From: cclear.cc/CLeaR/2022/Conference/Program_Chairs
title: Paper Decision
decision: Accept (Poster)
comment: This paper tackles an interesting but specialized problem in causal discovery for multiple (or multivariate) time series.  The _exact_ nature of the problem set-up is not at all clear from the paper unless it is read _very_ carefully, and understandably confused at least one reviewer, so if this is accepted the final version should have a significantly revised abstract and introduction.

The setting, as I understand it, is as follows.  We have multiple data sets, each of which consists of $N$ univariate time series, all of length $T$, and assumed to be regularly, and simultaneously, observed.  Each data set is coming from an independent system.  (These systems or data sets are what the MS. confusingly calls "samples".)  Not only are they statistically independent, but the graph of effective connectivity among the $N$ univariate processes is different from one system to another.  However, there is a common functional form, shared across all the systems, for how the future of each of the $N$ nodes is generated from its own past and the past of its neighbors.  This differs from system to system only up to a finite-dimensional set of parameters, which represents graph structure as well as, e.g., connection strength.  Thus for instance we might be looking at neuronal firing rates, and saying that $x_i^t = \mathrm{logit}^{-1}\left(\alpha + \sum_{j \neq i}{w_{ji} x_j^{t-1}}\right)$, where the assumption of a common functional form is showing up in the additivity and the inverse-logit parts.  (This isn't a great model of neuronal response but you get the idea.)
As the authors say in their replies to referees, and as the manuscript hints, an obvious application of this would be to multi-electrode-array neuronal recordings, where each data set from a different animal would be recording different neurons, with a different graph, but one might hope for a common neuronal response mechanism across experimental subjects.  (I can't come up with a second convincing application, especially not with fixed $N$ across "samples", which seems important to the encoding/decoding step.)

The innovation in this paper is to separate learning the common functional form of the vector autoregression from learning the graph, thereby allowing for pooling of information about the shared part of the model across data sets.  The reports agree that this is cleverly done, though it is not entirely clear what the limits on the expressive power of this method are, nor under what conditions it will converge on either the correct graph or the correct functional form.  Nonetheless, this is original and innovative work, and while text needs to clarify the intended application, the authors' replies to reviews make me fairly confident this can be done.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Reviewer_5AfY
title: Thanks for the discussion
comment: I would thank the authors for the discussion. After reading the other reviews and authors' responses, I believe the submission can be a good fit for the conference and bring certain interesting points after modifying the paper respectively. So I would suggest an acceptance.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Authors
title: Response on decoder structure
comment: We would like to thank the reviewer for their fast reply and the clarification. 

Indeed, equations (12-14) describe one specific way of implementing a decoding function $x_j^{t+1} = f(x_1^t, ..., x_N^t, z)$, but we believe that it is almost the most general one we could apply here. 

Overall, these equations follow the setup of a general graph neural network (GNN). Equation 12 describes an edge-propagation step, which allows different time-series to exchange information with one another and pushes the network to learn about the (causal) interactions between them. Equation 13 implements a node-propagation step, in which all incoming messages to a node are accumulated to determine the overall influence of other time-series on the current one. Both of these functions could be implemented with a wide range of neural networks. For example, we could replace the MLPs used in our experiments by recurrent neural networks which would enable the model to learn about causal relations of higher Markov order (i.e. with lag) as well.

Equations (12,13) contain three restrictions to this very general GNN setup:

1) As pointed out by the reviewer, in equation 13 we assume that the current step of a time-series is always a direct cause of its next step. As explained before, this assumption could be removed easily and is solely made to make training easier (in a similar way as residual connections do in other networks).

2) In Equation 12, we ensure that time-series can only propagate information along their edge, if this edge was predicted to exist by the encoder (i.e. if $z_{ij,e} > 0$ for some $e>0$). This restriction is a hard requirement for our results to be interpretable as Granger causal relations, but does not actually restrict the flexibility of the decoder.

