__ Summary and Contributions__: This paper considers the minimum entropy of a latent variable that renders two variables conditionally independent and explores the relevance of this quantity for causal discovery. The authors describe two ideas for employing this quantity:
first they use it to distinguish cause-effect relations from confounding subject to appropriate assumptions and second as a necessary condition for separating sets in PC type algorithms.

__ Strengths__: The theoretical claims are sound, assumptions are clearly stated.
I liked the estimation of common entropy for real-world cause-effect pairs, which supported the claim of the authors that it scales with minimum marginal entropy.

__ Weaknesses__: The main weakness is the missing contribution. The first part (telling confounding from cause-effect relation) is close to [23] (which is published at UAI 2011). The second part (necessary condition for PC type algorithms) is novel to the best of my knowledge, but its benefit remains unclear, despite some experimental evidence. The reason is that small sample problems also occur for the estimation of common entropy and there is no theoretical evidence that considering common entropy mitigates the risk of identifying the wrong separating sets.

__ Correctness__: The approach of stating strong assumptions regarding statistics and causality, which renders causal discovery problems solvable that ares unsolvable otherwise is an interesting direction. The authors state and motivate their assumptions explicitly and correctly derive mathematical conclusions.

__ Clarity__: I didn't see major issues with clarity, although I found the use of Renyi entropy as opposed to Shannon entropy comes with lack of motivation.

__ Relation to Prior Work__: I didn't see a discussion of [23] which contains already one of the two main ideas of the paper.
Another paper that sounds relevant to me:
Bastian Steudel, Nihat Ay: Information-Theoretic Inference of Common Ancestors
https://www.mdpi.com/1099-4300/17/4/2304

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: this paper applies common entropy for learning causal graphs from observational data. the paper demonstrates the use of common entropy for causal discovery with two examples.
In the first example, assume that there are two observed variables, X, Y. The paper show that if the latent confounder is simple, and there is no simple mediators in between two variable, this problem can be solved using common entropy.
In the second example, the authors show that common entropu can be used to improve any constraint-based causal discovery algorithm in the small sample regime.

__ Strengths__:
To the best of my knowledge, the applications of common entropy to causal discovery are original.

__ Weaknesses__: The problem & method are not well motivated. For example, the motivation for using common entropy is to avoid the issue of high dimensional covariate X and Y. However, in the first example, the authors used simple binary variables for both X and Y.

__ Correctness__: The empirical validation for ADULT dataset seems quite questionable. There is no ground truth in the dataset, but the authors assumed some causal relationships and validate the method based on that.
I did not thoroughly check the proofs of the theorems.

__ Clarity__: The writing can use substantial improvement.
For example, in line 65, the authors wrote "this problem can be solved using common entropy", without referring what the problem is.
There are also a lot of incomplete sentences, such as the caption of Figure 2.

__ Relation to Prior Work__: The connection to prior work is not clearly examined in the main manuscript.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: I thank the authors for addressing my concerns. I would urge the authors to include next time into their rebuttal additional experimental results when the reviewers ask for them. The rebuttal doesn't provide any additional supporting experimental results and as the experimental evaluation is the main weakness of this paper, I will not raise my score.
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This paper examines the problem of learning the simplest latent variable that make tow discrete variables conditionally independent.
It uses Renyi common entropy for this task and presents an iterative approach to estimating this.
Further, it shows how this quantity can be used for causal inference in conjunction with constraint-based approaches.

__ Strengths__: This paper addresses an interesting problem and applies their finding to causal inference which is an area of growing interest at the conference.
This paper is overall well written and manages to discuss a technical topic in an accessible way, making it a pleasant read.
It proposes a new class of entropy measures and provides theory for the use of these measures in causal discovery.
Further, it proposes a practical algorithm for the estimation of the in general intractable measures. Based on the theory it shows how this can be combined into a practical addition to any constraint-based causal discovery method which allows it to orient more of the otherwise unoriented egdes.
In summary, I believe that this paper would be of interest to the community and that it is novel enough.

