SlowGuess/NeurIPS_2023_OCR huggingface dataset:
A long-standing question has been whether exclusion restriction and independence axioms together could be complete, in that they capture all the inferential content of a given causal diagram  $\mathcal{G}$  (see, e.g., [39, 8, 31]). Answering such questions can help with the development of tractable inference methods. Partial completeness results for limited queries are known [35], and the method from Tian [39] supplies an algorithm that is complete with respect to all equality constraints [8]. Placing no limitations on queries beyond their expressibility in  $\mathcal{L}$ —and thus including inequality constraints as well—but making certain restrictions on  $\mathcal{G}$ , we answer this question in the affirmative:
Theorem 3. For any acyclic diagram  $\mathcal{G}$ , axioms  $\mathrm{T}(\mathcal{G}) + \mathrm{SCM}$  are sound for  $\mathcal{L}$  over  $\mathfrak{M}(\mathcal{G})$ , and also complete if the bidirected arcs in  $\mathcal{G}$  form a disjoint union of complete graphs.
Often the famous  $d$ -separation conditional independence criterion (Def. B.2) is used in place of our cf-sep. Since all instances of the latter are instances of the former, our Thm. 3 is stronger (see Cor. B.1). This completeness result implies that for such a  $\mathcal{G}$ , any known graphical conclusions—including do-calculus, identifiability results, and bounds—can be rederived in our calculus, e.g.:
Example 4 (Verma constraints). We derive the Verma constraint [40, 39] over the graph  $\mathcal{G}$  of Fig. 2a that  $\sum_{w} \mathbf{P}(y \mid z, w, x) \mathbf{P}(w \mid x)$  does not depend functionally on  $x$ :
$$
\begin{array}{l} \sum_ {w} \frac {\mathbf {P} (y , z , w , x) \mathbf {P} (w , x)}{\mathbf {P} (z , w , x) \mathbf {P} (x)} \stackrel {\mathrm {T} (\mathbb {C})} {=} \sum_ {w} \frac {\mathbf {P} (y _ {z w x} , z _ {y w x} , w _ {y z x} , x _ {y z w}) \mathbf {P} (w _ {x} , x _ {w})}{\mathbf {P} (z _ {w x} , w _ {z x} , x _ {z w}) \mathbf {P} (x)} \\ \stackrel {\mathrm {T} (\mathsf {E R} ^ {\mathcal {G}})} {=} \sum_ {w} \frac {\mathbf {P} (y _ {z} , z _ {w} , w _ {x} , x) \mathbf {P} (w _ {x} , x)}{\mathbf {P} (z _ {w} , w _ {x} , x) \mathbf {P} (x)} \stackrel {(9)} {=} \sum_ {w} \frac {\mathbf {P} (y _ {z} , w _ {x}) \mathbf {P} (z _ {w}) \mathbf {P} (x) \mathbf {P} (w _ {x}) \overline {{\mathbf {P} (x)}}}{\mathbf {P} (z _ {w}) \mathbf {P} (w _ {x}) \mathbf {P} (x) \overline {{\mathbf {P} (x)}}} = \mathbf {P} (y _ {z}). \\ \end{array}
$$
To summarize, for a wide class of cases, Thm. 3 tells us that the two principles encoded in  $\mathrm{T}(\mathcal{G})$  exhaust the types of algebraic statements that a researcher is committed to when venturing a graphical assumption. Putting this in algebraic terms facilitates a perspective on such assumptions that can be naturally interpreted with respect to RCMs, independent of any representation by an SCM.
2.3 Single-World Intervention Graphs
Another graphical framework that draws on ideas and concepts from both frameworks is that of single-world intervention graphs (SWIGs) [30]. In comparison to the usual formulation of do-calculus, SWIGs facilitate reasoning with a wider class of expressions by combining graphical and potential outcome notations (see, e.g., [20]). Here we show that this "hybrid" framework can also naturally be assimilated to the logical perspective adopted in the present paper. Assuming acyclicity:
Definition 14. Let  $\mathcal{D}$  be a dag over  $\mathbf{V}$  and let  $\mathbf{x}$  be an intervention. Let  $\mathbf{An}_V^{\mathcal{D}} = \{V' \in \mathbf{V} : V' \to \dots \to V$  in  $\mathcal{D}\}$  be the directed ancestors of a variable  $V$  in  $\mathcal{D}$ . Then the SWIG  $\mathcal{D}_{\mathbf{x}}$  has nodes labeled
I.e., the transitive closure of the converse of the parent sets  $\mathbf{P}\mathbf{a}_V^{\mathcal{D}}$ . Note that  $V\in \mathbf{A}\mathbf{n}_V^{\mathcal{D}}$  and  $\mathbf{P}\mathbf{a}_V^{\mathcal{D}}\subset \mathbf{A}\mathbf{n}_V^{\mathcal{D}}$ .
