SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Duligur Ibeling
duligur@stanford.edu
Thomas Icard
icard@stanford.edu
The aim of this paper is to make clear and precise the relationship between the Rubin causal model (RCM) and structural causal model (SCM) frameworks for causal inference. Adopting a neutral logical perspective, and drawing on previous work, we show what is required for an RCM to be representable by an SCM. A key result then shows that every RCM—including those that violate algebraic principles implied by the SCM framework—emerges as an abstraction of some representable RCM. Finally, we illustrate the power of this conciliatory perspective by pinpointing an important role for SCM principles in classic applications of RCMs; conversely, we offer a characterization of the algebraic constraints implied by a graph, helping to substantiate further comparisons between the two frameworks.
Toward the end of the twentieth century several frameworks arose for formalizing and analyzing problems of causal inference. One of these, associated with Rubin [34] and others (see [16]), takes the potential outcome—formalizing the effect of one variable on another—as a fundamental target of analysis. Causal assumptions in the Rubin causal model (RCM) are naturally encoded as algebraic constraints on potential outcomes, and research in this area has spawned a remarkable body of theoretical and applied work especially in social and biomedical sciences (see [14] for a review).
A second approach, associated with Pearl [26] and collaborators (see [27] for a textbook treatment; see also [37]), focuses instead on assumptions that can be encoded qualitatively, or more specifically, graphically, arising from a fundamental object known as a structural causal model (SCM). The do-calculus is one of the crowning achievements of this approach, and it has been shown derivationally complete with respect to a wide range of canonical estimation and inference problems [35, 2, 19].
Both frameworks have enjoyed considerable influence within recent causal machine learning in particular. As just one example, concern in reinforcement learning about the possibility of unobserved confounders—variables impacting both decisions and their outcomes—has generated a number of important advances, some employing tools and concepts from the RCM approach (e.g., [18, 25, 17]), others grounded in the SCM approach and typically involving graphs (e.g., [3, 38, 42]).
Despite the remarkable successes that both of these frameworks have engendered in machine learning and beyond, there remains substantial controversy over how to understand their relationship. In the literature it has been claimed, on the one hand, that the two are equivalent, that “a theorem in one is a theorem in the other” [27, p. 98]. On the other hand, some authors suggest that the two are only equivalent in a weak sense, one that “builds the limitations of SCM into the resulting comparison and likewise filters out aspects of the rival theory that do not readily translate to SCM” [21, p. 443].
At issue are two separable questions. The first is one of practical significance. Some argue that graphs give greater "conceptual clarity" [27, p. 103] and that SCMs more generally offer a "a flexible formalism for data-generating models" that helps ground causal inquiry [4, p. 514]; others argue that work in the RCM framework provides "transparent definitions of causal effects and encourages the analyst to consider individual-level heterogeneity as a first principle" [23, p. 91] as well as "guidance
to researchers and policy makers for practical implementation” [14, p. 1131]. While obviously very important, our goal is not to address these disputes about what theoretical primitives are most “natural” or “useful” for practitioners or applied researchers. Rather, the aim of the present contribution is to offer a number of new technical results that together shed light on a more basic question, namely, how precisely the RCM and SCM frameworks relate at a theoretical level. For example, are the two merely notational variants, or does one tactically enforce assumptions that the other does not?
In this paper we first endeavor, building on previous work, to elucidate the precise sense in which SCMs can be construed as representations of RCMs, provided the latter satisfy two key principles known as composition and reversibility [9, 10]. Interestingly, such principles (or their logical consequences) have been questioned in the literature (e.g., [7]). Our second goal is to help clarify the sense in which they may fail. Drawing from recent literature on causal abstraction (e.g., [33, 5])—broadened to cover both SCMs and RCMs—we suggest that failure of these principles plausibly results when causally relevant low-level details are elided in favor of more abstract variables. Our Thm. 1 buttresses this intuition, showing that every RCM is a constructive abstraction of a representable RCM (hence satisfying composition and reversibility). We furthermore remark on how the well known SUTVA assumptions [16] can be understood as conditions on good variable abstractions.
