Causal Identification with Matrix Equations

Abstract: Causal effect identification is concerned with determining whether a causal effect is computable from a combination of qualitative assumptions about the underlying system (e.g., a causal graph) and distributions collected from this system. Many identification algorithms exclusively rely on graphical criteria made of a non-trivial combination of probability axioms, do-calculus, and refined c-factorization (e.g., Lee & Bareinboim, 2020). In a sequence of increasingly sophisticated results, it has been shown how proxy variables can be used to identify certain effects that would not be otherwise recoverable in challenging scenarios through solving matrix equations (e.g., Kuroki & Pearl, 2014; Miao et al., 2018). In this paper, we develop a new causal identification algorithm which utilizes both graphical criteria and matrix equations. Specifically, we first characterize the relationships between certain graphically-driven formulae and matrix multiplications. With such characterizations, we broaden the spectrum of proxy variable based identification conditions and further propose novel intermediary criteria based on the pseudoinverse of a matrix. Finally, we devise a causal effect identification algorithm, which accepts as input a collection of marginal, conditional, and interventional distributions, integrating enriched matrix-based criteria into a graphical identification approach.


Reviews and author responses:

 Thank you very much for the clarification. Your responses are very helpful.

 We appreciate you taking the time to read our rebuttal and reconsider our paper, thank you. 

Regarding the first questions, the set differences will be evaluated from left to right.  

 For example, let $\mathbf{A}=\\{1,2,3,4\\}$, $\mathbf{B}=\\{3,4\\}$, $\mathbf{C}=\\{2,3\\}$, then 

$\mathbf{A}\setminus \mathbf{B}\setminus \mathbf{C} = (\mathbf{A}\setminus\mathbf{B})\setminus \mathbf{C} = \\{1,2\\}\setminus \\{2,3\\} = \\{1\\}$.

We will ensure that in the preliminaries the order of precedence among set operators will be clearly explained, and use parentheses whenever needed. Thank you for the note. 

Regarding the question about Lines 81--82, our goal is to describe the types of a distribution $P_{\mathbf{Z}}(\mathbf{V}’|\mathbf{W})$ based on $\mathbf{Z}$, $\mathbf{W}$, and $\mathbf{V}’$, disjoint subsets of endogenous variables $\mathbf{V}$. 
 For concreteness, consider the general distribution $P_{\mathbf{Z}}(\mathbf{V}’|\mathbf{W})$. 
 
Whenever the set $\mathbf{Z} = \emptyset$, $P_{\mathbf{Z}}(\mathbf{V}’\mid\mathbf{W})$ becomes $P(\mathbf{V}’\mid\mathbf{W})$, which is called _observational_. 
If $\mathbf{W} = \emptyset$, then the distribution becomes $P_{\mathbf{Z}}(\mathbf{V}’)$, which we call _unconditional_. 
Also, if the distribution is $P_{\mathbf{Z}}(\mathbf{V}\setminus(\mathbf{W}\cup\mathbf{Z})\mid\mathbf{W})$, that is, $\mathbf{V}’=\mathbf{V}\setminus(\mathbf{W}\cup\mathbf{Z})$, then it is said to be _nonmarginal_. 
Furthermore, _conditional_, _interventional_ (_experimental_), and _marginal_ are used to emphasize that the constraints mentioned beforehand do not necessarily hold. That is, the distribution is _conditional_ if $\mathbf{W}\supseteq \emptyset$, _interventional_ or _experimental_ if $\mathbf{Z}\supseteq \emptyset$, and _marginal_ if $\mathbf{V}’\subseteq\mathbf{V}\setminus(\mathbf{W}\cup\mathbf{Z})$.

In the definition of marginal (Line 82), we incorrectly wrote $\subsetneq$ instead of $\subseteq$, sorry. We will improve the text around these definitions using the examples above.