$\{V_{\pi_{\mathbf{A}}(\mathbf{x})}:V\in \mathbf{V}\} \cup \{\pi_V(\mathbf{x}):V\in \mathbf{X}\}$ , where  $\mathbf{A} = (\mathbf{An}_V^{\mathcal{D}}\cap \mathbf{X})\setminus \{V\}$ , with nodes in the first set in this union termed random, and those in the second termed fixed. This SWIG has edges
$$
\{\pi_ {X} (\mathbf {x}) \to V _ {\mathbf {p}}: X \in \mathbf {X}, \mathcal {G} \text {h a s e d g e} X \to V \} \cup \{V _ {\mathbf {p}} \to V _ {\mathbf {p} ^ {\prime}} ^ {\prime}: V \notin \mathbf {X}, \mathcal {G} \text {h a s e d g e} V \to V ^ {\prime} \}.
$$
See Fig. 2c for an example of this construction. Note that edges in the first set in the union above have fixed heads and random tails, while those in the second set have random heads and tails.
Definition 15. Define the following conditional independence schema  $\mathsf{sw - sep}^{\mathcal{D}_{\mathbf{x}}}$ , for any sets of random nodes  $\{(X_{i})_{\mathbf{p}_{i}}:1\leq i\leq l\}$  and  $\{(Y_j)_{\mathbf{p}_j'}:1\leq j\leq m\}$  that are d-separated (Def. B.1) given  $\{(Z_k)_{\mathbf{p}_k''}:1\leq k\leq n\}$  in the SWIG  $\mathcal{D}_{\mathbf{x}}$ :
$$
\begin{array}{l} \mathsf{sw - sep}^{\mathcal{D}_{\mathbf{x}}}. \quad \mathbf{P}\big[\bigwedge_{\substack{1\leq i\leq l\\ 1\leq j\leq m}}(x_{i})_{\mathbf{p}_{i}}\wedge (y_{j})_{\mathbf{p}_{j}^{\prime}}\big]\cdot \mathbf{P}\big[\bigwedge_{1\leq k\leq n}(z_{k})_{\mathbf{p}_{k}^{\prime \prime}}\big] \\ = \mathbf {P} \left[ \bigwedge_ {\substack {1 \leq i \leq l \\ 1 \leq k \leq n}} (x _ {i}) _ {\mathbf {p} _ {i}} \wedge (z _ {k}) _ {\mathbf {p} _ {k} ^ {\prime \prime}} \right] \cdot \mathbf {P} \left[ \bigwedge_ {\substack {1 \leq j \leq m \\ 1 \leq k \leq n}} (y _ {j}) _ {\mathbf {p} _ {j} ^ {\prime}} \wedge (z _ {k}) _ {\mathbf {p} _ {k} ^ {\prime \prime}} \right]. \tag{10} \\ \end{array}
$$
One notable model associated with SWIGs is the  $FFRCISTG$  [32]; given the same graph, FFRCISTGs are compatible with SCMs [30] but issue fewer (potentially controversial) implications:
Definition 16. Let  $\mathcal{R}$  be a full RCM. Then  $\mathcal{R}$  is a FFRCISTG over  $\mathcal{D}$  if every instance of  $\mathrm{T}(\mathsf{ER}^{\mathcal{D}})$  and  $\mathsf{sw - sep}^{\mathcal{D}}$  holds in its counterfactual distribution. Let  $\mathfrak{F}(\mathcal{D})$  be the class of FFRCISTGs over  $\mathcal{D}$ .
Proposition 3. Suppose the SCM  $\mathcal{M}\in \mathfrak{M}(\mathcal{D})$  represents the full RCM  $\mathcal{R}$ . Then  $\mathcal{R}\in \mathfrak{F}(\mathcal{D})$ .
Given that Def. 16 already defines  $\mathfrak{F}(\mathcal{D})$  in terms of  $\mathcal{L}$ -principles, while [30] have shown the soundness direction, the following is straightforward:
Theorem 4.  $\mathsf{RCM} + \mathsf{T}(\mathsf{ER}^{\mathcal{D}}) + \bigcup_{\mathbf{x}}\mathsf{sw - sep}^{\mathcal{D}_{\mathbf{x}}}$  is sound and complete over  $\mathfrak{F}(\mathcal{D})$
3 Conclusion
The task of this paper has been to clarify the senses in which the Rubin causal model and structural causal models are very closely related formalisms for encoding causal assumptions and deriving causal conclusions. We concur with [23], [14], [41] and others that “there are insights that arise when using each that are less transparent when using the other” [41, p. 8]. Our interest in this paper has been to elucidate the comparison from a theoretical (“in principle”) perspective.
We do not suppose that the present work will be the final word on theoretical connections between RCMs and SCMs. On the contrary, there remain numerous open questions. Perhaps chief among these is the generalization of Thm. 3 to encompass all possible causal diagrams (not just those in which the bidirected arcs form a disjoint union of complete graphs). Does the theorem hold with no further principles, or do additional algebraic constraints arise? This important open question [39, 8, 31] is a crucial step toward a complete theoretical synthesis of the two frameworks.
This work was partially supported by a seed grant from the Stanford Institute for Human-Centered Artificial Intelligence. We also thank Elias Bareinboim and Guido Imbens for helpful conversations.