Amortized Causal Discovery: Learning to Infer Causal Graphs from Time-Series Data
Authors: Sindy Löwe
David Madras
Richard Zemel
Max Welling
Venue:  (2021)

Abstract: Standard causal discovery methods must fit a new model whenever they encounter samples from a new underlying causal graph. However, these samples often share relevant information - for instance, the dynamics describing the effects of causal relations - which is lost when following this approach. We propose Amortized Causal Discovery, a novel framework that leverages such shared dynamics to learn to infer causal relations from time-series data. This enables us to train a single, amortized model that infers causal relations across samples with different underlying causal graphs, and thus makes use of the information that is shared. We demonstrate experimentally that this approach, implemented as a variational model, leads to significant improvements in causal discovery performance, and show how it can be extended to perform well under hidden confounding.


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AMORTIZED CAUSAL DISCOVERY: LEARNING TO INFER CAUSAL GRAPHS FROM TIME-SERIES DATA
Anonymous authors Paper under double-blind review
ABSTRACT
Standard causal discovery methods must fit new model whenever they encounter samples from a new underlying causal graph. However, these samples often share relevant information - for instance. the dynamics describing the effects of causal relations which is lost when following this approach. We propose Amortized Causal Discovery, a novel framework that leverages such shared dynamics learn to infer causal relations from time-series data. This enables us to train a single, amortized model that infers causal relations across samples with different underlying causal graphs, and thus makes use of the information that is shared. We demonstrate experimentally that this approach, implemented as a variational model, leads to significant improvements in causal discovery performance, and show how it can be extended to perform well under hidden confounding.
INTRODUCTION
Inferring causal relations in observational time-series is central to many fields of scientific inquiry (Berzuini et al.) 2012, Spirtes et al. (2000). Suppose you want to analyze fMRI data, which measures the activity of different brain regions over time - how can you infer the (causal) influence of one brain region on another? This question is addressed by the field of causal discovery (Glymour et al.. 2019). Methods within this field allow us to infer causal relations from observational data when interventions (e.g. randomized trials) are infeasible, unethical or too expensive.
In time-series, the assumption that causes temporally precede their effects enables us to discover causal relations in observational data (Peters et al., 2017); with approaches relying on conditional independence tests (Entner and Hoyer 2010), scoring functions (Chickering 2002), or deep learning (Tank et al. 2018). All of these methods assume that samples share a single underlying causal graph and refit a new model whenever this assumption does not hold. However, samples with different underlying causal graphs may share relevant information such as the dynamics describing the effects of causal relations. fMRI test subjects may have varying brain connectivity but the same underlying neurochemistry: social networks may have differing structure but comparable interpersonal relationships; different stocks may relate differently to one another but obey similar market forces. Despite range of relevant applications, inferring causal relations across samples with different underlying causal graphs is of yet largely unexplored.
In this paper, we propose a novel causal discovery framework for time-series that embraces this aspect: Amortized Causal Discovery (Fig. D. In this framework, we learn to infer causal relations across samples with different underlying causal graphs but shared dynamics. We achieve this by separating the causal relation prediction from the modeling of their dynamics: an amortized encoder predicts the edges in the causal graph, and a decoder models the dynamics of the system under the predicted causal relations. This setup allows us to pool statistical strength across samples and to achieve significant improvements in performance with additional training data. It also allows to infer causal relations in previously unseen samples without refitting our model. Additionally, we show that Amortized Causal Discovery allows to improve robustness under hidden confounding by modeling the unobserved variables with the amortized encoder. Our contributions are as follows:
We formalize Amortized Causal Discovery (ACD), novel framework for causal discovery in time-series, in which we learn to infer causal relations from samples with different underlying causal graphs but shared dynamics.

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Amortized Causal Discovery
Previous Approaches
Figure I: Amortized Causal Discovery. We propose train single model that infers causal relations across samples with different underlying causal graphs but shared dynamics. This allows us to generalize across samples and to improve our performance with additional training data. In contrast, previous approaches (Section 2 fit new model for every sample with different underlying causal graph.
We propose variational model for ACD, applicable multi-variate, non-linear data.
We present experiments demonstrating the effectiveness of this model range of causal discovery datasets, both in the fully observed setting and under hidden confounding.
2 BACKGROUND: GRANGER CAUSALITY
Granger causality (Granger 1969 is one of the most commonly used approaches to infer causal relations from observational time-series data. Its central assumption that causes precede their effects: if the prediction of the future of time-series Y can be improved by knowing past elements of time-series X, then "Granger causes" Y. Originally, Granger causality was defined for linear relations; we follow the more recent definition of Tank et al. (2018) for non-linear Granger causality:
Definition 2.1. Non-Linear Granger Causality: Given N stationary time-series [xII....N. across time-steps / = (1,... T} and non-linear autoregressive function gj. such that
++1 g ..... <t +
(1)
where 2j 2,.,, -111_ x denotes the present and past of series j and E/ represents independent noise. In this setup, time-series i Granger causes j. if gj is not invariant to St i.e. if E # x g ..... 15t 20) A g x
<t
.... (t)
Granger causal relations equivalent to causal relations in the underlying directed acvelic graph if all relevant variables are observed and no instantaneou connections exist (Peters et al. 2013 2017 Theorem 10.1).
