Summary
The paper presents a framework for causal reasoning that supports the simplification of large structural causal models (SCMs). One key operation that supports this simplification is consolidation of a SCM such that only a subset of endogenous variables are explicitly modelled, but in such a way that all possible interventions are still supported. Another key operation is the partitioning of an SCM into sub-models, in such a way that endogenous variables from one sub-SCM act as exogenous variables in another.
Potential reductions in complexity are then discussed by the use of these operations and associated constraints. For instance, replacing equations within SCMs with computationally simpler expressions, and dropping equations which do not relate to variables of interest. Throughout there is a focus on retaining information about the effects of interventions. The paper concludes with two examples: one on modelling tool wear on a milling machine and another, more complex example, exploring planning policies for a simple platformer game.
Strengths
* The paper very clearly presents its contribution and relates this well to existing work, justifying the relevance of the contribution to the field.
* The paper builds its arguments formally, and with intuition, and presents meaningful relevant results and constraints.
* The examples illustrate well the benefits of the proposed framework.
* The contributions appear meaningful to me and likely to be of relevance to other researchers, particularly those concerned with large scale causal modelling.
Weaknesses
* At times the arguments are a little vague or the explanations incomplete.
* Although mostly clear, the formal notation sometimes leaves a little to be desired.
* There are a few places where the authors appear to make errors or omissions in their explanations.
Issues with understanding:
1. In definition 1, the set of possible interventions is a little unclear. There appears to be 1 possible intervention, $I_i$ per endogenous variable, $X_i$. Or can there be multiple potential, but mutually exclusive, interventions per endogenous variable? What exactly is $I_i$?
2. On line 94 $f_i(\textbf{x}, x_0)$ is used to indicate the structural equation for variable $X_i$ under the intervention that $X_j$ is set to value $x_0$, but this seems a little under-defined to me.
3. On line 98, the authors state that $\mathcal{M}$ entails infinitely many intervened distributions, but this seems to conflict with earlier notation (see 1.).
4. Definition 2 could do with an explanation of $P^{\textbf{I}}_{\textbf{E}}$. I am assuming this means the distribution of target variables in $\textbf{E}$ under some intervention $\textbf{I}$ but I can't see this stated anywhere.
5. The notation $\rho(\textbf{U},\textbf{I})$ first introduced in Definition 2 does not refer to the subset of variables $\textbf{E}$ to which it refers. This would be good practice anyway, but becomes more problematic when partially consolidated SCMs are considered (Def 5) as the variables of interest $\textbf{E}$ are augmented with additional variables that act as exogenous variables in other sub-SCMs.
6. The caption for Figure 2 (right) states that the dotted line indicates explicit computation for $X_2$ but it isn't clear what is meant by this.
7. In section 3, in a number of places (lines 159, 165, 187), there appears to be repeated errors in the notation, e.g. $\textbf{V} \in \textbf{A}$. I think that $\textbf{V}$ is the complete set of endogenous elements while $\textbf{A}$ is a subset of $\textbf{V}$
8. In lines 175-183, the partitioning of an SCM appears to require that exogenous variables $\textbf{U}_i$ of a sub SCM must be endogenous variables of another sub-SCM in the same partitioned SCM. But could they be truly exogenous variables of the whole system? Also, there appears to be notational irregularities in this paragraph relating to what is a sub-SCM and what is a partitioned SCM.
9. Things get a bit messy around definition 5 with respect to $\textbf{E}$ and $\textbf{E}'$. The distinction between these two sets of variables could be clearer. For instance, if I am considering partially consolideted SCM $\mathcal{M}_{\mathcal{A},\textbf{E}'}$ then how do I know what the set $\textbf{E}$ is that is used to define $\textbf{E}'$.
10. In section 4, there is a chain of inference on lines 234-235 that is difficult to follow (what is the scope of the universal and existential quantifiers?) and some entity $D$ appears without being defined.
11. I got a little lost in section 4.1. In particular, the discussion of conditional branching and stacking was a little vague.
## Post rebuttal
After reading the rebuttal and individual responses to the above points, I am raising my recommendation to "accept".
Questions
My questions have mostly been articulated in the **Weaknesses** field.
Rating
7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Limitations
The authors discuss limitations and the possible relaxations of these effectively.