__ Weaknesses__: The main weaknesses are:
* experimental evaluation
* a little bit on the clarity of the writing (see below)
Experiments:
My main criticism is that experiments span a little over 2 pages, there is very little in terms of comparison to related work. It is very nice that the performance of the proposed iterative procedure for estimating common entropy is evaluated, but there is no comparison to any related methods. Please compare this (maybe to [30]?). A lot of the experimental section is spent in validating assumptions and conjectures. I would suggest that the authors put these experimental results in the appendix and focus on comparison of the proposed methods (latentsearch and entropicpc) to related work. For the evaluation of EntropicPC, I would strongly urge the authors to compare their methods against any of the many, many extension of the PC algorithm (and not just against PC itself). Further, I would suggest that the authors test on more real-world datasets than just the ADULT dataset.

__ Correctness__: To the best of my knowledge the theoretical results presented in this paper are correct.
To the best of my knowledge the empirical methodology used is correct.

__ Clarity__: This paper is overall clearly written.
There is a nice introduction to the examined problem and nice motivation for its usefulness in causal inference. Also, there is a nice explanation of common entropy in the introduction that sets up the rest of the paper nicely.
Also, I would like to complement the authors for clearly stating the assumptions used (page 4). For the theoretical results (section 3), I would suggest to the authors that only the central results are kept in the main text and the corollaries relegated to the appendix.
The main weaknesses of the writing are:
* a conjoined introduction and discussion related work
this does not make section 1 “flow” very nicely. i would suggest that the authors extract the discussion of related work into a separate section and put this section right after section 1
* missing references in many parts of the text — i would urge the authors to add references to all claims and findings that are not their own.
Some examples: line 50 “common entropy”, line 141 Reichenbach’s common cause principle, sentence in line 203/204
Also, please add additional explanations for Figure 1 and Figure 4.
Typos and suggestions:
* line 92 — shown -> denoted
* change subtitles in experimental section 5.1 -> LatentSearch and remove mention of Sections from subtitles

__ Relation to Prior Work__: From the exposition in the paper it is straightforward to infer how the proposed method differs from previous work, but this is never explicitly discussed.
I would suggest that the authors rework the discussion of related work (see suggestions above) and include explicit comparisons of previous work with their proposed method.
Also, as this paper examines the problem of causal discovery, I would suggest the authors include related work based on the principle of independence of cause and mechanism.

__ Reproducibility__: Yes

__ Additional Feedback__: The claim in line 33/34 “discover the causal graph with the fewest parameters” does not necessarily apply to all score-based methods. Please quantify when it applies/.
I would be happy to adjust my score upwards if the discussion of related work would be fixed and the proposed method EntropicPC compared with extensions of the PC algorithm (not just plain PC).

__ Summary and Contributions__: This paper explores the usefulness of common entropy for causal discovery purposes in small graphs of up to 3 variables. The authors contribute the first application of common entropy to any task in the field of causal discovery. In particular, the paper includes 3 main contributions: 1) definition of Renyi common entropy and an algorithm for the estimation of it in practice 2) the use of two special cases thereof for the purpose of distinguishing two fundamental types of causal graphs 3) the use of the distinguishing procedure above for the enhancement of the PC constraint-based causal discovery algorithm.

__ Strengths__: The paper proposes an original contribution to the field of causal discovery, by suggesting a new basic addition to the toolbox of causal discovery. While most approaches to causal discovery rely on a sequential search over nodes while applying a conditional independence constraint, or scoring based approaches, the authors originally propose to enhance some of those by using the notion of entropy from information theory, and especially the notion of common entropy. The idea is to search for a possible latent confounder to describe a correlation between two observed variables via thresholding based on the notion of Renyi common entropy.
The authors thus propose an interesting, novel and potentially useful idea that could become relevant across various applications in causal inference.