Our starting point in this work is theoretically neutral, taking for granted only the primitive, "probability of a counterfactual." In the second half of the paper we introduce a framework-neutral formal language for reasoning about such probabilities, which in turn facilities further comparison. With respect to this common language, we offer a completeness result for the class of all RCMs (Thm. 2) and, drawing on [10], the class of representable RCMs (Cor. 2). These results are illustrated with an example derivation of LATE (see [15, 1]), which also helps illuminate which assumptions are logically required in the derivation. Meanwhile, we offer a partial answer to the well-known open question [39, 31] of how to characterize the algebraic constraints implied by a particular graph (Thm. 3), a result that helps bring graphical assumptions into this neutral common language. Finally, we show how an existing result on single-world intervention graphs (SWIGs), a framework drawing from both perspectives, can be construed as a completeness result for the same language (Thm. 4).
Taken together, our results are largely conciliatory—in the same spirit as other important conciliatory work in this context; see, e.g., [30, 36, 20]—showing how the two frameworks are productively compatible, while also suggesting distinctive perspectives on problems of causal inference.
Proofs are deferred to supplementary appendices A, B, which contain additional technical material.
1 Modeling
We first introduce a formalization of the Rubin causal model [34, 11, 12] before then turning to structural causal models [26, 27, 4]. The relationship between these two is elucidated in §1.1.3.
Common to both frameworks will be a set  $\mathbf{V}$  of endogenous variables. Concerning notation:
Notation. The signature (or range) of a variable  $V$  is denoted  $\operatorname{Val}(V)$ . Where  $\mathbf{S}$  is a set of variables, let  $\operatorname{Val}(\mathbf{S}) = \times_{S \in \mathbf{S}} \operatorname{Val}(S)$ , the product of the family of sets  $\operatorname{Val}(S)$  indexed by  $S \in \mathbf{S}$ . Elements of  $\operatorname{Val}(\mathbf{S})$  represent joint valuations of the variables  $\mathbf{S}$ . Given an indexed family of sets  $\{S_{\beta}\}_{\beta \in B}$  and elements  $s_{\beta} \in S_{\beta}$ , let  $\{s_{\beta}\}_{\beta}$  denote the indexed family of elements whose object associated with the index  $\beta$  is  $s_{\beta}$ , for all  $\beta$ . The symbol  $\subset$  indicates any subset (or set inclusion) and does not imply a strict subset (or proper inclusion). For  $B' \subset B$  write  $\pi_{B'}: \times_{\beta \in B} S_{\beta} \to \times_{\beta \in B'} S_{\beta}$  for the projection map sending each  $\{s_{\beta}\}_{\beta \in B} \mapsto \{s_{\beta'}\}_{\beta' \in B'}$ ; abbreviate  $\pi_{\beta'} = \pi_{\{\beta'\}}$ , for  $\beta' \in B$ . Thus if  $\mathbf{s} \in \operatorname{Val}(\mathbf{S})$  is a joint valuation of variables  $\mathbf{S}$  and  $S \in \mathbf{S}$  is a single variable, then  $s = \pi_S(\mathbf{s}) \in \operatorname{Val}(S)$  is a value for  $S$ . If  $\mathbf{S}' \subset \mathbf{S}$  then  $\pi_{\mathbf{S}'}(\mathbf{s}) \in \operatorname{Val}(\mathbf{S}')$  is a joint valuation of  $\mathbf{S}'$ , namely the projection of  $\mathbf{s}$  to  $\mathbf{S}'$ . Upper-case letters like  $\mathbf{S}$  conventionally represent those sets of variables that the corresponding lower-case letters  $\mathbf{s}$  are valuations of,  $\mathbf{s} \in \operatorname{Val}(\mathbf{S})$ .
1.1.1 Rubin Causal Models, Potential Outcomes, Counterfactuals
The present formalization of the Rubin causal model [34, 16] loosely follows previous presentations; see especially [12]. It codifies experimental outcomes across individuals drawn from a distribution.
These are potential outcomes over a variable set  $\mathbf{V}$ , defined as expressions of the form  $Y_{\mathbf{x}}$  for an outcome  $Y \in \mathbf{V}$  and an intervention or treatment  $\mathbf{x} \in \mathrm{Val}(\mathbf{X})$  for some  $\mathbf{X} \subset \mathbf{V}$ , and interpreted as the value observed for  $Y$  in a controlled experiment where each  $X \in \mathbf{X}$  is held fixed to  $\pi_X(\mathbf{x})$ .