Many methods for Granger causal discovery, including vector autoregressive (Hyvärinen et al 2010)
and more recent deep learning- -aaningg-based approaches (Khanna and Tan 2020 Tank et al. 2018 Wu et al. 2020). can be encapsulated by a particular framework:
1. Define function fa (an MLP in Tank et al. (2018), a linear model in Hyvärinen et al. 2010 which learns to predict the next time-step of the test sequence æ.
2. Fit fe to æ by minimizing some loss C: 0, = argmin, L(x
3. Apply some fixed function (e.g. thresholding) to the learned parameters to produce the Granger causal graph estimate for æ:: G2 I((0). For instance, Tank et al. (018)) infer the Granger causal relations through examination of the weights 0.: if all outgoing weights wiy between time-scries and j are zero, then does not Granger-cause j.
connections between variables the same time step
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The shortcoming of this approach is that, when have S samples Bi....: ................................ ss with different underlying causal graphs, the parameters must be optimized separately for each of them. As a result, methods within this framework cannot take advantage of the information that might be shared between samples. This motivates us to question: can we amortize this process?
AMORTIZED CAUSAL DISCOVERY
We propose Amortized Causal Discovery (ACD), a framework in which we learn to infer causal relations across samples with different underlying causal graphs but shared dynamics. To illustrate: Suppose you want infer synaptic connections (i.e. causal relations) between neurons based on their spiking behaviour. You are given a set of S recordings (i.e. samples), each containing N time-series representing the firing of N individual neurons. Even though you might record across different populations of neurons with different wiring, the dynamics of how neurons connected by synapses influence one another stays the same, ACD takes advantage of such shared dynamics to improve the prediction of causal relations. It can be summarized as follows:
1. Define an encoding function fo which learns to infer Granger causal relations of any sample æ; in the training set Xuain- Define decoding function fo which learns to predict the next time-step of the samples under the inferred causal relations.
2. Fit Ja and fo to Mmin by minimizing some loss L: Je.; argmin fo.fe I(Xininn J Jo).
3. For any given test sequence Trest simply output the Granger causal graph estimate g
9mm.. fe. (Fies).
By dividing the model into two parts, an encoder and a decoder, ACD can use the activations of fo- infer causal structure. This increases the flexibility of our approach greatly compared methods that use the learned weights 0, such as the prior Granger causal discovery methods described in Section 2 this section, we describe our framework in more detail, and provide a probabilistic implementation thereof. We also extend our approach to model hidden confounders.
Preliminaries We begin with dataset X {ar.]s= of samples, where each sample æ, consists of N stationary time-series X, (E N.,N across timesteps = {l,..., T'}. We denote the I-th time-step of the i-th time-series of v. as ay.. We assume there is a directed acyelic graph G1:T = DV1.T g1:T3 underlying the generative process of each sample. This is structural causal model (SCM) (Pearl 2009). Its endogenous (observed) variables are vertices vi. E V!!T for each time-series and each time-step t. Every set of incoming edges to an endogenous variable defines inputs to a deterministic function g'., which determines that variable's value" The edges are defined
by ordered pairs of vertices E!:T {1 "" which make two assumptions about:
1. No edges are instantaneous (1 (1= (') go back in time. Thus, ( < l' for all edges.
2. Edges are invariant to time. Thus, if (vi, t+*) E E!:T then V1 s t' S T k (vt th E!:T_ The associated structural equations gh, are invariant to time well, i.c. gi,, I Vt,
The first assumption states that causes temporally precede their effects and makes causal relations identifiable from observational data, when hidden confounders are present (Peters et al. 2013 2017 Theorem 10.1). The second simplifies modeling: it is a fairly general assumption which allows us define dynamics that govern all time-steps (Eq.
Throughout this paper, we are interested in discovering the summary graph G. = (V., (Peters et al. 2017). It consists of vertices Ug E V, for each time-series i in sample s, and has directed edges whenever they exist in EL:T at any step, i.e. E, = {Vv,,,, Us,z) 3t, t' (v), ofnz) E E!TT). Note that while G!-T is acyclic (due to the first assumption above), the summary graph G. may contain (self-)eycled
Amortized Causal Discovery The key assumption for Amortized Causal Discovery is that there exists some fixed function g that describes the dynamics of all samples v. X given their past
"The SCM also includes exogenous (unohserved), independently-samplee error variable c, parent of each vertex u, which we do not model and thus leave out for hrevity.
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observations x.¹ and their underlying causal graph Gz:
x g (.) + +
(2)
There are two variables in this seataaagneraingg process that we would like to model: the causal graph GR that is specific to sample 2s and the dynamics that are shared across all samples. This separation between the causal graph and the dynamics allows us divide our model accordingly: we introduce an amortized causal discovery encoder S which learns to infer causal graph 9s given the sample Rts, and a dynamics decoder so that learns to approximate g:
fo(25" fo(2).)
We formalize Amortized Causal Discovery (ACD) as follows. Let be the domain of all possible summary graphs g G. E G. Let X be the domain of any single step, partial or full, observed sequence: xi, 25 I, E X. The model consists of two components: a causal discovery encoder fo X + G which infers causal graph for each input sample, and a decoder fo X x G - X which models the dynamics. This model is optimized with sample-wise loss é X x X + R which scores how well the decoder models the dynamics of æg, and a regularization term r : G G R on the inferred graphs, For example, this function r may enforce sparsity by penalizing graphs with more edges. Note, that our formulation of the graph prediction problem is unsupervised: we do not have access to the true underlying graph G.. Then, given some dataset Xinin with S samples, we optimize:
fe., fo. = argmin forfo L(Xuain fs.fel
(4)
8 T
where C( uain 0,
f(xtt)
++1 folæ fo(x)))) + (ff((x)))
(5)
8=1
See Appendix B for proof of the consistency of the loss f and discussion regularization r.