 Thank you for answering many of my questions and for clarifying the notations used in the paper. Overall, the paper is very interesting and novel and can be very impactful whenever the mentioned issues are taken into account in the revised version. Therefore, I increase my score from 5 to 6. I hope you can include a running example in the main body of the paper and improve the notational readability. For example, it is still not clear to me what X_j \ X_i \ Y_i means? Aren't there two possibilities to read this, namely as ( X_j \ X_i ) \ Y_i or as X_j \ ( X_i \ Y_i )? Moreover, my question about (line 81-82) was actually that I didn't understand that whole sentence. For example what does it mean that Z is interventional and it is observational if Z is empty? And what does the marginal condition in the if statement mean? Does it relate somehow to the distribution P_Z(V' | W)?


This paper provides a graphical identification approach that combines both graphical and matrical approaches. In particular, they apply certain matrix chain-rule and inversion criteria, previously used in for example certain proxy-based approaches, to a matrix version of the C-factorization to arrive at a new causal effect identification algorithm. They show that their algorithm applies to cases that could not be identified with certain previous algorithms and thereby rendering them incomplete.
  The paper is quite well written and motivated. The overall storyline of the paper is clear, however I had the impression that the balance of the amount of material the authors would like to treat was suboptimal. For example, the authors revisit different proxy based approaches, which take a lot of space. I would move either the single-proxy or the double-proxy setting to the appendix, which gives additional space for, for example, a running example of the identification algorithm, which would have been helpful. Although the authors explain their main algorithm in Section 6, it is very difficult to understand the motivation of the steps they are taking without having any intuition on what is happening. This also would give more space for explaining C-factorization better and the example in that section. In particular, it was not clear to me how this example relates to C-factorization and what the actual point is that the authors are trying to make.

This leads me to another point, and that is, the notation. The conventions used, such as lower and upper case, back and forward slashes, accents and double accents and the different combination of those make this paper very hard to read, especially if their meaning change if applied in different combinations (such as thinking of / as "complement/setminus" or as a/B defined in line 161; and \ defined as being consistent or as "complement/setminus" in line 144?), or if there are typos present (such as in line 158 which needs to be a forward slash?; or line 164 where there should be no sum over z, I think). For example, what does (x_{ij} \ y_{ij}) / ( X_j \ X_i \ Y_i ) mean? Because of this notational overload I was unable to understand the results in all their detail and I was not be able verify any of the proofs of the paper. 

The approach and results of this paper seem interesting and valuable. However, given the current state of the paper I would give it a marginally below accept, due to the imbalanced amount of material covered and the notational overload of the paper. I think that improving the notation and trading of some material for clarity would be a good idea to improve the overall readability and impact of the paper.

Minor comments:
 - Lemma 3: Why the graph in the lemma?
 - Footnote 1: Put this in the main text: writing a distribution in matrix notation is very confusing if one does not know that the variables are discrete. Moreover, the MGT criterion is used in the paper many times without being defined in the main text.
 - line 81-82: What are W, Z and the union here?
 - line 91: State that it is acyclic somewhere? Or, not?
 - line 299: Here, $X^+_k$ and $Y^+_i$ have an ambiguous definition. Does this relate to the C-factor $Y^+$ definition?

Update:
After reading the response from the authors, I feel they have clarified my main concerns. I'm updating my score.
 They clearly discuss the limits and implications of their work. Given their novel approach to causal identification by combining graphical and matrical approaches, I think this work has the potential of being a valuable contribution to NeurIPS.


 We are very thankful for your positive review. 

Presenting a table (or a figure) relating existing identification literature to the new identification result seems like a great idea and will help readers assess the differences among them and the contributions of the paper. We will implement this suggestion, thank you.

Regarding the examples and expositions, we will utilize an additional page to further provide more intuitive explanations with examples, enhancing the paper's readability. 

Regarding ID-ME completeness, we currently do not have any negative instance that would support this statement and proving it is our main focus. For the sake of context, we would like to note that there is no known complete algorithm even for sub-problems of what we investigated in this paper, including identifying a marginal, conditional, interventional query $P_\mathbf{x}(\mathbf{y}|\mathbf{w})$ from a marginal, conditional, interventional distribution $P_\mathbf{A}(\mathbf{B}|\mathbf{C})$. Even though we conjecture ID-ME is also necessary, the completeness is quite involved given the many moving parts and the combinatorial nature of the problem. 


