks of Granger and Pearl's causality as well as relate and extend approaches from debiased machine learning to structure discovery from temporal data. We hope that our approach enables important real-world applications bio-medicine where robustness confounding, sample efficiency, and case of use are important for causal discovery from observational time series.
12

---

Published in Transactions on Machine Learning Research (09/2025)
References
Constantin F Aliferis, Alexander Statnikov, Ioannis Tsamardinos, Subramani Mani, and Xenofon D Koutsoukos. Local causal and markov blanket induction for causal discovery and feature selection for classification part i: algorithms and empirical evaluation. Journal of Machine Learning Research, 11(1), 2010.
Abdul Rauf Anwar, Kidist Gebremariam Mideska, Helge Hellriegel, Nienke Hoogenboom, Holger Krause, Alfons Schnitzler, Günther Deuschl, Jan Raethjen, Ulrich Heute, and Muthuraman Muthuraman. Multi- modal causality analysis of eyes-open and eycs-closed data from simultancously recorded eog and meg. In 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Sociely, pp- 2825 2828. IEFE, 2014.
Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, 61(4))962 -73,, 2005.
David Benkeser, Marco Carone, MJ Van Der Laan, and Peter Gilbert. Doubly robust nonparametric inference on the average treatment effect. Biometrika, 104(4):863 880, 2017.
Wicher Pieter Bergsma. Testing conditional independence for continuous random variables. Report Eurandom, 2004.
Kenneth A Bollen. Structurol equations with latent variables, volume 210. Wiley, 1989.
Bart Bussmann, Jannes Nys, and Steven Latré. Neural additive vector autoregression models for causal discovery in time series. Discovery Science: 24th International Conference, DS 2021, Halifar, NS, Canada, October 11-13, 2021, Proceedings 24, pp. 446-460 Springer, 2021.
Gang Chen, J Paul Hamilton, Moriah E Thomason, Ian H Gotlib, Ziad S Saad, and Robert W Cox. Granger causality via vector auto-regression tuned for fmri data analysis. In Proc Intl Soe Mag Reson Med, volume 17, pp. 1718, 2009.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiase machine learning for treatment and structural parameters. The Ecomomelrics Journal, 21(1):C1-CO) -688, 01 2018.
Victor Chernozhukov, Whitney K. Newey, Victor debiased machine learning via neural nets for generalized linear regression. arXic:2104.14789 2021.
and Vasilis Syrgkanis. Automatic
Victor Chernozhukov, Whitney Newey, Victor Quintas-Martines and Vasilis Syrgkanis. RieszNet and ForestRicsz: Automatic debiased machine learning with neural nets and random forests. In Proc. of ICML, pp. 3901 3914, 2022.
David Maxwell Chickering. Learning bayesian networks is np-complete. Learning from data: Artificial intelligence and statistics V, pp. 121-130, 1996.
Max Chickering, David Heckerman, and Chris Meek. Large-sample learning of bayesian networks is np-hard. Journal of Machine Learning Research, 5:1287-1330, 2004.
Alexander M. Christgau, Lasse Petersen, and Niels Richard Hansen. Nonparametric conditional local independence testing. arXiv preprint urXiv:2203.13559 2022. URL https://arriv.org/aa/////2 12559
Vanessa Didelez. Graphical models for composable finite markov processes. Scandinavian Journal of Statistics, 34(1):169-185, 2007. doi: 10.1111/..467-94699222 20t..200000022...
Vanessa. Didelez. Graphical models for marked point processes based local independence. Journal of the Royal Statistical Sociely; Series B (Statistical Methodology), 70(1):245 264, 2008. doi: 10.1111//.466-9666. 2007.00634.x
Doris Entner and Patrik O Hoyer. On causal discovery from time series data using fci. Probabilistic graphical models, pp. 121-128, 2010.
13

---

Published in Transactions on Machine Learning Research (09/2025)
Kenji Fukumizu, Arthur Gretton, Xiaohai Sun, and Bernhard Schölkopf. Kernel measures of conditional dependence. In Advances in Neural Information Processing Systems, pp. 489-496, 2007.
Michele Jonsson Funk, Daniel Westreich, Chris Wiesen, Til Stürmer, M Alan Brookhart, and Marie Davidian. Doubly robust estimation causal effects. American journal of epidemiology, 173(7):761 767, 2011.
Andreas Gerhardus and Jakob Runge. High-recall causal discovery for autocorrelated time series with latent confounders. Advances in Neural Information Processing Systems, 33:12615-12622 2020.
Clark Glymour, Kun Zhang, and Peter Spirtes. Review of cansal discovery methods based on graphical models. Fronliers in genelics, 10:524, 2019.
Wenbo Gong, Joel Jennings, Cheng Zhang, and Nick Pawlowski. Rhino: Deep causal temporal relationship learning with history-dependent noise. CoRR, abs alss2220.14406, 2022.
Clive WJ Granger. Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society, pp. 424-438, 1969.
Clive WJ Granger. Some recent development in a concept of causality. Journal of econometrics, 39(1-)): 199-211, 211, 1988.
Luigi Gresele, Julius Von Küggelgen, Vincent. Stimper, Bernhard Schölkopf, and Michel Besserve. Independent. mechanism analysis, new concept? In Proc. of NewrIPS, volume 34, pp. 28233 28248, 2021.
Uzma Hasan, Emam Hossain, and Md Osman Gani. A survey on causal discovery methods for temporal and non-temporal data. arXiv preprint arXiv:2303. 15007 2023.
Miguel Ángll Hernán, Babotte Brumback, and James M Robins. Marginal structural models to estimate the causal effect of zidovudine on the survival of hiv-positive men. Epidemiology, pp. 561-511 1500 2000.
Craig Hiemstra and Jonathan D Jones. Testing for linear and nonlinear granger causality in the stock prico-volume relation. The Journal of Finance, 49(5):1639-16644 1994.
Biwei Huang, Kun Zhang, Mingming Gong, and Clark Glymour. Causal discovery and forecasting in nonstationary environments with state-space models. In International conference on machine learning, pp- 2901-2910. PMLR, 2019.
Aapo Hyvärinen, Kun Zhang, Shohei Shimizu, and Patrik O. Hoyer. Estimation of structural vector autoregression model using non-gaussianity. Journal of Machine Learning Research, 11:1709-1731, 2010.
Aapo Hyvärinen, Kun Zhang, Shohei Shimizu, and Patrik O Hoyer. Estimation of a structural vector autoregression model using nou-gaussianily. Journal of Machine Learning Research, 11()), 2010.
Dominík Janzing, Lenon Minories, and Patrick Blöbaun. Feature relevance quantification in explainable ai: A causal problem. In International Conference artificial intelligence and statistics, pp. 2907-29166 PMLR, 2020.
Saurabh Khanna and Vincent YF Tan. Economy statistical recurrent units for inferring nonlincar granger causality. arXie preprint or XrXiv:1911.09879 2019.
lles Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. Variational autoencoders and nonlinear ica: A unifying framework. In Proc. of ATSTATS, pp. 2207-2211, 2020.
Immun Kim, Matey Neykov, Sivaraman Balakrishnan, and Larry Wasserman. Local permutation tests for conditional independence. The Annals of Statistics, 50(6):3388 3414, 2022.
Sébastion Lachapelle, Pau Rodriguez, Yash Sharma, Katic Everett, Rémi Le Priol, Alexandre Lacoste, and Simon Lacoste-Julien. Disentanglement via mechanism sparsity regularization: new principle for nonlinear ica. In Proc. of CLearR, pp. 428 484, 2022.
14