Once we have completed optimization, we can perform causal graph prediction on any new input test sample Fiet in two ways
can feed Tess into the amortized encoder and take its output as the predicted edges (Eq. 6 ; or we can instantiate our estimate Grest G which will be our edge predictions, and find the edges which best explain the observed sequence Best by minimizing the (learned) decoding loss with respect to Gtest, which we term Test-Time Adaptation (TTA) (Eq. 7)
gEme fo. (Ires) ;
(6)
İTTA argmin argming....ec S. Claress, Gtest, So.)
By separating the prediction of causal relations from the modeling of their dynamics, ACD yields number of benefits. ACD can learn to infer causal relations across samples with different underlying causal graphs, and can infer causal relations previously unseen test samples without refitting (Eq. 6 ). By generalizing across samples, it can improve causal discovery performance with increasing training data size. We can replace either fs or fe with ground truth annotations, or simulate the outcome of counterfactual causal relations. Additionally, ACD can be applied in the standard causal discovery setting, where only a single causal graph underlies all samples, by replacing the amortized encoder fo with an estimated graph & (or distribution over G) in Eq. (4)
3.1 PROBABILISTIC IMPLEMENTATION OF ACD
We take a probabilistic approach to ACD and model the functions fo and fo using variational inference (Fig. 2) We amortize the encoder fo with a function ( æ)) which outputs distribution over z representing the predicted edges & in the causal graph; and learn a decoder po (ar 2) which probabilistically models the dynamics of the time-series under the predicted causal relations. We choose a negative log-likelihood for the decoder loss and KL--Divergence to prior distribution over G for the regularizer r. As result, our loss function L is a variational lower bound:
L (x)aa) [log po (z]=)) KI. (=
(8)
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of... 2) sj...aj
(z|2z)
(2) (2)
Figure 2: A Probabilistic Implementation of ACD. An amortized encoder (o (x)x)) predicts the causal relations between the input time-series æ. A decoder pe (a [=]=] learns to predict the next time-step of the time-series x++1 given their current values x' and the predicted relations z. This separation between causal relation prediction and modeling lets us train the model across samples with different underlying causal graphs but shared dynamics.
Encoder The encoder qo x)) applies graph neural network fenc. (Gilmer al. 2017 Kipf and Welling 2017 et al. (2066] Scarselli et al. 20088 to the input, which propagates information across fully connected graph [V,E). This graph includes vertices os E V for each time-series i, and each pair of vertices (v%, v2) connected by edge.
tb fenc, (ar)s
(9)
1s (Zig = So ftmax (zhi) /T)
(10)
To enable backpropagate through the samples of the discrete distribution go (2, during training, we relax by adding Gumbel distributed noise (Jang et al. 2017 Maddison et al. 201
Softmax ((bi + 9
(11)
The output Zij of the encoder represents the predicted edges ÉEs in the causal graph G We consider the possibility that there are ne different edge-types expressing causal relationships; for instance, inhibitory excitatory synaptic connections. Then, more specifically, Zij.e expresses that there a directed edge of type e from time-series to j. where E (0,..., - 1}.
Decoder The decoder (arl))) models the dynamics of the time-series under the predicted causal relations. It uses both the predicted causal relations žij and the feature vectors of the time-series at the current tine-step2 x' = {xf....x'' } as its input. First, it propagates information along the predicted edges by applying a neural network fe, using the zero function for fo:
h
2ig.efa( x(x),
(12)
e>0
Then, the decoder accumulates the incoming messages each node and applies neural network fu to predict the change between the current and the next time-step:
1 x²/ + fo
Enig
Σ iifi
hlg z
(13)
po( x², 2 2) N( ,0?1)
(14)
In other words, the decoder predicts Ad', which is added to the current value of the time-series to yield the prediction for the next time-step a++l x' + A.i".
Prediction of Causal Relations In order to align our model with the philosophy of Granger Causality, we include "no edge"-type edge function: If the encoder predicts the "no edge"-
edge = 0 by setting 2ij.0 1, the decoder uses the zero function and no information is propagated from time-series to (Eq. 12 Due to this, time-series will Granger cause the decoder-predicted
time-series only when the edge is predicted exist (see Appendix A). Hence, by the same logic that justifies prior Granger causal work (Section 2 we expect the predicted edges to correspond to Granger causal relations. Finally, since assume no hidden confounders and no instantaneous edges, these Granger causal relations will correspond to relations in the underlying SCM (Peters et al. 2017 Theorem 10.3).

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3.2 HIDDEN CONFOUNDING
Hidden confounders causality is not guaranteed to correspond to the true causal graph anymore (Peters et al. 2017 Theorem 10.3 Inspired by proxy-based methods from causal inference (c.g. Louizos al. 2017).
critical problem in the time-series context: when they exist, Granger
see Section 4) we present method for applying ACD to the hidden confounding setting. First, we extend the amortized encoder (o(2)x;) to predict an additional variable. Then, we encourage this variable to model the hidden confounder by applying a structural bias depending on the type of unobserved variable that we want to model, its predicted value is utilized differently by the remaining model. The decoder remains responsible for modeling the dynamics, and now also processes the predictions for the unobserved variable. While this setup might not allow us to identify the hidden confounders, the data-driven approach underlying ACD can benefit our model: by pooling the statistical strength across samples with different underlying causal graphs, our model can learn to mitigate the effects of the hidden confounders.
