Scalable do-Shapley Explanations with Estimand-Agnostic Causal Inference

We thank the reviewer for their review. We will address each of the concerns in the following:

W1: The reason for choosing DCGs over CNFs was, mainly, because they admit latent confounders (provided identifiability) while CNFs do not. Even if experiment 5.2 did not have latent confounders, for consistency with the experiment in Appendix E.1.2 (the semi-Markovian case), and applications in Appendices F.1 and F.2 (the two real-world datasets), all including latent confounders, it was preferable to always employ the same model for ease of readers wanting to reproduce the code (which will be included with the camera-ready version). More importantly, estimation time was a crucial consideration for experiment 5.2, given the number of replications and the increasing number of features. As stated, CNFs were orders of magnitude slower than DCGs during estimation, which would result in a prohibitively expensive experiment time-wise in comparison to DCGs, more so when the choice of model was not important in this particular experiment (longer SCM estimation times would simply multiply by the number of $\nu$ computations, which is reduced by FRA). We deliberately chose a fast SCM so as to compare estimation time ($\nu$ evaluation) against the FRA algorithm (the tests required to identify irreducible sets); since FRA is negligible time-wise compared to a faster SCM, it is only advantageous to apply it for slower SCMs. Besides, the objective in this experiment was to compare between no-cache, cache and FRA-cache, and therefore the choice of model was not particularly relevant. Indeed, as you say, FRA may make more powerful SCM architectures like CNFs feasible for do-SHAP estimation, but there was no need to try one such architecture for this demonstration.

W2: Thank you for pointing this out; the choice of words in the abstract may misrepresent our contribution (which is better stated in the main text, see line 60). The intended meaning was that we propose the estimand-agnostic approach for do-SHAP, which is a significant contribution given that the original do-SHAP paper did not contemplate such a possibility, as stated by the authors (see line 58), proving it impractical. By identifying this connection and proving its feasibility and accuracy in estimating these attributions through our experiments, we can make do-SHAP feasible for arbitrary graphs, which, in our opinion, is an important contribution to the literature, especially since it could help promote the use of causality-aware attribution methods.

Regarding your second comment, when we mention that we make do-SHAP feasible for arbitrarily complex graphs, we refer to the fact that, if we employed the estimand-based approach, it would require to manually employ estimation techniques for $2^{|X|}$ estimands, one for each query $\nu(S)$, which is simply infeasible in practice. Instead, by training one single model and employing general procedures for interventional queries like these $\nu(S)$, the estimand-agnostic approach can automatically adapt to any kind of graph. It is true that the exponential nature of SHAP remains, which is why it is important to employ the approximation approach (section 3.4) and our FRA algorithm and the derived properties (section 4.2) to further reduce computations. We do not claim to avoid the exponential complexity of SHAP, but we do provide a way to compute these terms feasibly in practice (estimand-agnostic) and more efficiently (FRA), as proven in section 5.2.

Communicating these nuances has proven to be hard, but we will rectify the specific wording of the abstract to avoid misrepresenting our contribution. Thank you for pointing this out.

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We thank the reviewer for their review. We will address each of the concerns in the following:

W1. This specific instance is not a mistake, since both terms are not the same. The first, the set of nodes $**V**$ in the causal graph, consists of the SCM’s measured variables $\mathcal{V}$, their corresponding exogenous noise variables $\mathcal{E}$ and (if any) the latent confounders $\mathcal{U}$. $**V**$ includes all random variables in the SCM (not only $\mathcal{V}$), despite not all of them showing explicitly in the causal graph representation, as discussed in footnote 2, hence why the notation is different. We were very particular about the use of notation, so any perceived inconsistencies may not be so, but there may have been some errors in writing that escaped our notice. If you identified any others, we would greatly appreciate you notifying us so we can rectify them.

W2. We will address this concern in detail, since it seems to be the most pressing point.

Regarding the first contribution, while this work does not propose the estimand-agnostic framework in itself (Parafita & Vitrià 2022), it does: 1) make do-SHAP feasible (since it was impractical due to its reliance on estimands) by employing the estimand-agnostic approach, and 2) prove that SCMs can indeed estimate do-SVs effectively, despite not employing estimands like the original authors proposed.

The FRA algorithm contribution has many facets. Firstly, it is a novel set of properties employing the rules of do-calculus to derive these irreducible sets. Note that this approach is completely different from the work in (Luther et al., 2023), since theirs was exclusively conditional and did not consider the causal structure in the data. We mention Luther’s work because of its connection in employing conditional independence to identify instances where $\nu(S \cup \{k\}) = \nu(S)$, but we go further by considering: 1) the causal framework, where these $\nu$ entail causal queries with $do(.)$ interventions, and 2) by identifying the irreducible set $S’$ such that $\nu(S) = \nu(S’)$, thereby not reducing by one variable, but potentially many, resulting in more computations avoided since they all collapse on the same irreducible set, already cached. Secondly, we derive an algorithm to efficiently identify these irreducible sets, which is vital in the context of SHAP given its exponential complexity (albeit alleviated by the approximated approach, discussed in section 3.4).