3) In our decoder, we apply only a single node-edge propagation step, while in a general GNN one might employ several of these steps (essentially repeating equations (12,13)) to allow for higher-order interactions between nodes to be modeled. We believe that it might be possible to extend our decoder accordingly, but that similar effects might be achievable more easily in our setting with time-series by using recurrent neural networks.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Reviewer_5AfY
title: A further discussion about the flexibility of the decoder using Eqn. (12-14)
comment: As shown in Eqn. (12-14), let's put aside the probabilistic formulation, for now, Eqn. (12-14) are used for modeling the aggregation/pooling process which collects the information at time $t$ and use the {$x_1^t,...,x_N^t$} for predicting $x_j^{t+1}$. If we write it in the form of function, it will be $x_j^{t+1} = f(x_1^t,...,x_N^t)$. And Eqn. (12-14) is one specific way (or a model choice) to implement such a process. I was trying to summarize some characteristics of it in the review, but essentially it takes the sum form for each variable pair in Eqn. (12) and the sum of $h$ in Eqn. (13).

Then the question is that how flexible is the designed pooling/aggregation process with Eqn. (12-14) or how it can be justified to be a suitable/proper one in general?
It seems that (3) can be a reason to consider such question/point, e.g., why it requires an explicit modeling choice for the self-edges.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Reviewer_5AfY
title: Thanks for the response to my concerns and questions ! 
comment: I would thank the authors for the effort in the response. 
(1) It would be very good to include this example; (2) Yes, the factorization fixed the ambiguity issue very well; (3) It sounds pretty good and it could be beneficial for the readers if including such discussion somewhere in the paper; (4) I think such experimental results and discussion would be helpful and useful for the causal discovery field as well. (5) I will elaborate on it in another comment window.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Authors
title: Reply to Reviewer 5P7m
comment: We would like to thank the reviewer for their feedback. We will first clarify some aspects around ACD before tackling the individual points raised by the reviewer.

To better understand the motivation underlying ACD as well as the novelty in its problem set, we would like to outline an example application. Suppose we want to infer the synaptic connections (i.e. Granger causal relations) between neurons based on their spiking behaviour. With existing causal discovery methods, we would take one sample consisting of $N$ time-series which represent the spike trains of $N$ neurons, and train a model on this sample to infer the underlying causal graph. If we were to encounter another sample which represents the spike trains of a different set of neurons (e.g. because the electrode measuring the spiking activity was moved), previous approaches would have to fit an entirely new model to this new sample. In our paper, we point out that this is rather wasteful, since the two samples share information -- namely the underlying dynamics of how neurons influence one another -- which is lost with this approach. We propose to make use of this shared information by training a model across samples. In the example above, this would allow us to train a single model on the two samples to infer the different causal graphs that are underlying them, while improving the model performance by learning about the causal dynamics in a shared, i.e. amortized approach.

**To the best of our knowledge, ACD is the first method that can take advantage of such shared dynamics and the first method to tackle this problem set.**

To reply to the points raised by the reviewer on “quality and clarity”:

(1) In our paper, we assume that the underlying causal graph changes across samples, but stays constant throughout an individual sample (and thus throughout time). We leave it to future work to also consider the case in which the underlying causal graph changes across time as well. 

(2) Besides the description in section 3, we provide details on the implementation of our method to improve the robustness against hidden confounders in section 5.2, and explain further in Appendix C. Additionally, we will make our code available with the camera-ready version of our paper. If the reviewer has any further questions regarding the implementation we are happy to answer them here.

(3) To explain why ACD is more robust against hidden confounding, we would like to reuse our example from above. In a recording of neural spike trains, it is nearly impossible to record all neurons that are causally influencing one another. As a result, hidden confounders -- for instance, neurons that are not part of the recording, but still influence the neurons we observe -- will always be present in this setting. As described above, previous approaches would retrain a new model for each new recording (i.e. sample) from a different neural sub-population. However, similarly to the dynamics that describe the way in which neurons causally influence one another, the hidden confounders will behave in a similar way across samples. By training a single model across samples, ACD can make use of this shared information and learn to become more robust against hidden confounders. 