We consider two types of hidden confounders: 1) a temperature variable that confounds all endogenous variables by influencing the strength of their causal relations. This temperature is sampled separately for each sample, and remains constant throughout each sample. 2) hidden variable that behaves just like the observed variables, i.c. may affect or be affected by the observed variables through the same causal relations, and its value changes across time. In both cases, we extend the encoder to predict this hidden variable, and feed that prediction into the decoder. We provide more details in Section 5.2
RELATED WORK
range of approaches causal discovery in both temporal and non-temporal data exist (Heinze-Demil eet al. 2018 Peters et al. 2017 Spirtes et al. (000). One common class is constraint- relying on conditional independence testing to uncover an underlying DAG structure or equivalence class (Spirees al. 2000). These methods predict a single graph 9 (or equivalence class) for all samples. There is no notion of fitting dynamics model for time-series methods in this class (Entner and Hoyer 2010). Another common class of methods for causal discovery score-based (Bengio et al., 2019 Chickering 2002). Here, score function h is chosen, and the methods perform a search through graph space to optimize this score, i.c. 9 = argming f(()). Our proposed decoder-based inference (Eq. (7) can be seen as score-based causal discovery with a learned score function L fe. A third class of methods fits (possibly regularized) dynamics model and then analyzes its form to produce a causal graph estimate, by using linear dynamics (Hyvärinen et al.) 2010)), recurrent models (Khanna and Tan| [020 Nauta al.
2019
Tank al. (018), or other deep-learning based 2020: Zheng et al. 2020). See Section 2 for discussion.
approaches (Lachapelle et al. 2019 Wu et al.
Other approaches causal discovery in temporal data use independence or additivity assumptions (Eichler 2012, Peters et al. 2013). A number of works have explored the idea of jointly learned causal structure across examples, including range of papers in the setting where number of related datasets are collected, possibly with different columns (Dhir and Lee, 2020 (Huan et al. 2019 2020 Shimizu 20122 Tillman and Eberhardi, (2014). et ( (2018) proposes learning linear mixed effects model across samples, and concurrent work explores amortized deep learning of differing types of causal structure (Ke et al., 22020] Li et al., 2020). Very few papers systematically study the hidden confounding setting. Some empirical work shows that encoder- models with enough proxies (variables caused by hidden confounders) can improve causal inference under hidden confounding (Louizos et al.. 2017 Parbhoo et al. 2020), and theoretical work proves the identifiability of latent variables from proxies under some assumptions (Allman et al.
2009
Kruskal
1977).
Several works have used graph neural networks (Battaglia et al. 2017) or attention mechanisms (Fuchs et al. (0099/Goyaa et al 2011 Van Steenkiste et al.) 2018
2016
Kipf et al. 2018 Santoro et al.
Vaswani et al. 20177) infer relations between time-series. Alet et al. (2019) propose meta-learning algorithm to additionally model unobserved variables. While these approaches model object relations in number of ways, they are not explicitly designed infer causal graphical structure.
For instance, if an unobserved time-series U causes both time-series X and Y, then the past of X can help predict the future of Y, even though there causal link between them.

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1.0
1,0
(Enc)
0.9
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0.8
AUROC
0.8
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0.7
0.7
NGC
ESRU
0.6
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Figure 3: Causal discovery performance (in AUROC) on the particles dataset (A-left) and Kuramoto (B-right). ACD improves with more training data, outperforming previous approaches with as few as 50 available training samples on Kuramoto. In the high-data regime, encoder inference (Enc) is best, while test-time adaptation (TTA and Enc+TTA) is superior in low-data settings.
The probabilistic implementation of ACD is based the Neural Relational Inference (NRT) model (Kipf et al. 2018), extended by a special zero-function to allow for a causal interpretation of the inferred edges, new inference methods using test-time adaptation and new algorithm handle confounders. Moreover, the new model is applied to a different problem than NRI, namely (Granger) causal discovery, and we show that outperforms the current state of the for this type problem.
EXPERIMENTS
Implementation We measure causal discovery performance by area under the receiver operator curve (AUROC) of predicted edge probabilities over test samples. We compare to recurrent models (Khanna and Tan (2020); Tank et al. (2018)), and a mutual-information (MI) based model by Wu et al. (2020) and several baselines implemented by those authors, including MI (unmodified), transfer entropy (Schreiber 2000), and linear Granger causality. More details in Appendix C
5.1 FULLY OBSERVED AMORTIZED CAUSAL DISCOVERY
We test ACD on three datasets: two fully-observed physics simulations (Kuramoto and Particles) and the Netsim dataset of simulated fMRI data (Smith et al. 201T). Note, contrast to the physics simulations used in Kipf et al. (2018), we generate data with asymmetric connectivity matrices to represent causal relations.