Finally, the third contribution, theorem 4.8, may seem a simple, intuitive result, but requires an extensive demonstration employing the rules of do-calculus (see Appendix D.3). Thanks to this proof, we can now explain these inaccessible DGPs in a human-understandable manner, since the SHAP values now add up to the actual value for $Y$, whereas before it simply added to $\mathbb{E}[Y \mid \widehat{**x**}]$, which need not be close to the actual, factual value of $Y$, proving confusing for the end-user when receiving these explanations. Intuitively, we might have attributed this difference on the exogenous noise signal; our theorem, however, proves this fact (under certain assumptions). That is the importance of the claim.

Finally, all of these contributions are tested with our experiments and showcased within two real-world datasets, in section 5 and Appendixes E-F.

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We thank the reviewer for their review. We will address each of the concerns in the following:

W1: We would be grateful if the reviewer were more concrete regarding the claim that the paper is poorly written and not meeting the required standards. We are willing to improve upon the text and, apart from the bold text issue mentioned in W5, no further measures are mentioned on how to address this complaint.

Regarding the need to incorporate causal structure into Shapley, we do address it, albeit briefly, in the paragraph starting on line 41. By focusing on a simple example (Figure 1), we can see how the two non-causal alternative definitions to the Shapley game fail to address the behavior of the underlying mechanism. In marginal SHAP, consider for example $\nu(\{E\})$, we would marginalize A and S regardless of the assigned value to E, thereby ignoring the impact that education level may have on the seniority level of the employee (their standing in the company), and producing out-of-distribution configurations (a, e, s). Therefore, it would ignore an important mechanism in the data. Alternatively, with conditional SHAP, we would operate with $P(A, S \mid E=e)$, thereby including this dependency between E and S, but also taking whichever value of A would have generated this specific value e, introducing in turn an anti-causal effect from E to A, which we know is not possible. Both approaches, because they ignore the causal structure, incorporate non-causal effects that fail to reflect the real-world data generating process, and would therefore lead to unreliable explanations. In contrast, do-SHAP does take into account this causal effect, by using the intervention do(E=e), therefore affecting S (E → S) while not affecting A (E ← A). This is why, only by considering the causal structure, and therefore employing do-SHAP rather than the non-causal alternatives, we can obtain reliable explanations that will translate to the real world if we applied these interventions.

This example is explained very briefly in the paper because of space concerns. However, we will extend this explanation along these lines for the camera-ready version.

On the other hand, on the topic of real-world applications, we did include two real-world examples in appendix F; although reviewers are not asked to read the appendix, we do include them and both are mentioned in the paper, in line 412.

W2: The required assumptions are stated in the paper. Note that we are not operating from the Potential Outcomes perspective, where the assumptions you mentioned (SUTVA, consistency) are standard and always mentioned, but from the Causal Graph perspective, where the assumed causal graph G itself encodes the required assumptions. Our main assumption is that the graph is known and from there we can infer whatever assumptions are necessary. See Assumption 4.1 (line 245). “We assume $P(\mathcal{V})$ to be strictly positive,” (positivity) “resulting from an unknown SCM $\mathcal{M}$, but whose graph $\mathcal{G}$ is known, a DAG, and s.t. its do-SVs are identifiable in $\mathcal{G}$ (i.e. all $\nu(**S**)$ terms, with $**S** \subseteq **X**$, are identifiable). Note that $\mathcal{G}$ may include latent confounders as long as its do-SVs are identifiable.” This means that we do not need to assume, for example, ignorability (there may be latent confounders). As long as we have a DAG G (assumed), and this G results in $\nu(S)$ queries that are identifiable (assumed) we do not need to add any further assumptions. Depending on the shape of the graph, this do-SV identifiability assumption entails, for example, strong ignorability for some of its queries $\nu(S)$, but it is not required to assume it explicitly, as the graph itself can entail these assumptions through its structure. See also the semi-Markovian graph in Figure 3, for the experiment in section 5.1 and Appendix E.1.2, where a latent confounder appears between X and B and still results in identifiable do-SVs. SCMs are still able to compute do-SVs on this graph despite the confounder, thanks to the fact that the queries are non-parametrically identifiable (proved through do-calculus), as explained in section 3.2 and (Parafita & Vitria, 2022). Regarding SUTVA, as far as we know, it is not a standard assumption explicitly stated in the Causal Graphs literature (Pearl’s perspective, not Rubin’s); consistency is entailed from the SCM framework directly, while no-interference may require further assumptions. See [1] below for an in-depth discussion of the relationship between both frameworks.

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