We want to point out that our paper is the first to show improved robustness in this setting empirically. We leave theoretical explorations to future work. 

(4) To the best of our knowledge, all compared methods (including ACD) are capable of inferring causal summary graphs with cycles.

Finally, on “originality and significance”: the reviewer notes that the paper “lacks the details of the method and the theoretical explanation”. If the reviewer can give more details about what specifically is missing from the paper, we are happy to answer their questions here. In response to the specific point raised about summary graphs:

(5) We agree that summary graphs contain less information than full-time graphs and that inferring the latter would therefore be more desirable. However, the aim in our paper is different: we want to demonstrate that we can use the information that is shared across samples with **different underlying causal graphs**. Our results show that this allows us to substantially outperform previous causal discovery methods for time-series, all of which also focus on summary graphs. As a result, we consider it an interesting path for future work to extend ACD (as well as previous nonlinear Granger causality approaches) to infer full-time graphs.
The referenced theorem 10.3 in Peters 2017 applies to summary graphs, specifically stating that edges in the summary graph can be inferred using Granger causal reasoning if no hidden confounders or instantaneous connections exist.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Authors
title: Reply to Reviewer U1gF
comment: We thank the reviewer for the helpful feedback. Here, we hope to clarify some of the questions around the strengths and weaknesses of our method.

_The nature of this problem should have been better justified_: We are not totally sure of the type of justification the reviewer is seeking (some clarification would be helpful), however we will explicate some of the ideas here. We consider a two-level data generating process, where first a causal graph is sampled from some meta-distribution over causal graphs, and then a trajectory is sampled from that causal graph. To generate $S$ samples, we run this two-step process $S$ times. For instance, we can imagine that each sample represents the spike trains of $N$ neurons. The connectivity between the subpopulation of neurons underlying each sample is unique, and drawn from some broad distribution over possible connectivity graphs. Then, the neural spiking data is generated from that given connectivity graph. We hope this provides the justification the reviewer is seeking for our setup - if not, we welcome further inquiry on this topic.

_The relevance of this method depends very much on the assumption of "shared dynamics"_: We agree that testing the assumptions which drive causal inference is an important research area for practical trustworthiness. However, we note that causal inference tends to be built on a whole host of unverifiable assumptions, such as ignorability, monotonicity or consistency; these assumptions are usually supported by domain knowledge rather than observed data. Further, we suggest that the assumption of shared dynamics is in line with our intuitions for a number of domains: for instance, many samples from a physical system will share the same laws of motion, even if their causal relationships differ.

_​I expected more details about the encoder and decoder function_: In Appendix B.1.2 we provide more detailed information about the exact architecture we use for our encoder and decoder, and in Appendix C we give even more details about our implementations in the unobserved confounding settings. Our code will also be made available after the deadline. If the reviewer has any questions clarifying the role or structure of the encoder and decoder function we are happy to answer them here.

_Minor point in equation (2)_: Thank you for pointing this out, we have made this change.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Authors
title: Reply to Reviewer 5AfY
comment: We thank the reviewer for their overall positive feedback, and would like to use this rebuttal to reply to their raised concerns:

(1) We will include an example for what samples can represent in the preliminaries section for the camera-ready version. More concretely, we will describe that the $N$ time-series within sample $\mathbf{x}_s$ may represent the recording of the spiking behaviour of $N$ neurons. Then, the $S$ samples within a dataset $\mathbf{X} = \\{ \mathbf{x}_s \\}_\{s=1\}^S$ may represent the recordings of the spiking behaviour across different neural sub-populations. As a result, the dynamics of how the time-series influence one another, i.e. how neurons cause one another to spike, stays the same across the samples, but the underlying causal graphs differ, since we look at different sub-populations of neurons.