__ Weaknesses__: The paper is trying to cover a fair amount of ground, and as such seems to glance over important details, and does not substantiate its claim with an equal amount of attention to all parts. Here are my main concerns:
1) Figure 1 considers 3 scenarios of interests to describe a causal graph for a distribution p(x,y) where x and y are correlated. However, the paper proceeds to consider another scenario, where X and Y are *mediated* by a latent, which we didn't consider in the introduction, and the significance of which is not fully discussed. Furthermore, the triangle graph is often discussed in contrast to the latent graph, but the direct graph is often forgotten. For certain applications, the difference between the direct graph and triangle graph might make a big difference (when interested in direct effects, fairness concerns, etc.). The paper seems to slip constantly between grouping triangle/direct graphs or simply dropping direct graph all together (see lines 156-157, algorithm 2, corollary 2 vs. 149, Theorem 2, 162-163). In fact, assumption 2 seems to relate to direct graphs and not triangles, but used for some reason in algorithm 2 to describe triangle graphs. I might be missing something, but I believe more care is needed in distinguishing these cases and when/why a distinction is made or not. If triangle and direct graphs are indeed interchangeable in certain cases, consider using a global term to describe both to clarify things.
2) Section 2 and 3 move back and forth between Rnenyi_0 and Renyi_1 common entropies, without much justification. While Theorem 2 is clearly established for Rejnyi_0, Assumptions 1 and 2 are stated as a general case for Renyi entropy. I am not sure why that is a fair statement (especially as Assumption 1 is never justified elsewhere and Assumption 2 is established via empirical experiments). Theorem 3 and Conjecture 1 are stated in terms of Renyi_1, and in practice, Algorithm 2 and 3, which are the main contributions, seem to be stated in terms of Renyi_1. Why include Renyi_0 at all then? Am I missing something? Perhaps clarifications or smoother movements should be included between the two? Furthermore, most readers are likely to be familiar with Shannon Entropy, so why not make the connection between Renyi_1 and Shannon Entropy more explicit?
3) The method seems to rely on tuning various parameters (\beta for the loss function (2), T as dependency threshold in Algorithm 2, \alpha in Conjecture 1, which is hardcoded to 0.8 in Algorithm 3) while searching over candidate *latent* variables. Moreover, Assumption 1 and 2 seem to be the key to much of what this paper proposes, and are mostly established via empirical results. That seems problematic and casts some doubts as to the general usability of the method. While I still think it is a very interesting proposal, clear indication of limitations, and explicit caution needs to be stated clearly.
For a more detailed list of my comments and questions, see additional feedback section below.

__ Correctness__: To the best of my knowledge and understanding, the proposed methodology seems correct, including its expansions and derivations in the appendix. However, occasional inaccuracies in presentation and writing makes correctness more obscure at times (see comments and suggestions for authors below)

__ Clarity__: The paper could use rewrites in various places, and while its main ideas are pretty much clear, organizational as well as explanatory choices could be improved on in my opinion. Please refer to my comments and question in the additional feedback section below.

__ Relation to Prior Work__: Common entropy has also been established elsewhere, notably in https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6874815, but without a connection to causal inference or discovery.
The use of common entropy for causal inference and causal discovery seems novel -- I believe it differs substantially from previous contributions in the field. Previous works in causal discovery, while some use information theoretic notions, did not consider common entropy before for the identification of latent variables to the best of my knowledge.