---

Published in Transactions on Machine Learning Research (09/2025)
Phillip Lippe, Sara Magliacane, Sindy Löwe, Yuki M. Asano, Taco Cohen, and Efstratios Gavves. icitris: Causal representation learning for instantaneous temporal effects. CoRR, abs/ Abs/2206.06169, 2022a.
Phillip Lippe, Sara Magliacane, Sindy Löwe, Yuki M Asano, Taco Cohen, and Stratis Gavves. Citris: Causal identifiability from temporal intervened sequences. In Proc. of ICML, pp. 13557-13603, 2022b.
Sindy Löwe, David Madras, Richard Z. Shilling, and Max Welling. Amortized causal discovery: Learning to infer causal graphs from time-series data. In Proc. of CLeal, pp. 509 525, 2022.
Zheng Lu, Chen Zhou, Jing Wu, Hao Jiang, and Songyue Cui. Intograting granger causality and vector auto-regression for traffic prediction of large-scale wlans. KSII Transactions on Internet and Information Systems (TITS), 10(1):136-151, 2016.
D. Marbach, T. Schaffter, C. Mattiussi, and D. Floreano. Generating realistic in silico gene networks for performance assessment of reverse engineering methods. Journal of Computational Biology, 16(2):229-330, 2009.
Daniele Marinazzo, Wei Liao, Huafu Chen, and Sebastiano Stramaglia. Nonlinear connectivity by granger causality, Neuroimage, 58(2):330 338, 2011.
Atalanti-Anastasii Mastakouri, Bernhard Schölkopf, and Dominik Janzing, Necessary and sufficient conditions for causal feature selection in time series with latent common causes. In Proceedings of ICML, volume 139, pp. 7502 7511, 2021.
Søren Wengel Mogensen and Niels Richard Hansen. Markov equivalence of marginalized local independence graphs. The Annals of Stalistics, 48(1):539 559, 2020. doi: 10.1214/19- AOS1182..
Søren Wengel Mogensen, Daniel Malinsky, and Niels Richard Hansen. Causal learning for partially observed stochastic dynamical systems. In Proceedings of the 34th Conference on Uncertainty in Artificial Intelligence (UAI), pp. 350-360, 2018a.
Søren Wengel Mogensen, Daniel Malinsky, and Niels Richard Hansen. Cansal learning for partially observod stochastic dynamical systems. In UAT, pp. 350 350-360, 2018b.
Raha Moraffah, Paras Sheth, Mansooreh Karami, Anchit Bhattacharya, Qianru Wang, Anique Tahir, Adrienne Raglin, and Huan Liu. Causal inference for time series analysis: Problems, methods and ovaluation. Knowledge and Information Systems, 63:3041-3085, 2021.
Meike Nauta, Doina Bucur, and Christin Seifert. Causal discovery with attention-based convolutional neural networks. Machine Learning and Knowledge Extraction, 1(1):312-33 340, 2019.
Jerzy Neyman and Elizabeth L. Scott. Asymptotically optimal tests of composite hypotheses for randomized experiments with noncontrolled predictor variables. Journal of the American Statistical Association, 60: 699 721, 1965. ISSN 1537274X.
Roxana Pamfil, Nisara Sriwattanaworachai,, Shaan Desai, Philip Pilgerstorfer, Konstantinos Georgatzis, Paul Beaumont, and Bryon Aragam. Dynotears: Structure learning from time-series data. In International Conference on Arlificial Intelligence and Statistics, pp. 1595-1605. PMTRR, 2020.
Junhyung Park and Krikamol Muandet. A measure-theoretie approach to kernel conditional mean embeddings. In Advances in Neural Information Processing Systems, 2020.
Judea Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, 2000.
Jonas Peters, Joris M. Mooij, Dominik Janzing, and Bernhard Schölkopf. Identifiability of causal graphs using functional models. In Proc. of UAI, pp. 589-598, 2011.
Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Causal inference on time series using restricted structural equation models. In Proc. of NIPS, pp. 154-162, 2013.
15

