Venue: CLeaR 2022 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: 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 $f([x_1, ..., x_i, ... x_N]) = f_v(\sum_{i=1}^N f_e^i(x_i))$.
review_title: null

final_score: rating: 3: Clear rejection
review_content: 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.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: 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.
review_title: null

Meta-review: 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.
Decision: Accept (Poster)