(2) We can make eq. (8) and eq. (13-14) consistent with one another by using the chain-rule of probabilities, i.e. by writing
$p(x|z) = p(x^0|z) * p(x^1|x^0, z) * p(x^2|x^1, z) * … = p(x^0|z) * \prod_{t=1}^T p(x^t|x^{t-1}, z)$. Then, $p(x^t|x^{t-1}, z)$ corresponds to the terms used in eq. (13-14) and we ignore the $p(x^0|z)$ term in our modelling.

(3) This is correct - we use the assumption that the current step of a time series is always a direct cause of its next step. This makes modelling easier in our case since the systems we consider contain items which maintain individual state between time steps (e.g. physical systems). We note that in evaluation, we do not include the assumed self-edges in AUROC estimation. This assumption could be removed within our implementation by removing all instances of $x^t_j$ from the right hand side in Eq (13).

(4) In terms of more standard causal discovery (CD) datasets, we note that we include experiments on the Netsim dataset, which has been explored in the CD literature. It contains one shared graph between all samples and is particularly challenging due to inherent noise. We show that here, despite relatively few training samples (~50) and a single underlying causal graph, ACD can match or improve on the performance of many baselines. Otherwise, we show that on more challenging simulations, ACD can make large gains over previous CD baselines.

(5) We are not quite sure we understand the question the reviewer has around the flexibility of the model - some clarification would be appreciated. We note that if we adjusted the decoder to the simplification (we think) the reviewer is proposing, we would lose the causal edge structure provided by the $z_{ij}$ edge variables. As discussed at the end of Sec 3.1, using these variables in the decoder is essential for establishing the correspondence between the output of the encoder and (Granger) causal relationships.

From: cclear.cc/CLeaR/2022/Conference/Paper11/Reviewer_5AfY
confidence: 4: The reviewer is confident but not absolutely certain that the evaluation is correct
Summary: An interesting problem formulation and proposed method for Granger causality 
Main review: # Summary 
The authors aim at solving an interesting problem for Granger causality, i.e., for different objects/environments/specific applications, the causal graphs can be different but the causal relationships/functional relationships/dynamics are in the same form, then how to infer the causal structures for different cases while utilizing the shared dynamics. 

They propose a probabilistic implementation for modeling the graph structure with an encoder and the shared dynamics with a decoder.
The experiments are based on simple physical synthetic data and investigate the influence of the unknown confounding, especially, the authors propose a solution for the case with unobserved time-series data.

# Pros
(A) **An interesting view to see the problem.**  
There are works in causal discovery focusing on solving the problem that the causal graph is shared in different environments while the causal relationships are in different forms. It is also important to consider the scenario where the causal graph is not shared. 
Although the ideal case would be that both of the causal graphs and the causal relationships are not fully shared, the work indeed provides a great value for the further development of causal discovery.

(B) **A well-written paper.**  
The paper is well written which makes the idea and method easy to be understood. Especially, first, introduce the model and high-level idea and then introduce the probabilistic implementation. 

(C) **Investigation for the presence of unknown confounding.**  
The work also provides an extension for the case where there are unobserved time series. The solution is straightforward, makes sense, and has reasonable results.


# Concerns / Questions  
(1) In the preliminaries, after a while, I understood what does the sample mean. I would suggest elaborating the concept sample with examples or further explanation. Different samples seem to be corresponding to different applications which can have different causal graphs.

(2) Eqn. (8) doesn't seem to be consistent with Eqn. (13-14). Because the decoder seems to output the next time value $x_{t+1}$ based on the current time value $x_t$, but Eqn. (8) is only conditioning on $z$ without indicating the actual process.

(3) Eqn. (13) use "hard code" to model $x_j^t$. It means that it must be the case $x_j^t$ is the direct cause of $x_j^{t+1}$; however, it may not always be the case. What if $x_j^t$ is not the direct cause?

(4) One minor concern: I think it would be interesting to see how the model performs on the data generated with (non-)linear autoregressive models (which are used more often in causal discovery) instead of physical synthetic data.

(5) It remains unclear for me how flexible is the model with its decoder (12-14) or how rich is the functional class? 
It seems like modelling the function with vector as input by assuming something which can be simplified as $f([x_1, ..., x_i, ... x_N]) = f_v(\sum_{i=1}^N f_e^i(x_i))$.