---

Published in Transactions on Machine Learning Research (09/2025)
Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.
R. J. Prill, D. Marbach, J. Sadriizzz, P. K. Sorger, L. G. Alexopoulos, X. Xue, N. D. Clarke, G. Altan- Bonnet, and G. Stolovitzky. Towards a rigorous assessment of systems biology models: the dream3 challenges. PloS one, 5(2))e9202 2010.
Francesco Quinzan, Ashkan Soleymani, Patrick Jaillet, Cristian R Rojas, and Stefan Bauer. Drefs: Doubly robust causal feature selection. In International Conference on Machine Learning, pp. 28468 28491, 2023.
James M Robins, Andrea Rotnitzky, and Lue Ping Zhao. Estimation of regression coefficients when some regressors are not always observed. Journal of the American statistical Associalion, (947))88846 866, 1994.
James M Robins, Miguel Angel Hernan, and Babette Brumback. Marginal structural models and causal inference in epidemiology. Epidemiology, pp. 550 560, 2000.
A Rotnitzky, E Smucler, and J M Robins. Characterization of parameters with mixed bias property. Biomelrika, 108(1):2311255) 238, 08 2020.
Paul K Rubenstein, Stephan Bongers, Bernhard Schölkopf, and Joris M Mooij. From deterministic odes dynamic structural causal models. arXic preprint 1rXi::1608.088: 2016.
Jakob Runge. Causal network reconstruction from time series: From theoretical assumptions to practical estimation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(7):075310, 2018.
Jakob Runge. Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets. In Conference on Uncertainty in Artificial Intelligence, pp. 1388-1397. PMLR, 2020.
Jakob Runge, Sebastian Bathiany, Erik Bollt, Gustau Camps- Valls, Dim Coumou, Ethan Deyle, Clark Glymour, Marlene Kretschmer, Miguel D Mahocha, Jordi Muñoz-Mari, et al. Inferring causation from time series in earth system sciences. Nature communications, 10(1):2553, 2019a.
Jakob Runge, Peer Nowack, Marlene Kretschmer, Seth Flaxman, and Dino Sejdinovic. Detecting and quantifying causal associations in large nonlinear time series datasets. Science advances, :eaa496,,,, 2019b.
Rajen D Shah and Jonas Peters. The hardness of conditional independence testing and the generalised covariance measure. 1rXio:1804.0720300 2020.
Tianhong Sheng and Bharath K. Sriperumbudur. On distance and kernel measures of conditional independence, 2020.
Shohei Shimizu, Patrik O Hoyer, Aapo Hyvärinen, Antti Kerminen, and Michael Jordan. A linear non- -ou-gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7(10), 2006.
Ali Shojaie and Emily B Fox. Granger causality; A review and recent advances. Annual Review of Statistics and Its Application, 9(1):289-311 319, 2022.
Tymon Sloczyúski and Jeffrey M Wooldridge. A general double robustness result for estimating average treatment effects. Econometric Theory, 34(1):112-13) 2018.
Ashkan Soleymani, Anant Raj, Stefan Bauer, Bernhard Schölkopf, and Michel Besserve. Causal feature selection via orthogonal search. Transactions on Machine Learning Research, 2022.
Peter Spirtes, Clark N Glymour, and Richard Scheines. Causation, prediction, and search. MIT press, 2000.
Adolf Stips, Diego Macias, Clare Coughlan, Elisa Garcia- and X San Liang. On the causal structure between co2 and global lemperature. Scientific reports, 6(1):21691, 2016.
George Sugihara, Robert May, Hao Ye, Chih-hao Hsich, Ethan Deyle, Michael Fogarty, and Stephan Munch. Detecting causality in complex ecosystems. science, 338(6106):496- 5500, 2012.
16

---

Published in Transactions on Machine Learning Research (09/2025)
Alex Tank, lan Covert, Nicholas Foti, Ali Shojaie, and Emily Fox. Neural granger causality for nonlinear time series. stat, 1050:16, 2018.
Sebastian Weichwald, Martin E Jakobsen, Phillip B Mogensen, Lasse Petersen, Nikolaj Thams, and Gherardo Varando. Causal structure learning from time series: Large regression coefficients may predict causal links better in practice than small p-values. In NeurIPS 2019 Competition and Demonstration Track, pp. 27-36. PMLR, 2020.
Williams. Probability with Martingales. Cambridge University Press, 1991.
Tailin Wu, Thomas M. Breuel, Michael Skuhersky, and Jan Kautz. Discovering nonlinear relations with minimum predictive information regularization. CoRR, abs/2001.01888) 2020.
Weiran Yao, Yuewen Sun, Alex Ho, Changyin Sun, and Kun Zhang. Learning temporally causal latent processes from general temporal data, 2021.
Kui Yu, Xianjie Guo, Lin Liu, Jiuyong Li, Hao Wang, Zhaolong Ling, and Xindong Wu. Causality- feature selection: Methods and evaluations. ACM Computing Surveys (CSUR), 53(5):1-35 36, 2020.
Kui Yu, Lin Liu, and Jiuyong Ti. A unified view of causal and non-causal feature selection. ACM Transactions on Knowledge Discovery from Data (TKDD), 15(4):1-40 46, 2021.
Kun Zhang, Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Kernel-based conditional independence test and application in causal discovery. In Conference on Uncertainty in Artificial Intelligence, pp. 804-813, 2011.
17

---

Published in Transactions on Machine Learning Research (09/2025)
A Direct Causal Effects and Interventions
In this section, we clarify the notion of an intervention and direct causal effects. We will also introduce some notation that will be used later on in the proofs.
We consider interventions by which a random variable X is set to a constant X f- Yr do(X) z) the outcome time series Y time step T, after performing an intervention as described above. We can likewise perform multiple joint interventions, by setting group of random variables I at different time steps, pre-determined constants specified by array i. We use the symbol Yr do i) to denote the resulting post-interventioall outcome, and we denote with P (Yr do(I i)) the probability of the event {Yr do(I = i) = y).
We denote with
Using this notation, a time series X' has a direct effect on the outcome Y, if performing different interventions on the variables X', while keeping the remaining variables fixed, will alter the probability distribution of the outcome Y. Formally, define the sets of random variables IT {Xt,... ................................ X Y)tET, which consists of all the information before time step T. Similarly, define the random variable Xy- (X)) consisting of all the information of time series X before time step T. Define the variable I = Ir X7; which consists of all the variables in T except for X².. Then, a time series X has direct effect on the outcome Y if it holds
P PVr= y do X7 = x', I = +P) = y do = x=-=x",10 (x>="",T--))
Yr
(6)
We say that a time series step T.
X' causes Y, there a direct effect between X' and as in Eq. for any time
B Necessity of the Statistical Independence of E
We provide counterexample to show that if there are dependencies between the noise and the historical data, then the causal structure may not be identifiable from observational data. To this end, we consider a first dataset {Xe, Yihıez, defined as
X₁-1, ~ (0,
Here, E) is zero-mean joint Gaussian distribution with covariance matrix
Σ 1
We also consider a second dataset (W,, Zi defined
W ~ NO0,1), 1), -
The parameter Σ is defined as above. Both datasets entail the same joint probability distribution. However, the causal diagrams change from one dataset to the other. Hence, the causal structure cannot be recovered from observational data, the posterior additive noise E, correlated with some the covariates.
C Necessity of No Instantaneous Causal Effects between Y and the Potential Causes X
Here, we provide counterexample to show that without the no instantaneous causal effect, the causal structure may not be identifiable from obsorvational data. Consider the following two modols:
Model 1: We consider time series (X.), (Yr} of the form X, = [[X,-]]+ and Y; =
In this model, e is a random variable drawn from a Gaussian distribution with a mean of and covariance of 1.
Model 2: We consider time series (X,), (12) of the form Y. E|Y,- and X, = Y,-I++Y, In this model, is a random variable drawn from Gaussian distribution with a mean of and covariance of 1.
18