First, we test our method on the Kuramoto dataset, which contains five 1-D time-series of phase-coupled oscillators (Kuramoto 1975). We find that ACD greatly outperforms all ap- proaches for Granger causal discovery that we compare against (Table T In contrast to these approaches, ACD achieves this result without fitting to the test samples. Additionally, we find that ACD can indeed utilize samples with differ- ent underlying causal graphs its performance improves steadily with increasing training data size (Fig. 3) Nonetheless, it is also applicable to the low-data regime: when applying ACD with test-time adaptation (TTA), it requires less than 50 training samples to outperform all previous approaches. We note that the baseline performance here is worse than presented elsewhere in the literature this is because evaluate accuracy without considering prediction of celf-connectivity, as celf-connections are the easiest type to predict.
Method
AUROC
MPIR Wu et al. 2020
0.502 + 0.006
Transfer Entropy (Schreiber
2000
0,560 = 0.005
NGC (Tank èt al. 2018
0.574 + 0.018
eSRU (Khanna and Tan
2020
0.607 + 0.001
Mutual Information
0.616 + 0.000
Linear Granger Causality
0.647 + 0.003
Amortized Causal Discovery
0.952 + 0.003
Table 1: AUROC for causal discovery on Kuramoto dataset. 95% confidence interval shown.
In our second experiment, we apply ACD the particles dataset. This dataset models five particles that move around wwo-dimensional space, with some particles influencing others mii-irectiooally by pulling them with spring. Since all previous methods were intended for one-dim mi-diinesionna time-series,
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0.95
1.00
Latent
0.90
AUROC
AUROC
0.95
Latent
0.85
Observed
0.90
Supervised
0.80
None
0.85
Mcan
0.75
0
Alpha
#time-serles influenced by unobserved one Figure 5: AUROC with unobserved time- series. As more time-series are influenced by the unobserved one (x-axis), the benefit of using additional latent variable for model- ing its effects grows.
Figure 4: AUROC with unobserved tempera- ture. ACD with latent variable outperforms a baseline which imputes a mean tempera- ture, and learned Exed-temperatur decoder (None).
we were unable to evaluate them in this domain. ACD, on the other hand, readily applicable to higher-dimensional data, and performs almost perfectly on this dataset with 0.999 AUROC.
In both experiments, causal relation prediction with the learned encoder (Enc Eq. 6 performs best in the high-data regime, while test-time adaptation (TTA Eq. (7)) improves the performance in low-data settings (Fig. 33. This benefit of TTA can be largely attributed to two effects. First, TTA closes the amortization gap of the encoder (Cremer al. 2018). Second, TTA overcomes the encoder's s overfitting on the training data (as seen the training curves in Appendix C.2) by being adapted the individual test samples. On the particles dataset, initializing TTA with the encoder's prediction TEnc+TTAI improves over random initialization (TTA) as the encoder improves; but we do not observe this effect on the Kuramoto dataset.
Finally, we apply ACD to the Netsim dataset (Smith al. 2011) of simulated fMRI data. Here, the task is to infer the underlying con- nectivity between
Method
AUROC
MPIR (Wu 2020
0.484 1 0.017
brain regions across 50 samples. A single graph underlies all samples, allowing us to demonstrate ACD's s applicability to the classical setting, We replace the amor- tized encoder (z x)) with a global latent dis- tribution q( optimize it through the decoder, and then use test-time adaptation (TTA). Even though model cannot benefit from its data- driven design here, it performs comparably to methods that are intended for use in the single- graph setting (Table 2
Transfer Entropy (Schreiber
2000
0.543 1 0.003
NGC (Tank ct al. (2018
0.624 + 0.020
cSRU (Khanna and Tan 2020)
0.670 + 0.015
Mutual Intormation
0.728 + 0.002
Linear Granger Causality
0.503 1: 0.004
Amortized Causal Discovery
0.688 + 0.051
Table 2: AUROC for causal discovery on Netsim dataset. 95% confidence interval shown.
5.2
AMORTIZED CAUSAL DISCOVERY UNDER HIDDEN CONFOUNDING
5.2. LATENT TEMPERATURE
In this experiment, we use the particles dataset and vary unobserved temperature variable, which modulates how strongly the particles exert force on each other higher temperatures result in stronger forces and more chaotic system. For each x., sample an independent temperature c ~ Categorical(( a, 2ox) from a categorical distribution with a E R and equal probabilities. We predict this unobserved temperature by extending the amortized encoder with additional latent variable which models a uniform distribution. Then, we add a KL-Divergence between this posterior and uniform prior the interval [0, 4aj our variational loss. To allow for learning in this setting, we introduce an inductive bias: we use decoder which matches the true dynamics given the predicted temperature and causal relations. See Appendix D.I for more details and additional results.
Results Fig. 4 shows the causal discovery results across different values of a. ACD enhanced with an additional latent variable (Latent) outperforms both tested baselines across all temperatures: Mean, which uses the same ground-truth decoder as Latent and fixes the decoder temperature to be
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the mean of the categorical distribution, and None, which does not model explicitly and trains MLP decoder. Additionally, this method achieves high predictive performance on the unobserved temperature variable: for 2, temperature prediction obtains 0.888 R 0.966 AUROC and 0.644 accuracy. These results indicate that we can model an unobserved temperature variable, and thus improve robustness under hidden confounding.