Definition 1. A Rubin causal model (RCM) is a tuple  $\mathcal{R} = \langle \mathcal{U},\mathbf{V},\mathfrak{D},\mathfrak{F},P\rangle$  where  $\mathcal{U}$  is a finite set of units or individuals,  $\mathbf{V}$  is a finite set of endogenous variables,  $\mathfrak{D}$  is a set of potential outcomes over  $\mathbf{V}$ ,  $\mathfrak{F}$  is a set of potential response functions, to be defined shortly, and  $P:\mathcal{U}\to [0,1]$  is a probability distribution on  $\mathcal{U}$ . A potential response for  $Y_{\mathbf{x}}\in \mathfrak{D}$  is a function  $f_{Y_{\mathbf{x}}}:U\to \operatorname {Val}(Y)$ . For each  $Y_{\mathbf{x}}\in \mathfrak{D}$  we require that  $\mathfrak{F} = \{f_{Y_{\mathbf{x}}}\}_{Y_{\mathbf{x}}\in \mathfrak{D}}$  contain exactly one such function.
RCMs are often specified in a tabular form as in, e.g., Fig. 1 below. We adopt the notation  $y_{\mathbf{x}}(i)$  or  $Y_{\mathbf{x}}(i) = y$  as a shorthand for  $f_{Y_{\mathbf{x}}} (i) = y$ : for  $\mathcal{R}$  as in Def. 1, write  $\mathcal{R} \models y_{\mathbf{x}}(i)$  iff  $Y_{\mathbf{x}} \in \mathfrak{D}$  and  $f_{Y_{\mathbf{x}}} (i) = y$ . This means that in the above controlled experiment, outcome  $y \in \mathrm{Val}(Y)$  is observed for individual  $i$ . Each  $Y_{\mathbf{x}}$  can be thought of as a variable with range  $\mathrm{Val}(Y_{\mathbf{x}}) = \mathrm{Val}(Y)$ . We call the set  $\mathrm{Val}(\mathfrak{D})$  of joint valuations of these variables counterfactuals. A set of potential responses  $\mathfrak{F}$  then maps units to counterfactuals,  $\mathfrak{F}: \mathcal{U} \to \mathrm{Val}(\mathfrak{D})$ , by defining  $\mathfrak{F}(i) = \{f_{Y_{\mathbf{x}}} (i)\}_{Y_{\mathbf{x}} \in \mathfrak{D}}$ , and:
Definition 2. Where  $\mathcal{R}$  is as in Def. 1, the counterfactual distribution  $P_{\mathrm{cf}}^{\mathcal{R}}:\operatorname {Val}(\mathfrak{O})\to [0,1]$  induced by  $\mathcal{R}$  is the pushforward  $\mathfrak{F}_*(P)$  of  $P:\mathcal{U}\rightarrow [0,1]$  under  $\mathfrak{F}:\mathcal{U}\to \operatorname {Val}(\mathfrak{O})$
The reason we call  $P_{\mathrm{cf}}^{\mathcal{R}}$  a counterfactual distribution (and  $\operatorname{Val}(\mathfrak{O})$  counterfactuals) is because such joint probabilities over multiple potential outcomes appear in the usual ratio definition of counterfactual probabilities. For instance,  $P(y_x|y_{x'}') = P(y_x,y_{x'}') / P(y_{x'}')$  gives the probability that a unit who was withheld treatment and did not recover would have recovered if assigned treatment. But  $P_{\mathrm{cf}}^{\mathcal{R}}$  also answers via marginalization all questions (whenever defined by  $\mathcal{R}$ ) about "interventional" probabilities like  $P(y_{\mathbf{x}})$ , as well as purely "observational" probabilities such as  $P(\mathbf{x})$ ; see, e.g., [4].
Some authors submit that "probability will mean nothing more nor less than a proportion of units" [11, p. 945], thereby assuming a uniform distribution on a finite population  $\mathcal{U}$  (cf. also [1]). Of course, in the infinite population size limit we recover all RCMs as in Def. 1 (see Prop. A.1).