---

Published in Transactions on Machine Learning Research (09/2025)
Both models entail the same joint distribution. However, in Model 1, X has causal effect on Y, whereas in Model 2, Y has causal effect on X. Hence, in this example, the causal structure is not. identifiable.
D Proof of Theorem 4.2
We prove the following result.
Theorem 4.2. Consider a causal model as in Aziom (A) Ariom (C) Then, it holds X3 E pa, (Y) for some t, E Z if and only if (iff-) (2 holds. That is, time series X has a direct causal effect on Y iff. x
Granger causes Y.
Proof. We first prove that it holds
PP (rr do do 1 = -1) YYr xx = = 1 =
(7)
for any non-zero event {Yr = y). To this end, define the group P consisting of all the causal parents of the outcome. Note that P S (X1, (y), By Axiom (A) the outcome can be described Y == f(P) + e, where is independent of (X;, 1- Hence by Rule of the do-calculus (see Pearl 20000) page 85) Eq. holds, since
Y becomes independent of (Xt, } once all arrows from to Y are removed from the graph of the DGP.
We now prove the claim using Eq. 7 To this end, assume that Eq. holds and suppose that X does not Granger causes Y, i.e, it holds
P
Vr y 1 x = (Yr=yl = 1 = =t)
(8)
for any non-zero event (Yr = y}. Then,
PP (Vr = do = x. 1 = i) = Yr = y/ x; = x, 1. =
Fqq. 7
= Yr = y 1 =)
LEq. g
P Yy y x = x', I =
Eq. g
P Yr y do( = x' 1 = =i)
Eqg 7
Hence, causality implics Granger causality.
We now prove that Granger causality implies causality. To this end, suppose that X is not potential cause of Y. By the definition of direct effects, it holds
Yr = y do( = x, 1
E
P
Yr = do = 2', I.. =()) 1 =
(9)
Hence,
Yr = 1/ do X x. 1 i))
P
Yr y x = x 123 =
Eq.
[P Yr = y do(x); x} 1 =
E
Eq.
E P Yg = y x =3-²) x =
Eq.
P
Yr = 1332 = i)
and the claim follows.
19

---

Then, the following conditions are Published in Transactions on Machine Learning Research (09/2025)
E Proof of Theorem 4.3
We prove the following result.
Theorem 4.3. Consider the notation as in Eg. 2 and fir time step T. For any square-integrable random
variable 9 (X <T,
Sor,I1, 12r), consider the moment functional mv(V:g") = Yr gg. Similarly, for any square-
integrable random variable g²(T22) consider the moment functional mv = Yr g'. Assuming Ariom
A Axiom (C) X Granger causes Y ifʃ. it holds
E [V: VV [i)] E [ VV :b)] * 0,
(4)
with (0 8) E[yr X
x- = x, 12 i], and ob(+) E
i <T s].
In order to prove Theorem 4.3 use the following auxiliary lemma.
Lemma E.1. Consider causal model as in Ariom (A) Ariom (C)
equivalent:
1. E Yr Xi x. 1 E YT Xi = x', 17 a.s.
2. P Yr = y x; = x. x = 1) = P (Yr
x = xx' -1- a.s.
Proof. Clearly, Item
2 implies Item
We now prove the converse, ie., we show that Item [ implies Item 2 To this end, define the group Pr
consisting of all the causal parents of Yr. Note that it holds Pr S (
X;} S Xi}. Hence, the joint
intervention (Xi, dit 1 {z, i} define an intervention the parents Pr t p. Further, write the potential outcome
Yr do( J. 1 =) f()) +e.
(10)
Similarly, the joint intervention (Xi, - {z, i} define an intervention on the parents Pr p'. We write the potential outcome
Yr do(X; x 12 i) = f(p') +
(11)
Hence, it holds
((p) E [c]
E
Yr do J. 12 i)
Eqq.
Yr Xi = x, 1 = $]
E
Eq. Theorem 4.2
E
Yr x; = x' + =
bb assumption]
= E
Yr Xx; =  1
Eq.
Theorem 4.2
f(p') + [e].
LEq.
By Axiom (A) the variable exogenous independent noise. From the chain of equations above follows
that f(p) f(p'). Hence,
P Yr - do(x) x, 1 =)) P p(fp))++== +5 +5=
-P((p'))+ y) P (Y do = = =i)
(12)
20

---

Published in Transactions on Machine Learning Research (09/202))
We conclude that it holds
P Yr X I, 17 =
P Yr x I, 1 = i)
Eq.
Theorem 4.2
P Yr do(X; x', 1 = i)
E4q. 12
P YT x = x' , = =i)
Eq. Theorem 4.2
as claimed.
We can now prove the main result.
Proof of Theorem 4.3 We first prove that X Granger causes Y iff. it holds
Yr {3] 2
E
E
Yr
0.
(13)
First, suppose that Eq.
13 does not hold. Then, it holds [Yr X; = 1 =]] = B|YT X; = x 1 = i].
a.s. Combining this equation with Lemma
E.1 yields
P (Yr y X x, 1 i) E P Yı = X 12
1=== = i)
=P (Yr= 1 =t),
a.s. Hence, if X Granger causes Y, then Eq. 13 holds.
Yr x = x. 1
*
Yr
x
x 1 i
(14)
for triple {I, w). By combining Eq. with Lemma E.1 we see that Eq. 13 implies causality. However, by Theorem 4.2 Granger causality is equivalent to causality in this case.
We now prove the claim. By the tower property of the expectation Williams 1991 that
E Yr I 1))
E
Yr
X
Yr x 27 X
E
E
Yr
E
E
E
Yr
x
Yr
E
E
E
Yr
E XX
Yr
E
Yr
XX
E
Yz
E
Yr
[2]
E
Yr
E
Yr
E
Yr
Yr
X +
YT
Yr
r31]
claimed.
21

