Venue: CLeaR 2022 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: Summary: The paper proposes a category theory formalization of causal model equivalence, i.e., when one model is an abstraction of another one.
Main review: Strengths:
The paper uses category theory to formalize the idea of causal model equivalence which is rather elegant and consistent with previous works in category theory applied to causal modeling.
The theory derivation seems sound.
This model equivalence formalizes the notion of abstraction and thus has applicability to more fundamental ideas like coarse-graining and emergence.

Weaknesses:
The categorical formulation only applies to discrete causal models, whereas in causal ML practice, people more often work with continuous causal models.

The categorical formulation only covers Bayesian networks and not full causal models: In Bayesian networks, only the conditional probabilities of a variable given its parents are known, but not the deterministic function mechanism and the noise specific to this variable, i.e., only P(A|PA_A) is modeled and not the combination of deterministic function and noise A := f(PA_A, N_A).
According to the ladder of causation from Pearl, this makes Bayesian networks only applicable at level 2 of the ladder (interventional), they cannot answer counterfactual (level 3) because counterfactual queries involve knowing the noise and the deterministic function.

It has the strengths and weaknesses of applied categorical approaches: fairly elegant construction that seem to reveal profound structure in the problem of interest, the promise of lifting and porting properties from other areas into the problem of interests via the categorical abstraction. Yet, it appears difficult to find concrete, directly applicable findings. The criteria in section 6. are not easily convertible into operational algorithms.
What does this newly presented derivation allow us to do? What applicable insights does it bring?

I don't think the limitations described above are enough to reject the paper, because building such an understanding of the causal problems is important and might turn out to be useful. In particular, this specific paper on the equivalence of causal models also tackles one particularly important aspect: causal model abstraction which is very related to coarse-graining and emergence.
review_title: null

final_score: rating: 6: Marginally above acceptance threshold
review_content: Summary: Review of Paper#14
Main review: 
This paper develops a category-theoretic criterion for determining the equivalence of causal models having different but homomorphic directed acyclic graphs over discrete variables. One of the contributions of this paper is that it interprets a structural causal model in the context of category theory. This allows one to translate a causal diagram into a corresponding string diagram, which appears to be instrumental in determining the homomorphic transformation between different structural causal models.

The authors develop a complete condition for an isomorphic transformation between two causal models using causal string diagrams.  While I have not checked the details of the proof, the results in Theorem 6 seem reasonable. Since I am not familiar with the related work on this specific topic, I cannot comment on the significance and novelty of these results.

Overall, I think this paper studies an interesting problem and proposes a reasonable characterization. However, I do have some questions, summarized as below:
The characterization in Definition 2 and Theorem 7 rely on the detailed parametrization of the underlying causal models. This assumption seems quite strong since these parametrizations are mostly not identifiable in practical applications.

This paper also assumes that all variables in the causal model are observed. Or equivalently, there exists no unobserved confounder (UCs). This brings me back to the previous point: when UCs are present, the underlying causal mechanisms (i.e., the functors) are not generally identifiable from the observational data. 

I think the writing of this paper could be improved. I would appreciate one or two possible applications of the proposed method or some practical challenges that involves identifying the homomorphic causal diagrams. This could help the readers to understand the significance of the results.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: Summary: This paper discusses equivalence between causal models. Somewhat unually, the causal models is represented in a causal string diagram, after which an equivalent abstraction of a causal model is defined by a isomorphism between functors. The motivation for using string diagrams is the allow for preserving different causal mechanisms in the abstraction. The approach is compared to several other approaches for abstractions and it is shown that it generalises those other proposals.
Main review: Generally, the paper introduces a very elegant method for both thinking and representing abstractions of causal models. While
the mathematical language is somewhat unusual in this field, it is not difficult to understand the main points that the paper
tries to make. I think it also convincingly shows that the approach provides a systematic criterion for the existing of a
causal abstraction. As such, though theoretical, I believe that is a very relevant contribution to this field.

One thing I would like to suggest is the title of the paper: if I understood correctly, the contribution is limited to providing
equivalence between models and their abstractions (the 'micro' and 'macro' level), and not equivalence 'in general' (whatever that 
may mean). While discussed in the introduction, I was expecting a somewhat different in the paper at first.
review_title: null

Meta-review: This paper studies some notions of equivalence between and abstraction of causal (Bayesian network) models in category-theoretic terms. The basic idea is to treat a causal model as a functor from a DAG-induced syntax category to the category of stochastic matrices, and define one model being an abstraction of another in terms of the existence of a natural transformation between the functors, and equivalence in terms of the natural transformation being a natural isomorphism. The approach is original, and the writing is reasonably clear for papers of such abstract nature. The main limitation pointed out by reviewers is that it is unclear whether the constructions and results in this paper can yield any application to address more concrete problems. However, all reviewers agree that it is an interesting and novel paper that is suitable for this conference. 

In my view, another drawback of the paper is that it omits technical details (due probably to lack of space) that make parts of the paper elusive. For example, when defining the induced syntactical functor on p. 4, the authors seem to suggest that each generating morphism in Syn_{G} is mapped to a generating morphism in Syn_{H}. But it is unclear why this is always feasible. In theory, a graph homomorphism \phi from G to H allows that \phi(Y) has a parent in H that is not \phi(X_k) for any parent X_k of Y in G. When this is the case, it seems that the relevant generating morphism in Syn_{G} would involve more nodes than images of the parents of Y in G. In other words, the image in Syn_{H} of a generating morphism in Syn_{G} is in general not a generating morphism in Syn_{H}, but one constructed from the generators in Syn_{H}. Such a construction seems to be omitted in this paper. I believe it is also relevant when the authors talk about morphisms corresponding to "single causal arrows". I understand that space is limited, but if possible, I suggest the authors should briefly describe the relevant constructions.  

Overall I agree with the reviewers that this paper is acceptable. 
    
Decision: Accept (Poster)