5.2.2 UNOBSERVED TIME-SERIES
Here, we treat one of the original time-series in the particles dataset as unobserved. It exhibits the same dynamics the observed time-series, evolving and causally influencing others the same way before. This challenging setting has received little attention the literature; Alet al. (2019) tackled it with mixed suc- cess. We model the unobserved time-series by extending the amortized encoder with an additional latent variable and applying a suit- able structural bias: the latent prediction z,, for time-steps t (1...., T) is treated in the same way the observed time-series æ. Its entire tra- jectory is used by the encoder to predict causal relations, and its value at the current time-step is fed into the decoder. See Appendix D.2 for more details and additional results.
Figure 6: Trajectory prediction with unobserved time-series (TS). Faded: ground truth. Bold: pre- diction, starts after observing the first half of the ground truth. Dots denote end of TS. Top: ACD with Latent, bottom: None baseline does not model unobserved TS. Left: unobserved TS, middle: TS directly influenced by unobserved, right: remaining TS. Though we underestimate the unobserved TS, observed TS prediction improves.
Results Fig. 5 shows how the causal discov- ery AUROC depends on the number of ob- served time-series directly influenced by the unobserved one. When this number is zero, all tested approaches perform the same. With growing numbers of influenced time-series, the baselines that either ignore the missing time-series (None) impute its value with the average of the observed time-series over time (Mean) deteriorate strongly. In contrast, the proposed ACD with Latent variable stays closer the performance of the fully Ob- served baseline. As shown in Fig. 66 also improves the future trajectory prediction of the observed time-series. A Supervised baseline that uses the (usually unavailable) ground-truth trajectories op- timize the prediction of the unobserved time-series, improves only slightly over our approach. These results indicate that ACD can use latent variables improve robustness to unobserved time-series.
6 CONCLUSION
In this paper, introduce ACD, framework for causal discovery in time-series data which can leverage the information that is shared across samples. We provide probabilistic implementation and demonstrate significant performance gains when predicting causal relations, even under hidden confounding. Exciting future directions include interventions, more flexible graph structures, or methods that adapt dynamically the type of hidden confounder hand.
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12


Reviews:
[{"review_id": "lH959uMqOtu", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors proposed a framework called Amortized Causal Discovery (ACD) for recovering causal relationships in time series where samples are generated from models with different underlying causal graphs but shared dynamics. This framework is applicable in settings such as modeling neural spiking trains where the dynamics of how neurons react to the activities of other neurons remain the same. In the proposed framework, they considered a causal discovery encoder $f_{\\theta}$, which tries to extract causal relationships and map it to a latent space. Moreover, there is a dynamic decoder $f_{\\phi}$, which provides one-step predictions. The proposed architecture is similar to a variational auto-encoder and the encoder part is based on graph neural networks.\n\nAlthough experimental results show that a considerable improvement in some dataset (even in the presence of a latent variable), there are some concerns about the contributions of this work and also its presentation:\n\n1- About the two main assumptions: The authors considered two assumptions on the causal model of time series (no instantaneous effect and invariant causal graph in time). We know that both assumptions are not satisfied in many practical applications. For instance, we have instantaneous effects if the sampling rate is low. Moreover, most causal mechanisms are changing in time.  As an example, in (Pfister et al., 2019), this problem has been addressed by detecting change points. I am not sure whether that approach is promising or not but it is expected from an experimental work to handle a more general setting.\n\n2- About the shared dynamic: Is it possible to check whether the assumption of shared dynamic is valid in the observed data? What if there are some small changes in the dynamics across different training samples.\n\n3- In Eq. (12-14), it seems that we are using the values of other time series at only time $t$ to give a one-step prediction of a particular time series. Why do not we consider the history of previous values from time $1$ to $t$?\n\n4- About handling latent confounders: As noted by the authors, recovering causal relationships is challenging in the presence of latent variables. Although experimental results show that the proposed method has better performance than baselines, it is still unclear whether there are some theoretical guarantees in recovering the correct causal graph using the proposed method.\n\n5- The presentation of the paper could be improved:\n*Please provide the exact definition of $\\hat{\\mathcal{G}}_x$, $\\mathbb{G}$, $\\mathbb{X}$, $r$,....\n*Please explain the two methods (the amortized encoder and TTA) in more detail on page 4.\n\n(Pfister et al., 2019) Pfister, Niklas, Peter Bühlmann, and Jonas Peters. \"Invariant causal prediction for sequential data.\" Journal of the American Statistical Association 114.527 (2019): 1264-1276.", "extra_scores": {}, "extra_text": {}}, {"review_id": "QYQca3YOlC4", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper makes an observation that signal dynamics common to a class of causal systems may contain strong information to enable the use of the encoder of the Neural Relational Inference (2018) for extracting (Granger) causal graphs. It minimally extends the NRI model with an empty edge type and demonstrates that the observation holds in a few cases of dynamical systems. Additionally, to handle the unobserved common causes the paper shows improvements in ROCAUC when the encoder is modified to model them directly.\n\nThe paper discusses a potentially interesting use of the Neural Relational Inference for (Granger) causal discovery in multivariate time series. Although lacking in detail, it is very clearly written. The proposed method of causal discovery for Markov order one system does demonstrate impressive performance on smooth dynamical systems.