---

Published in Transactions on Machine Learning Research (09/2025)
F Intuition on Paro-Masking
We provide intuition why masking reasonably good idea. Assume that the function I is e-close estimator of the true function f in the L (Px, norm, where the functions f, f and the joint probability distribution Px Xm. are defined on the set {X1, X2........ J.),
1f 1w
Now, let's mask the random variable X1. We are interested to see how close is the estimator Ex, to the true function Ex, f in the (P norm, where the functions Ex, J, Ex, I and the marginal probability distribution P......... are defined on the rest of variables {X2,... Xm²} By Jensen's inequality, we infer that for any realization of X2 F2, X3 I3,.... . Xm Em
(Ex, f(x, Tm) - fx (X1, _mmm)
kmm)²
S Sxx [[f((x 22, ................................ {m) l (2..,
Plugging it in the e-closeness assumption leads to,
[x -xx, Ex, ExcPss xn) -l S.
which guarantees that Ex, fis is also e-closs to fx and hence it's a good estimator. In the sequel, a natural solution would be to estimate Ex, f by taking averages of f over different samples of X1. However, for the linear regression problem that the estimator has a linear structure the input, it is straightforward to show that it is enough to evaluate ī at E[Xi]. And finally due to the zero-centering step data preprocessing, E[X]] = O. Thus, the aforementioned procedure is equivalent to aro-mmskinn.
G A Note on the Number of Partitions
The number of partitions k affects the performance of our algorithm in practice since a larger number of partitions will help in removing bias in the estimates. However, in experiments, observe that small number of partitions is sufficient to achieve good results. Furthermore, excessive number of random partitions may have detrimental effect, due to the possible small number of samples in each partition. Hence, we believe that the number of partitions will not drastically affect performance in practice. Reasonable choices of & for our experiments range between 3-7, hence O(1) w.r.t. parameters of the problem. Thus, the resulting runtime can be reported as (mdd).
Extension to Full Causal Discovery
It possible use Algorithm 1 for full causal discovery, for fully-observed acyelic auto-regressive models with no instantaneous effects (see Peters al. (2013): Löwe et al. (2022) for a precise definition of this restricted framework). In fact, under these more restrictive assumptions, we can identify the causes of each random variable of the model by testing Granger causality. For these models, can learn the full summary graph, by identifying the causes of cach variable with Algorithm [ The resulting run time can be quantified as (mmak), where d is the time complexity of performing a regression, as outlined above, m is the number of time series considered, and & is the number of dataset partitions used for double cross- cross-fitting, Since = O(1) w.r.t. parameters of the problem (see Appendix G). the runtime of DR-SIT O(md).
Computational Complexity and Comparison
As discussed in Section compared to DR-SIT, conditional independence-based approaches such as PCMCI (Runge et al.] 2019b), PCMCI+ (Runge) 2020), and LPCMCI (Gerhardus & Runge (020) face exponential computational barriers. Tt is widely known that even endowed with a perfect infinite sample
22

---

Published in Transactions on Machine Learning Research (09/2025)
Table 3: Table with runtime means and standard deviations for DR-SIT and PCMCI+ (in seconds).
20
30
40
100
200
400
DR-SIT 42 13
35 +
29
30 +
35 15
41 + 6
62 + 6
77 + 10
PCMCI+
4.2 0.4
18.6 =
0.8
48.6 + 1.4
99.2 +
216.4 = 42.2
1091 + 65
5678 + 264
8 hours
independence lesting oracle, learning Bayesian Networks becomes NP-Hard (Chickering et al. 2004 Chick ering, 1996). Consequently, computational challenges arise not only due to the nature of the conditional independence lests themselves but also from the computational intractability of searching through the expo- nentially large space of possible network structures. Hence, the runtime of the O(m) number of regressions that DR-SIT demands is negligible compared to the exponential number of conditional independence tests from lengthy time-series. To support this argument practice, we provide a runtime comparison between DR-SIT and PCMCI+ w.r.t. the number nodes m in Table 3
J Additional Experiments
J.1 Results for Synthetic Experiments
Table 4: Accuracy of our method for increasing number of potential causes m, and different noise-to-signal
ration (NSR). We observe that method maintains good accuracy, in challenging settings with many potential causes and high noise.
Accuracy
NSR = 0
NSR = 0.05
NSR = 0.1
NSR 0.15
NSR 0.2
NSR = 0.25
NSR = 0.3
0.60
-
0.90 + 0.11
0.80
0.94 + 0.12
10
0,99
t 0.02
0.92 = 0.08
0.97 = 0.04
0.79 + 0.18
0.89 = 0.06
0.91 0.04
0.90 = 0.08
0.89
0.05
0.89
0.03
0.88 0.88
0.85 + 0.02
0.79 0.11
0.79 # 0.09
0.81 + 0.03
0,83
0.07
0.73 0.05
0.72 0.07
0.77 1 0.10
0.73 0.05
0.75 0.05
0.69 1 0.06
25
0.76 + 0.04
0.72 + 0.10
0.71
0.63 + 0.08
0.68 - 0.04
0.71 + 0.07
0.66 0.04
0.76 0.01
0.74 0.04
0.07
0.70 0.10
0.00
35
0.68 +.8
0.65
+ 0.07
0.72
0.65 1 0.03
0.03
0.66 1 0.05
0.61 + 0.07
0.64 0.09
0.64
+ 0.03
0.69
0.05
0.62 + 0.03
0.63
0.07
0.60 0.07
45
0.65
0.01
0.68 0.08
0.58 # 0.04
0.61 - 0.03
0.61 0.04
0.59 0.04
0.61 = 0.06
50
0.68
0.05
0.63
0.03
0.64 0.06
0.63 1 0.05
0.66
0.05
0.39
0.08
0.64 1 0.07
Table 5: CSI Score of Algorithm 1 for incrcasing number of potential causes m, and different noise-to-signal ration (NSR). Again, we observe that our method is robust to increasing NSR.
CSI Score
773
NSR =
NSR = 0.05
=
NSR 0.15
0.2
NSR = 0.25
NSR = 0.3
0.57
0.07
0.71 0.28
0.63 0.19
0.86 1 0.12
0.69
0.10
0.73 0.12
0.91 + 0.17
10
0.98 + 0.04
0.86 0.14
0.95 0.06
0.69 + 0.21
0.80 = 0.09
0.87 + 0.07
0.82 0.14
0.75
0.06
0.59
20
0.66 + 0.12
0.52
- 0.12
0.46 1 0.07
0.56 1 0.10
0.46 2 0.08
0.31 1 0.07
0.39 1 0.09
0.51
0.06
0.45
0.12
0.09
0.37 + 0.09
0.38 0.05
0.41 0.08
0.33 + 0.06
0.47 0.05
0.50
- 0.07
0.46
0.38 + 0.08
$ 0.35
0.34 = 0.05
0.31 0.07
0.42 1. 0.04
0.31
1 0.06
0.39 1.39.00000
0.29 1 0.07
0.30 0.09
0.22 1 0.09
0.26 1 0.09
0.06
0.38
0.03
0.33
0.29 1 0.06
0.24 2
0.07
0.19 0.11 0.11
45
0.33 0.10
0.34
- 0.07
0.20 0.02
0.22 - 0.05
0.25 + 0.03
0.20 + 0.05
0.19 0.06
J.2 Performance in Low-Sample Regimes
The double robustness property enables our algorithm to rely on simple estimators with low statistical complexity. As a result, our method shows more consistent performance in low-sample regimes as opposed to existing approaches that are based on overparameterizee models demanding so many data points (Figure
Moreover, in Figs.
we compare the progression of AUROC score vs the total runtime for DR-SIT RHINO in various combinations of tasks and training sizes. The hardware specifications are described in
23