__ Reproducibility__: Yes

__ Additional Feedback__: Abstract
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1) The abstract claims the paper “propose[s] a modification to these constraint-based causal discovery methods”. In practice, it only deals with the PC algorithm. While the authors do claim the method can be easily extended FCI and others as well, the statement in the abstract is misleading.
Section 1
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2) The paper is concerned with a search over latent variables. While the text refers to such a latent as Z, much of literature on causal inference would use U to denote latent variables. For consistency’s sake, consider changing Z to U.
3) More on notation: G is used for half of the paper to refer to common entropy, but refers to Graph in the other half. Notice how in algorithm 3 for example, G refers exclusively to the graph and not to common entropy anymore. Why not choose a different character for common entropy to begin with?
4) Figure 1 can be drawn to be more helpful, including the cases and distinctions this paper actually cares about (see weaknesses section).
5) Consider adding q to notation sub-section, given how often it is used later.
6) Line 27-8: Learning a causal graph is not the first step for any causal inference task. Perhaps moderate this statement?
7) Given lines 53-58, one might expect we want the latent not simply be simple, but how simple it is should be relative to the covariates X and Y in question. Perhaps clarify? (this repeats in section 3 as well).
Section 2
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8) Section 2, line 129: qi(z|x) appears twice. I assume the second should be qi(z|y)?
9) Section 2, line 128-129: This line is very helpful. Why is this step not shown explicitly in Algorithm 1?
10) Algorithm 1: why is beta >*=* 0? Can we actually allow \beta = 0? Wouldn’t that mean we are not enforcing anything about the entropy of the latent anymore?
11) line 131-2 is helpful in understanding where Algorithm 1 comes from. Consider expanding on the derivation in the appendix and explicitly leading the readers to the relevant part in the appendix?
12) Line 138: for \beta <= 1? Or bounded away from 0 and 1? Perhaps make the interval more explicit for clarity?
Section 3
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13) While the section is distinguishing between only spurious association and some level of causation, the triangle graph still includes some spurious association, i.e. it does not cleanly separate causation from spurious correlation. Consider changing the section title to be more faithful to this fact.
14) Line 157 states the common entropy of the observed variables is “large” for almost all distributions coming from the triangle graph. Line 168 explained that according to assumption 1, a latent confounder has to be “simple”. These adjectives seem too global -- don’t these “size” or “complexity” statements relative to the entropy of the observed variables? Perhaps clarify that these indications are relative to a specific system in question?
15) Line 182 refers to section 7.6. That is the appendix. Perhaps change that to link directly to the appendix, and make it clearer?
Section 4
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16) Line 198: typo, “fo” at the end of the line should be “to”
17) Lines 202-205 point out the weaknesses in existing constraint based causal discovery algorithms, but do not clearly state (and neither is it stated later on, as far as I can tell), that entoptic enhancement solves all these concerns.
18) Algorithm 3 uses overcomplicated and crowded notation compared to other presentations of the PC algorithm (see Spirtes et al 2001, section 6.2, or https://arxiv.org/pdf/1502.02454.pdf#:~:text=The%20PC%20algorithm%20is%20the,data%2C%20e.g.%20gene%20expression%20datasets. , algorithm 1). Consider simplifying accordingly, given that only lines 10, 13, 15 seem like the modifications.
19) Hardcoding 0.8 in Algorithm 3 and line 217 doesn’t seem properly explained.
20) Line 222, line 10 is mentioned for the second time and is bolded. Consider dropping, I think it’s clear enough given line 220.
Section 5
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21) the description in section 5.1 makes it sound like bigger n values were used unsuccessfully. Is that the case? If so, consider giving full details on what worked and what didn’t.
22) Why is assumption 1 never tested empirically like assumption 2?
23) The figures are out of order and do not follow the text order. That is unnecessarily confusing.
24) Figure 2a: Having y axes differ makes it hard to assess at first sight. Furthermore, left: why would we want the recovered entropy to be smaller than real? Don't we want to see some measure of loss (i.e. via loss function you defined?)
25) Why do Figure 2b and c seem to indicate worse performance for smaller n? I’m probably misinterpreting it, but perhaps clarification would be helpful.
26) Figure 3(c): the choice of Y axis seems arbitrary and not explained. Why is it not comparing directly to assumption 2? And where did the \hat{M} notation come from?
27) Section 5.5: Entropic-C seems hardly helpful given the performance reported in Figure 4. Is that true? If so, why include it? In what cases is it helpful? If not, where am I reading it wrong?
28) The synthetic data experiments were carried out on samples of size 1000, 5000 and 80000. Is the last one a typo? Should it say 8k? And if not, why choose these unexplained jumps between sample sizes?
29) Line 272 claims a “very significant” improvement. I don’t know how to base levels of significance in this case, this seems like a qualitative statement. Consider qualifying this statement?
Appendix
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30) Like the Figures, the appendix is all out of order. E.g. Why is Theorem 1 in section 7.8-9? Please reorder the appendix chronologically + consider referring to relevant sections in the appendix from the main text?