\n\nPossibly, the greatest weakness of the NRI approach in this formulation is the seeming reliance on relatively smooth noiseless dynamics. While for the NRI in the original paper (2018) it was within the proposed scope, the claimed use of NRI in this paper is not fully supported by experiments. All of them are from a smooth dynamical system. The Particles experiment is no different from NRI (2018) with the same reported AUC result. Even the most realistic dynamical system from Smith et al. 2011, which is also quite simplistic by the way, still model hemodynamic lag and thus a relatively smooth signal. Notably, this last less smooth of all presented signal is where the NRI model performs the worst.\n\n-   How would NRI in this proposed application behave if we use a still simplistic but potentially less smooth VAR model? Would be good to test it with different SNR.\n-   In general, even in dynamical systems used for method demonstration, what happens at various noise levels?\n-   How would it behave when the assumption of the model is slightly violated and the signal is generated by an SVAR model?\n\nGiven the above limitations, it is unclear if the paper presents something substantially novel/interesting. Technically it is still the NRI model of 2018, so not much contribution there. That would not be a problem if the paper would manage to indeed demonstrate that the model is useful for causal discovery in multivariate time-series. The experiments are limited to very specific kinds of signals and thus not convincing of the approach generality.\n\nIt is not immediately clear from preliminaries, whether the proposed model is capable to recover relations of a Markov order higher than one. The described \"summary graph\" as a compressed representation only holds true to the Markov order one underlying dynamic Bayesian network-like structure (see D. Danks and S. Plis. Learning causal structure from undersampled time series. In NIPS 2013 Workshop on Causality, 2013.), yet the description of the model is a bit vague and may indicate that the method works with the full unrolled graph and then folds it into Markov order one. This part, despite the extensive use of notation, is not clear. It would be simpler to say that indeed an unrolled representation of a graph of limited Markov order is used. It is not clear though if there are any restrictions on the Markov order as discussed since the encoded graph holds just Markov order one.\n\nWhile the problem of unobserved common causes is discussed, only a single type is considered - fully missing random variables (or particles). Yet, a simple mismatch in measurement and causal time scales will violate some of the assumptions (e.g. generate samples from the VAR model and drop every other time sample for all variables and the model can be fit with SVAR and not VAR). See:\n\n-   M. Gong, K. Zhang, B. Schoelkopf, D. Tao, and P. Geiger. Discovering temporal causal relations from subsampled data. In Proc. ICML, volume 37 of JMLR W&CP, pages 1898–1906. JMLR.org, 2015\n-   S. Plis, D. Danks, C. Freeman, and V. Calhoun. Rate-agnostic (causal) structure learning. In Proc. NIPS, pages 3285–3293. Curran Associates, Inc., 2015\n\nFrom the paper it is unclear what encoder and decoder models were used and can be used to perform the experiments or use the approach. How does the graph enter the decoder together with a variable-length input? If the Decoder is MLP the input layer has a fixed size. Possibly, the MLP is fed one sample at a time with a vector representation of the encoded graph, but that contradicts the paper description. Generally unclear. Supplement mentions that the models are exactly the same as for the NRI paper, but this is still not enough information and does not make the paper self-contained.\n\nFrom the description of experiments it is unclear how many random variables were used and how many can the NRI handle.\n\nWhat was the experiment in Figure 3? How many particles/random variables? Was the training done on a single causal generative graph while testing performed on a variety of different unseen graphs? Was it a random sample of a large number of different graphs for both testing and training? Is NRI even capable of learning transferable information from data generated from a single graph?\n\n-   Note, reference P. Spirtes, C. N. Glymour, R. Scheines, and D. Heckerman. Causation, prediction, and search. MIT Press, 2000. - Heckerman is *not* a co-author of [this book](https://mitpress.mit.edu/books/causation-prediction-and-search-second-edition)\n-   Last paragraph of page 3 $x\\in \\mathbb{X}$, the $\\mathbb{X}$ is not defined.", "extra_scores": {}, "extra_text": {}}, {"review_id": "MC51xMN_aiC", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "# Introduction\n\nYou should have praise for your first sentence in your second paragraph. It is excellent. A neat summary indeed. Emphasise it if you can.\n\nRefer to figure one in paragraph one to allow the novice reader to get a quick understanding of what you are trying to do. Better yet, move figure one into the first page and put it at the bottom, it is a great eyecatcher.\n\n# Background\n\nN should be italicised in definition 2.1.\n\n# Amortized causal discovery\n\nPresumably like all work of this kind you assume, there are no cycles or self-cycles in the SCM. That is pretty common fare for causal work (and a huge limitation) but it should still be noted at the very top of this section or indeed expanded upon in the preliminaries. I take it thus that you cannot handle SCMs with self-cycles and cycles in the endogenous variables? A far harder problem of course, but just to be clear from this reviewer's point of view.\n\nWhat your second point is saying in the preliminaries is that you assume that your data is sampled from a stationary process. You may want to add that key-word as that says a lot to people from outside this field, and again, emphasises the limitations of this causal discovery (and most in fact) method. It also gives you plenty of scope for future work! Indeed you go on to talk about the summary graph which is by definition only defined for stationary processes.\n\nA common theme throughout this paper is that you just insert figures without referring to them beforehand. This is very confusing. Please stop doing this and re-write the paper so that your figures follow the discussion of them. You introduce an unnecessary break in the flow of the reading otherwise, like between pages 1 and 2 and 3 and 4. \n\nPlease explain what you mean by this: \"Note, that we do not apply a supervised loss on the graph prediction\"\n\nUse \\text{} in your subscripts in your equations, it reads better i.e. $\\hat{\\mathcal{G}}_{\\text{TTA}}$ is preferable.\n\nPlease explain what you mean by \"expresses that there is a directed edge of type e from time-series i to j\" -- what is type e?