---

Published in Transactions on Machine Learning Research (09/2025)
Table 6: Score the DR-SIT for increasing number of potential causes m, and different noise-to-signe ration (NSR). Interestingly, our method maintains good F1 score for increasing NSR.
F1 Score
NSR
NSR 0,05
NSR
NSR 0.15
NSR
0.2
NSR 0.25
NSR 0.3
5
0.73 - 0.06
U.79 = 0.20
0.75 Σ 0.12
0.92 t 0.07
0.81 = 0.07
0.81 : 0.08
0.95 = 0.11
15
0.89 1: 0.05
0.86
+ 0.08
0.85 1 0.11
0.81 1 0.04
0.73 1 0.15
0.72 + 0.12
0.74 1 0.03
20
0.79
0.08
0.67
0.10
0.62
0.07
0.71 + 0.09
0.63
0.07
0.67 0.06
0.36 + 0.09
0,68
0.05
0.61
0.11
0.60 0.09
0.53
1 0.11
0.55
0.57 0.08
0.49 1 0.07
30
0.64 + 0.05
0.66
= 0.06
0.62 0.10
0.54 + 0.09
0.51 = 0.09
0.50 + 0.06
0.46 + 0.09
0.59
0.08
0.46
0.11
40
0.49
+ 0.07
0.55
+ 0.05
0.49 + 0.08
0.45 + 0.U7
0.38 = 0.08
0.33 + 0.09
0.30 + 0.15
50
0.50
4 0.03
0.44 + 0.07
0.44 4 0.08
0.38 4 0.05
0.41 0.07
0.34 0.11
0.40 + 0.09
AUROC vs N. feat for noise level 0.0
AUROC vs feat for noise level 0.1
AUROC vs N feat for noise level 0.2
1.00
RHINO DR.SIT POWO
RHND
DR.STT
PCKC
0.35
0.55
0.55
0.30
0.00
0.00
AUDKC
0.55
0.65
0.85
RHNO DR.SIT
PCNCI
0.70
0.70
N, NLaeat
N
vs
1.00
1.00
1.00
RMND
0.55
0.55
0.95
AUNOX
0.50
0.00
0.75
035
0.75
0.70
0.70
0.70
10
9.Seat
Seat
Figure 4: AUROC metric for DR-SIT, RHINO and PCMCT for various noise levels on the synthetic dataset..
Section 6.4 and the training hyperparameter settings for RHINO in Section 6.2 The runtime of DR-SIT is always less than minute (so afterwards AUROC curve is plotted constant) where each epoch for RHINO takes about 30 seconds independently of the training dataset size.
24

---

Published in Transactions on Machine Learning Research (09/2025)
AUROC
AUROC
RHINO
RHINO
RHIND+g
DR-SIT
DR-SIT
Number trajectories
Number trajectories
(a) Task: E.Coli
(b) Task: E.Coli
RHINO RIIINO+g
RHIND
664
RJIINO+gg
DR-SIT
DR-SIT
842
AUROC
AUROC
354
Number of trajectories
Number of trajectories
(c) Task: Yeast.
(d) Task: Yeast. 2
RHINO
RHINO++
DR-SIT
AUTOC
0:2
328
Number trajectories
(c) Task: Yeast 3
Figure 5: This figure demonstrates the consistent performance of DR-SIT w.r.t number of observations compared to state-of-the-art methods Rhino and Rhino+g. Note that Rhino and Rhino+g are built on neural networks. DR-SIT significantly outperforms Rhino and Rhinotg in E.Coli 1 and E.Coli 2 and shows competitive results Yeast 1. Thanks the double robustness property of DR-STT, the dependence of our algorithm on the estimator much lower than the well-established approaches. In this regard, DR-SIT with simple kernel regression with polynomial kernels has superior performance compared to stale-of-the-ard methods Rhino and Rhino+g. This superiority gets magnified in the low number of observation regimes due to the high sample complexity required by Rhino and Rhinotg.
25

---

Published in Transactions on Machine Learning Research (09/2025)
Task: Ecoli1
Nm5
N-10
0.650
0.66
0.625
0.600
AUROC
0.575
0.60
0.550
058
0.525
033
0.500
0.54
0.52
200
600
800
200
600
800
N-15
N-20
0.66
0.655
AUROC
0.62
0.625
0.60
0.58
0.575
0.56
0.50
0.52
0500
N=25
N=30
0.70
0.68
0.66
AUROC
aigorthm
0.64
0.64
DR SIT
0.62
RHINO
0.60
0.60
0.58
058
N-35
N-40
0.70
0.70
0.68
0.68
0.66
0.66
AUROC
0.62
0.64
062
0.58
0.56
058
0.54
200
800
600
800
Nm43
N=46
0.70
0.68
0.68
0.66
RUROC
0.64
0664
0.62
0.60
058
200
400
600
800
200
400
600
800
(sec)
(sec)
Figure 6: AUROC time (in secs) for DR-SIT (blue) RHINO (orange) for Ecoli 1 task and various mumbers of training observations (number of trajectories).
26

