Paper:
Causal Discovery from Discrete Data using Hidden Compact Representation
Causal discovery from a set of observations is one of the fundamental problems across several disciplines. For continuous variables, recently a number of causal discovery methods have demonstrated their effectiveness in distinguishing the cause from effect by exploring certain properties of the conditional distribution, but causal discovery on categorical data still remains to be a challenging problem, because it is generally not easy to find a compact description of the causal mechanism for the true causal direction. In this paper we make an attempt to find a way to solve this problem by assuming a two-stage causal process: the first stage maps the cause to a hidden variable of a lower cardinality, and the second stage generates the effect from the hidden representation. In this way, the causal mechanism admits a simple yet compact representation. We show that under this model, the causal direction is identifiable under some weak conditions on the true causal mechanism. We also provide an effective solution to recover the above hidden compact representation within the likelihood framework. Empirical studies verify the effectiveness of the proposed approach on both synthetic and real-world data.

Review:
UPDATE after authors' rebuttal: Thanks a lot for your detailed answers, quite convinced by most of your arguments, you played the game well within the one page constraint! I increase my score from 6 to 7, hoping it would be enough to see your paper being presented at NIPS! If not (competition is strong here!), I strongly encourage you to still fight to publish it, clearly a key topic (I'm biased :)).  This work presents a method to decipher a causal link between 2 variables. The method relies on a re-coding of one of the variables (the cause) to obtain an 'optimal' link with the effect variable. The paper reads well, is well structured and easy to understand. I am puzzled as to whether it is a mere simple implementation of existing work or whether it is a truly novel (and then would well be worth presenting) method. The numerical results are not very convincing.  Comments:  - abstract: as a first pass, I was intrigued by the use of a hidden variable mentioned here, wondered whether it was unobserved. OK once reading the paper.  - introduction: l36 "right" -> define what the right order is. Is the variable necessarily ordered? Cannot be purely qualitative? This paragraph is quite interesting. Do you have a ref? If this is original work, may need further explanation. l45 (typo): ...maps THE cause to...of lower cardinality  - Before you introduce your work from l43, is the motivation strong enough? I am not entirely convinced at this stage...Moreover, it reads that a (simple) re-coding of the (true) cause is enough to distinguish the true causal relationship. It is not very intuitive. Careful, the Table of Figure 2 is not causal, only probabilistic (see text l60)  - Section 2, l72: f needs to be a surjection, right? You could enumerate them (how many are there for each unknow |Y^{\prime}| ? l79: not sure your example is a misdiagnosis example. l83: LOG-likelihood. l96: ...$ P(X) $, THE deterministic...  - l103: is the alternate maximisation equivalent to a joint maximisation or an approximation? Quality of the approximation then? l117: ...with $ y_0 $, where...Could you also explain the choice for y0?  - Bottom of p4: so in the end, (i) re-coding either X or Y and finding a 'good' probabilistic relationship with the target variable is enough? (ii) Moreover, it quantifies the 'correctness' of the model? (iii) What if neither X->Y, nor Y->X are correct?  - Section 3: does Th 1 generalise  the additive noise model assumption? THIS IS INTERESTING. So is Th 4.  - Section 4: l174, remove (HCR in short), l175 ...on both synthetic and real world data. Rephrase the next sentence.  - Your choice is synthetic data looks 'easy' as it is exactly your model. May need more general framework! You could then do some sensitivity of your method to breaking the assumption of Th 1?!  - l188: notation issue, do you mean {3,...,15} instead? Fig2: why don't you show performance below m=500? Strange performance for SA, ANM and DC. Worse than random guess??!! Define the decision rate.  - real world data: I am sure these data sets were used in some public challenges (see the work by I. Guyon), I doubt they get results as poor as the ones you show. How do you explain some method perform more poorly when m increases?  - References: consistency, with or without first names? (e.g. Jonas Peters, but B. Sch\"{o}lkopf)