Overall score: 7: Good paper, accept

From: cclear.cc/CLeaR/2022/Conference/Paper11/Reviewer_5P7m
confidence: 2: The reviewer is willing to defend the evaluation, but it is quite likely that the reviewer did not understand central parts of the paper
Summary: This paper proposes Amortized Causal Discovery, a method for estimating causal relationships from time series data using neural networks, based on the idea of Granger causality.
Main review: # Quality and clarity
There seem to be many important points that are not explained in this paper.
- In this paper, although it says "we learn to infer causal relations across samples with different underlying causal graphs but shared dynamics" on page 1 and 3, it also says "Edges are invariant to time" on page 4. I have been confused when I read this paper because I first thought that the underlying causal mechanisms change over time. I still don't understand what each graph corresponds to.
- Section 3 lacks the details about the method to identify latent confounders, so the method is not likely reproducible.
- I don't understand why the authors' proposed method is robust even if there are hidden confounders. The details about the method and a theoretical explanation for it is needed.
- I wonder if the previous methods compared in this paper generates causal graphs with cycles. If not, the authors need to provide an explanation about the details about how they compared their method with the previous methods.

# Originality and significance
There is no novelty in the problem set addressed in this paper. This paper proposes a new method for an existing problem. However, it lacks the details of the method and the theoretical explanation to support the effectiveness of the method.
- I wonder why the authors propose to learn a causal graph in the form of aggregated time series causal graphs (summary graphs). The graphs inferred by the method proposed by Hyvärinen et al. (2010) contains more detailed information about causal relationships. In my opinion, the causal graphs proposed in this paper are inferior to the graphs generated by the methods proposed in existing research in terms of the detail of the information. The authors need to clarify this point. The authors refer to (Peters et al., 2017) but I could not find any descriptions about summary graphs in the book.

Overall score: 3: Clear rejection

From: cclear.cc/CLeaR/2022/Conference/Paper11/Reviewer_U1gF
confidence: 3: The reviewer is fairly confident that the evaluation is correct
Summary: The main contribution of the paper is the proposal of a new "framework" to infer causal relations across samples with different underlying causal graphs, but with a shared dynamics. The paper does not present a specific and explicit algorithm, but a probabilistic implementation of the framework with a good theoretical background and an application to simulated data.
Main review: Originality: 

The paper provides something new, namely it faces the problem, probably not yet well-addressed in the existing literature, that causal discovery methods may infer a different causal graph for each sample when the data are time series. The nature of this problem should have been better justified, from a statistical theoretical point of view. For example, is this due to non-ergodicity or non-stationarity of the underlying stochastic process? The idea of "amortization" across different causal graphs through the "encoding" and "decoding" function is very original. At least, it is new to me. Related works are correctly discussed.

Significance: 

The paper addresses surely an important problem, namely causal heterogeneity across samples, which are relevant to the CLeaR community. The proposed method significantly advances the state-of-the-art, although the relevance of this method depends very much on the assumption of "shared dynamics". It is difficult to assess whether this assumption holds or does not hold in empirical applications. (The authors present applications to simulated data only). The method proposed in the paper, if better motivated and justified, is likely to have some impact outside the CLeaR community, e.g. in neuroscience, finance, and perhaps economics. The paper hinges on the Granger causality framework, therefore it  may be more interesting for fields of studies with high frequency data.

Technical quality: 

The proposed approach is technically sound, but I expected more details about the encoder and decoder function. Minor point: in equation (2) the time index of the noise variable should probably be "t+1" instead of "t".

Clarity: 

The paper is clearly written and well-organised. The motivation of the paper could be improved. It should be clarified under which conditions we get different graphs from different samples  and which settings support the "shared dynamics" assumption. 

Overall assessment: nice contribution in terms of new method / framework, but the paper needs to be better motivated. It should be better clarified why it is interesting or even sound to consolidate causal discovery output over other samples.


Overall score: 7: Good paper, accept