---

Published in Transactions on Machine Learning Research (09/2025)
Task: Ecoli2
5m5
N-10
0.65
0.050
0.60
0.625
0.55
AUROC
0.555
0.50
0325
0.40
0475
200
800
200
600
800
N-15
N-20
0.650
0.05
0.625
0.60
AUROC
0.575
0.55
0.525
0.500
0.475
N=25
N=30
0675
0.65
0.600
066
AUROC
0.600
DR SIT
0.555
RHINO
0350
0.50
0.25
0.500
N-35
N-40
0.650
0.600
0.625
0625
AUROC
0.575
0.75
0.550
0350
0.500
0500
0.475
200
800
600
800
N=43
N=46
0.650
AUROC
0.625
0.600
0.75
0550
200
400
600
800
200
400
600
800
(sec)
(sec)
Figure AUROC time (in secs) for DR-SIT (blue) RHINO (orange) for Ecoli 2 task and various mumbers of training observations (number of trajectories).
27

---

Published in Transactions on Machine Learning Research (09/2025)
Task: Yeastl
N
N=10
0.58
0.62
0.56
0.60
AUROC
0.54
058
0.5
0.50
054
200
400
600
800
400
800
N=15
N-20
0.62
0.64
0.62
AUROC
0.58
0.60
058
0.54
056
N=25
N=30
0.66
0.64
0.64
AUROC
0.62
DR SIT
0.60
0.60
0.5
N-35
N-40
0.66
0.64
AUROC
0.02
0.62
0.60
0.58
038
800
400
800
N=43
N=46
066
AUROC
0.62
0.62
0.60
0.60
038
200
400
600
800
200
400
600
800
Figure AUROC time (in sees) for DR-SIT (blue) vs RHINO (orange) for Yeast. 1 task and various mumbers of training observations (number of trajectories).
28

---

Published in Transactions on Machine Learning Research (09/2025)
Task: Yeast2
N
N=10
0.56
0.56
0.54
AUROC
0.52
0.52
030
0.50
0.48
400
600
800
400
800
N-15
N-20
0.56
0.5
AUROC
0.52
0.5
051
0.50
N=25
N=30
0.9
0.57
038
0.56
0.5
AUROC
0.55
aigorthi
0.5
DR SIT
0.54
RHINO
0.53
033
N-35
N-40
0.9
0.50
0.7
0.56
AUROC
0.55
0.5
055
0.5
0.53
0.52
052
0.1
800
400
800
N=43
N=46
0.5
0.58
0.6
AUROC
0.57
0.56
0.4
0.54
052
0.53
030
200
400
600
800
200
400
600
800
Figure AUROC time (in sees) for DR-SIT (blue) vs RHINO (orange) for Yeast 2 task and various mumbers of training observations (number of trajectories).
29

---

Published in Transactions on Machine Learning Research (09/2025)
Task: Yeast3
N
N=10
0.54
0.54
0.3
AUROC
0.2
0.51
051
0.50
0.49
050
200
400
600
800
400
800
N=15
N-20
0.545
0.54
0.535
AUROC
0.53
0.525
0.520
0.51
0.510
N=25
N=30
0.5
0.5
AUROC
0.53
aigorthi
DR SIT
0.52
RHINO
0.3
0.51
052
N-35
N-40
0.57
0.57
0.56
AUROC
0.55
0.53
053
0.52
0.2
800
400
800
N=43
N:46
0.57
0.7
0.6
AUROC
0.55
0.54
053
053
052
0.52
200
400
600
800
200
400
600
800
Figure 10: AUROC time (in secs) for DR-SIT (blue) RHINO (orange) for Yeast task and various mumbers of training observations (number of trajectories).
30