 Thanks for reviewing our paper and providing us constructive feedback. 

In general, we hope to be able to improve the organization and readability of the paper by utilizing the additional page, in case our paper is accepted to the venue. We will pay special attention to the c-factorization (Lemma 2) and the algorithm section (Section 6). For example, to help readers understand Lemma 2, we would be able to visualize the three matrices involved in Lemma 2 more clearly so as to explain better what variables should be fixed and why. Please note that we currently have an example in the appendix demonstrating how different parts of the algorithm play roles in identification. We also tried to add illustrations and more details there given the space constraints of the body of the paper. We also consider some of the tradeoffs you pointed out interesting and will experiment with them in the next version of the paper and see how the flow goes, thank you. 

Regarding the two types of slashes, we used the backward slash $\setminus$ as "set minus" (i.e., conventional notation) consistently, which can be not only applied to variables but also to values (e.g., $\mathbf{a}\setminus \mathbf{B}$ = values of $\mathbf{a}$ excluding variables in $\mathbf{B}$). The forward slash / is to transform a part of values as variables. Line 158 is correct "set minus". Line 164 asserts that the left-hand side is c-factorized as the right-hand side, which implies that $\mathbf{Z}$ will satisfy the explanation made in Line 148--151.

We understand that there is a certain trade-off when introducing a new notation. For example, Line 146 introduces a compact form of union, and Line 161 introduces a shorthand notation for "part values and part variables". We considered the benefit of shortened expression outweighs the notational overloading (e.g., Line 165). We can consider additionally presenting the result without newly introduced notations. Further, if the similarity between two slashes can confuse readers, we can employ a different notation for the forward-slash that does not look similar to $\setminus$.

Let us address the ‘minor comments’ you raised one by one.

(Why Lemma 3 has $\mathcal{G}$):
Thanks for pointing this out. The lemma can be generalized to any conditional independence structure. Hence, constraining probability distributions to be compatible with a causal diagram $\mathcal{G}$ can be unnecessary. Similarly, we can drop the constraint in Lemma 4. However, we would like to note that, in the paper, whenever we write probabilistic expressions, we are assuming that they are derived from the same model $\mathcal{M}$, which induces a causal diagram $\mathcal{G}$.

(Footnote 1):
Previous literature (e.g., Miao et al.) similarly employed "distribution in matrix notation": they used typical "$P$" for a matrix and "$\mathrm{pr}$" for a distribution. We believe that "bold sans-serif P" and "plain italic P" are sufficiently distinguishable from each other. If other reviewers also find that the current notation can cause confusion, we can change to a different letter, perhaps "$M$". We will emphasize that the variables are discrete in the main text.

(Lines 81–82) $\mathbf{W}$ and $\mathbf{Z}$ are sets of variables as shown in Line 80 where $\mathbf{Z}$ is the set of intervened variables and $\mathbf{W}$ is the set of conditional variables. Union is the set union. We are afraid that we misunderstood your question. If you can specify your question a bit more, we will be able to answer better.

(Line 91): 
We will constrain ourselves to recursive SCMs, which will lead to acyclic causal diagrams. We will make this explicit.

(Line 299): They are not related to c-factors. Here, $X_k^+$ is to highlight the elements of $X_k$ that do not belong to the other five sets (i.e., exclusively included in $X_k$). We can employ a different notation to avoid confusion.

We hope the reviewer improves his/her assessment as most of the issues are addressable in the revised version following the discussion above (improving notation, moving some content to/from appendix, etc).


 Thank you for your positive assessment.

We certainly agree that statistical inference (estimation) is a crucial next step once a query is identified. We believe that developing an efficient and robust estimator involving matrix (pseudo)inverse will be extremely challenging. It is hard for us to address both identification and estimation in a single paper given the complexity of the identification conditions alone. After all, we do agree with your suggestion and will mention it in the paper, thanks. 