\n\n## Hidden confounding\n\nThis section could do with a diagram. I find it very confusing. I think you are saying that you allow the decoder to predict the additional variable (in the summary graph?) which confounds _all_ endogenous variables in the summary? Or some? or one? Which is it? What if the confounding variable does not affect all endogenous variables?\n\n# Related work\n\nSolid section. Well done.\n\n# Experiments\n\n## 5.1\n\nIt would be helpful if you could show the inferred adjacency matrix of the particle dataset, alongside the GT adjacency matrix.\n\nThese experiments feel very selective. Your goal is to demonstrate ACD's superiority over the other methods, granted, that's the process and we all know it. However, this does seem like an odd comparison. Take Khanna and Tan, 2020 (not 2019 like you've quoted it): on a Lorenz-96 simulation, they reach an average AUROC of 0.95. Yet here they get 0.607, and this ought to be because you have generated asymmetric connectivity matrices which renders a structure which eSRU cannot handle. But how we do know? You haven't run ACD on say the Lorenz-96 system which is pretty common fare in causal discovery. If your model does not outperform SOTA methods in that space you may have to relax some of your claims. What is more, Tank et al do very well on Lorenz-96 too. They are not particularly good at the DREAM-3IN SILICO NETWORK INFERENCE CHALLENGE, and that's where ACD could shine through, rather than taking these models 'out of context' as it were. Point being, please demonstrate ACD on the _simpler_ operating space where the likes of NGC and eSRU live. You appear to have started that process with Table 2 but that is a complex setting, perhaps too complex to understand where ACD is failing. Again, I advise starting simple with the standard Lorenz-96 model just to ensure that your method performs well in that easy space.\n\n# 5.1\n\nIt is not okay for you to bold ACD's performance value in table 2. Remove it. MI is clearly higher and putting anything else is bold is confusing.\n\n## 5.2.2\n\nThis experiment is very interesting. The figures are good too. My advice would be to make the axis text larger, and set alpha=0.75 or something when you plot the prediction -- that way the reader can better see how the prediction tracks the GT.\n\nHowever, I am a bit confused about the deterministic chaos that you briefly mention at the start of 5.2.1 -- the parameter setting you use in this section (5.2.2) presumably does not give rise to deterministic chaos?\n\n---\n\nNone of your experiments contains fully identifiable noise models from what I can tell? You mention \"noise\" twice in the manuscript. None is in the experimental section. Now since you are only using simulations, I find it odd that you do not discuss this nor ACD's robustness against noise. As you are fully aware of course, most of these models break-down in the presence of a low SNR. And for this to be useful at all in the 'real' world, how does ACD fare?\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "9m1jKWy51v-", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper studies the problem of learning causal graphs from time-series data. A new framework, called Amortized Causal Discovery (ACD), is proposed based on the observation that samples often share identical or similar dynamics. ACD is then implemented with a variational model, mostly following existing works. Overall, I think it is a very interesting idea to consider shared dynamics and, in some sense, to 'learn a score function' from data, compared with existing score-based methods that usually need to put assumptions on data generations to make the score function sound. However, there is also a key problem, identifibility and consistency, to ACD, for which I do not recommend an acceptance for now. I would be happy to increase my score if authors can address my concerns. \n \nIdentifiability and Consistency: (a) identifiability is an important problem to causal discovery from observational data. I find that only Peters et al, 2013 is cited for such an issue. However, in that work, the identifibility result requires a much stronger condition than the assumption used in this paper. It would be good to discuss this issue in more details. (b) as stated in Section, 4, the proposed approach can be seen a score based method with learned score functions. However, it is unclear when this score function could be 'consistent', i.e., maximizing the score function would lead to a consistent estimate of the true graph (possibly under some model assumptions). (c) Another part, related to this learned score function, is whether the overall approach is consistent. In particular, Eq. (5) has a regularization term (which is often a sparsity term, as stated in the paper). The question is how to pick this penalty term, including its weight, in practice? This is crucial for causal discovery and I do not find that this issue is explained in the current version. \n \nWriting: the overall writing is very good, but could be improved for some places. (a) Eq (12) uses $z_{ij, 0}$ which was not introduced until the next paragraph; (b) it seems that the problem size (how many variables of the problem) is small. It would be good to introduce the problem size explicitly in the main content. (c) as mentioned above, the identifiably and regularization term are not clearly discussed; they are however important to score based causal discovery.\n \nOther questions: (a) why the BIC score used by Chickering 2012 is called 'heuristic score'? (b) Gumbel technique may not give exactly zero or one, so how was the 'no edge' type processed? This part was somewhat omitted. (c) what if there are instantaneous effects? I do not see why ACD does not apply to this case.\n\n*********after reading rebuttal *******\nI have increased my rating, but may still have some concerns. See below.\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes an interesting approach that leverages shared dynamics across causal systems for improved joint causal discovery.  The reviewers and AC all agree that the approach is interesting, promising and that the paper is well written. 

While theoretical validation would be an exciting thing to have, it is perfectly acceptable for the paper to focus on an empirical study. But in this case, it is very important to provide convincing evaluation experiments.  As several reviewers have pointed out, in order to convincingly demonstrate the value of the approach, it would be very important for the experiments to go beyond noiseless systems.  We strongly encourage the authors to address this point as this will significantly strengthen the significance of their contributions.
Decision: Reject