Reviews:
[{"review_id": "RJ0VEUxxIj", "reviewer": "Reviewer_4TwL", "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:\nWell written (some minor clarifications could help)\nStrongly motivated and well presented.\nRelevant / sensible evaluations\n\nWeaknesses:\nSimple and limited choice of estimators (kernel ridge regression with 3rd order polynomials)\nNo ablation (see comments below)\n\nOther comments:\nMain / notable assumptions are: additive / independent noise - ‘key for identifiability’ and no instantaneous effects. Nonetheless the method is designed to handle hidden confounders and cycles.", "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": "The authors tackle the challenging task of causal discovery / structure identification from time series data using double-robust approaches which have root-n consistency, and situate and distinguish their method with the landscape of Pearl’s causality vs Granger causality. Their method does not require the assumption of faithfulness or sufficiency.The theoretical guarantees are backed up with empirical tests.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The graph is markov-1 but that about auto correlative lags from Y or X_t to X_{t+d} with arbitrary d?  Is the phrase “Note that the time series… different lags.” trying to communicate this? If so, why only mention X->Y, what about Y->Y  and X->X? I think I know whats intended here but perhaps the authors could make this slightly clearer.\n\n\nGood computational efficiency compared to other methods - could the authors confirm whether this runtime scales linearly with the number of edges?\n Typo: section 4.1 second paragraph line 3 ‘I this work’\nTypo: Foot note page 6 citations not formatted properly\nTypo; Definition 4.1 citation not formatted properly\n\n\nClarifications:\n\nWhat is the closest method to DR-SIT and/or is there a version of DR-SIT which can be tested without the doubly robust component? I mention this because in practice I find the empirical benefits of DR / semi-parametric to be hit and miss (rests heavily on the quality of the trained estimators). My intuition would have been that the DR component contributed only marginally to the performance gap, but it looks like I might be wrong there. Interested to hear the authors’ comments, particularly regarding  the sensitivity of the method to estimator quality/convergence (https://openreview.net/forum?id=dQxBRqCjLr) . Indeed, in appendix J.2. the authors state ‘the double robustness property enables our algorithm to rely on simple estimators with low statistical complexity’ (but I think this is quite a strong statement because it surely increases the probability of misspecification).  Some kind of ablation would be useful here.\n\nI feel like some of the dependence of the performance of the method is illustrated in Fig.4 in the appendix - DR-SIT does equivalently or worse than alternatives on E.Coli 1 and Yeast 3. Although its performance on the other datasets is somewhat convincing, it does remind me a bit of this paper: https://datasets-benchmarks-proceedings.neurips.cc/paper_files/paper/2021/hash/2a79ea27c279e471f4d180b08d62b00a-Abstract-round2.html where, as a domain we have arrived at a set of benchmarks, certain/many of which interact preferentially with certain methods (e.g. because of complexity or whatever). I don’t get the impression that that is what is happening here, but it seems worth discussing. \n\nRelating to the ablation, the authors use (unless I missed something) kernel ridge regression as estimator(s). I wonder what the sensitivity of the results is to this choice and how they arrived at this (apologies if I missed a discussion on this point).", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "yYx2Ndf34R", "reviewer": "Reviewer_Edd4", "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\n**S1**: The problem is well motivated, with a clear review of related work and a clear introduction to the DML framework.\n\n**S2**: The application of DML to temporal causality inference (via Granger causality) is novel and conceptually interesting.\n\n**S3**: The theoretical foundations are strong, and the empirical results, particularly on the DREAM3 dataset, are compelling.\n\n**S4**: The authors provided code to reproduce the experiments.  \n\n**Weaknesses**\n\n**W1**: While the theoretical application of DML to Granger causality is clearly presented (e.g., Theorem 4.3), the practical details of the learning step are lacking. In particular, the estimation of the nuisance parameter $\\eta$, which is crucial to the method's performance, is not described in sufficient detail. Key aspects such as the choice of regression algorithms, and input features (lag) are not discussed. Additionally, the paper does not describe how the data is split for the DML procedure, which is a critical component.\n\n**W2**: There are issues with the notation that hinder readability. Specifically, the definitions of $\\mathbf{X}_T$​ and $\\mathbf{I}_T$​, which are essential to understanding the method, are relegated to the appendix, despite being used throughout the main text. These definitions should be brought into the main text. Furthermore, the notation is confusing: using $\\mathbf{X}_T​$ to denote the history of a time series $X$, while using $X_T$ to denote the value at time $T$, leads to unnecessary ambiguity.\n\n**W3**: The synthetic experiments are not sufficiently detailed and are only reported in the appendix. These experiments should be included in the main paper to empirically validate DR-SIT. Furthermore, they lack a comparison to state-of-the-art methods: the tables only report DR-SIT F1 and CSI scores without benchmarking against existing algorithms.", "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 paper introduces a novel method, DR-SIT, based on the Double Machine Learning (DML) framework, to infer the causes of a given time series. Under certain assumptions, the authors show that Granger causality is equivalent to causality, and leverage DML to test for Granger causality. The method is evaluated on both synthetic and real datasets (specifically, the DREAM3 dataset) and compared against state-of-the-art methods such as PCMCI and RHINO. Experimental results indicate that DR-SIT outperforms existing methods in terms of accuracy, computational efficiency, and scalability to high-dimensional settings.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Add implementation details of the learning step (estimation of $\\eta$ and details on the split).\n\n- Fix and clarify notation.\n\n- Improve synthetic experiment section:\n     - Move at least part of the synthetic experiments into the main body of the paper.\n     - Include comparative results against state-of-the-art methods (e.g., PCMCI, RHINO) to more thoroughly demonstrate the performance of DR-SIT.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Thx0HUVhf8", "reviewer": "Reviewer_qs2B", "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- This work studies an important problem in causal discovery with time-dependent (time-series) data.\n- The application of double machine learning to this setting is potentially innovative. \n\nWeakness: \n- Novelty of this work appears limited and some claims seem speculative and overstated. In particular, the main contribution of this work is a new way to test for the \"Granger causality condition'', all theoretical properties seem entirely rely on the existing double machine learning (DML) theory. However DML theory as cited in this work is developed under the assumption of iid data, yet in this work data is not iid. It is unclear to me how the results can be carried over so straightforwardly. I recommend the authors improve their Section 4.1 to make this clear, and at the end of Section 4.3, clarify under what \"mild convergence conditions on the nuisance parameters'' can DML ensure the claimed fast convergence. Moreover, even if the estimation of the parameter theta achieves root-n consistency, the authors still need to prove that their proposed test is indeed valid (asymptotic control of type-I error rate and power), as claimed in contribution 2 in Section 1. \n- Can the authors give intuitions on why faithfulness assumption is not needed for causal structure identification? If one does not require faithfulness, then some conditional independencies in the data are not d-separations in the graph. \n- In Section 5.1 step (4) Should it be paired t-test instead of two-sample t-test (which is for two independent samples)?\n- The authors should comment on the regression methods they use in the experiments and how they choose or tune the hyperparameters, e.g., why did they choose KernelRidge(alpha=1,kernel='poly',degree=3,coef0=1) in their code for the experiment in Section 6.1? \n- Definition of Granger causality in Section 3.4 is very imprecise, e.g., Y can only discrete? This also appears in proofs e.g., of Theorem 4.2.\n- In Section 4.2, the authors claim that in their case Granger causality corresponds to \"true causation'', and in Theorem 4.2 the authors call X^i a \"potential cause''. Can the authors define what they mean by true and potential causes, and resolve the conflict between these two sentences?", "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 proposes a method for causal structure identification with time series data  using Double Machine Learning (DML), called DR-SIT. The key proposal here is a new way to test for Granger causality which can take advantage of the DML framework. The authors claim that this work propose first non-parametric Granger causality test that achieves the semi-parametric root-n rate, the method allows confounding and cycles among covariates, and does not require faithfulness.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Beginning of Section 3.1: why i.i.d. observations? Isn't it a single realization of a time series?\n- What is T in Axiom (A)? Missing quantifiers?\n- In Axiom (C), there can be instantaneous causal effects between the X^s right? The authors should refine the axioms in mathematical terms to ensure their clarity.  \n- Definition of faithfulness in footnote 5 on page 4: Here faithfulness is somehow defined based on the Markov property, do the authors assume the Markov property always hold? \n- Some sentences don't make sense: e.g., \"VarLiNGAM (Hyvärinen et al., 2010) takes a step further and uses much more restrictive assumptions on the functional form of the structural equations'', this paper is 11 years older than the paper cited before it, how do they \"take a step further\" and \"use much more restrictive assumptions\"?...\n- Typos and grammatical errors e.g., page 5 \"I this work, [...] as follows Chernozhukov et al [...]''; Definition 4.1 inconsistent font for \"D''; \"Since both $m_0 (V; g_0^0)$ and $m_i(V; g_0^i)$ we can use DML as in Definition 4.1 to estimate $\\theta_0^0$ and $\\theta_0^i$.''", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)