Regarding the rank condition in Miao, et al., it requires that $P(W|Z,x)$ has rank $k$ (the number of categories of $U$). In the case that $Z$ and $W$ have more categories than $U$, coarsening them can achieve the condition. This rank condition can be derived from the fact that the invertibility of $P(W|Z,x)$ implies that of $P(W|U)$ and of $P(U|Z,x)$, as well. Note that the coarsening can be achieved as far as the variable to be coarsened does not involve d-separation as a part of the separating set. Interestingly enough, it’s also the case for Theorems 1 and 2 (where a full rank condition is required for every invertible matrix and coarsening) can be applicable to the sets of variables.

We will streamline and improve the presentation of the assumptions to increase clarity and transparency, thanks. 


This paper proposes an identification criterion which combines approaches from the graphical modeling community and matrix-based proxy variable based approaches which leverage matrix based criterion. The authors do so by first providing a characterization of the implied matrix equations of the distributions resulting from graphical criterion, then leverage these insights to define new criteria for matrix based approaches which appear to me to be a strict generalization of prior approaches and then unify them into a larger identification theory.   I found this paper to be a very strong contribution to the literature, with strong framing and a commendable level of theoretical rigor. The paper presents no experimental evidence, but given that this is clearly a theory focused paper I don't think that can be held against the work. My largest concern is that some of the entailed statistical inference procedures may be exceedingly difficult to estimate in practice. While I wouldn't recommend rejection based on this, perhaps the authors could make mention of it. Also, in Miao, et al. there is a requirement on the rank of the proxy variables, are no such assumptions necessary here? In general, it would be helpful if the authors placed necessary assumptions in a more well delineated fashion. As it stands, they are strewn throughout and often made opaque. The authors should make more plain the limitations of the proposed methodology in terms of what is practically realizable from data. I would also perhaps suggest a small motivating example to provide intuition.

This submission presents a unified theoretical view of graphical and matrix-based approaches to nonparametric causal identification. The authors use this unified view to produce a new identification algorithm that is sound, and covers a number of existing proxy-variable and observational/experimental identification algorithms.   While some theoretical arguments were challenging to follow (likely because of the condensed format of an 8 page conference paper), overall this submission makes a relevant and significant contribution to the field of causal inference. Moving from identification results based on the full observable joint distribution (as in the do-calculus) to sets of observational and interventional distributions is of practical value. My understanding is that this extends the existing identification algorithms using observational and interventional data to settings where the sets of observational variables are disjoint, as in the Kuroki and Pearl proxy variable setting.

While my impression of the work is generally very positive, I think some changes could be made to make the content easier to contextualize in the literature. For example, I would suggest adding a table that describes how ID-ME relates to recent identification algorithms.

Based on the summary of reviews at ICML 2021, it appears that previous reviewers encouraged the authors to emphasize the main theoretical results in the main body at the expense of exposition and examples. I respectfully disagree, and would rather see greater emphasis placed on intuition and examples throughout the main body of the paper and the theoretical details reserved for the appendix. Specifically, it would be helpful to clarify how graph structure relates to the resulting matrices a bit more intuitively.

As ID-ME is proven to be sound, but not complete it would helpful to understand these limitations in a bit more detail. Are there example graphs/queries the authors could point to where ID-ME does not provide complete identification results? Alternatively, do the authors expect that ID-ME is complete but that a proof of completeness is left to future work? The authors have not addressed the societal impact, but that is appropriate given the theoretical content of the submission.

Meta-review: The paper contains a solid and novel theoretical contribution. Authors should make sure that the remarks that convinced a reviewer to raise the score are sufficiently accounted for in the final version. Please take remarks on readability serious to help readers understand the challenging material. Abstract and introduction should be more explicit about the fact that not the full joint distribution of the observed variables are given, hence this is a method of merging information from distributions of subsets.  

Decision: accept