While practitioners do not typically consider potential outcomes  $Y_{\mathbf{x}}$  when  $Y \in \mathbf{X}$ , instead maintaining a strict dichotomy between cause and effect variables (e.g., [11, 12]), it is natural to impose the following requirement (known as effectiveness) whenever such potential outcomes are defined. An intervention is always assumed to be a successful intervention: whenever  $Y \in \mathbf{X}$ ,
$$
\text {E f f e c t i v e n e s s .} \quad Y _ {\mathbf {x}} (u) = \pi_ {Y} (\mathbf {x}) \tag {1}
$$
for every  $u \in \mathcal{U}$ . In fact, practice in the RCM framework reflects this same assumption, in the sense that violations of it are taken to signify a poor choice of variables. As a classic example, the possibility of non-compliance in an experimental trial motivates the introduction of instrumental variables, and specifically a separation between, e.g., treatment and assignment to treatment (cf. Ex. 3). Crucially, we recover effectiveness with respect to the latter. We will assume any RCM to meet (1) unless otherwise specified; let  $\Re_{\mathrm{eff}}$  be the class of such RCMs. $^3$
1.1.2 Structural Causal Models
An important feature of RCMs is that potential outcomes are cleanly separated from assignment mechanisms [16]. A different starting point is to assume that potential outcomes and their algebraic behavior are rather derived from an underlying formal representation of causal structure. These putatively "deeper mathematical objects" [27, p. 102] involve concrete functional dependencies, and an operation of function replacement known as intervention:
Definition 3. A structural causal model (SCM) is a tuple  $\mathcal{M} = \langle \mathbf{U},\mathbf{V},\mathcal{F},P(\mathbf{U})\rangle$  where  $\mathbf{U}$  is a finite set of exogenous variables,  $\mathbf{V}$  is a finite set of endogenous variables,  $\mathcal{F}$  is a family of structural
<sup>1</sup>By generalizing to allow multiple such functions one arrives at a class of models closely related to the generalized structural equation models (GSEMs) of Peters and Halpern [29], or the equivalent class of causal constraint models (CCMs) introduced by Blom et al. [6]. Rather than mapping each potential outcome  $Y_{\mathbf{x}}$  to a value in  $\operatorname{Val}(Y)$ , GSEMs map each (allowable) intervention  $\mathbf{x}$  to a (possibly empty) set of values for all the variables, that is, to elements of the powerset  $\wp(\operatorname{Val}(\mathbf{V}))$ . RCMs, by contrast, allow that, e.g.,  $Y_{\mathbf{x}}$  may be defined for all  $\mathbf{u}$ , while  $Z_{\mathbf{x}}$  is undefined because  $Z_{\mathbf{x}} \notin \mathfrak{D}$ . The two are thus incomparable in expressive power. <sup>2</sup>That is,  $P_{cf}^{\mathcal{R}}(\mathbf{o}) = P(\mathfrak{F}^{-1}(\{\mathbf{o}\}))$  for any  $\mathbf{o} \in \operatorname{Val}(\mathfrak{D})$ .
<sup>3</sup>Not only does this assumption reflect practice, but it is also without loss with regard to comparing RCMs and SCMs, as the latter also satisfy effectiveness: see footnote 4.
functions, to be defined shortly, and  $P: \operatorname{Val}(\mathbf{U}) \to [0,1]$  is a probability distribution on (joint valuations of)  $\mathbf{U}$ .
A structural function for  $V \in \mathbf{V}$  is a function  $f_V: \operatorname{Val}(\mathbf{U}_V \cup \mathbf{P}\mathbf{a}_V) \to \operatorname{Val}(V)$ , where  $\mathbf{U}_V \subset \mathbf{U}$ ,  $\mathbf{P}\mathbf{a}_V \subset \mathbf{V} \setminus \{V\}$ . For every  $V \in \mathbf{V}$  we require that  $\mathcal{F} = \{f_V\}_{V}$  have exactly one such function; the entire collection  $\mathcal{F}$  thus forms an exogenous-to-endogenous mapping.
Interventions come as a derived notion, replacing structural functions with constant functions [22, 27]:
Definition 4 (Intervention on SCMs). Let  $\mathbf{x} \in \mathrm{Val}(\mathbf{X})$  for some  $\mathbf{X} \subset \mathbf{V}$  be an intervention and  $\mathcal{M}$  be the SCM of Def. 3. Then define a manipulated model  $\mathcal{M}_{\mathbf{x}} = \langle \mathbf{U}, \mathbf{V}, \{f_V'\}_{V}, P(\mathbf{U}) \rangle$  where each  $f_V': \mathrm{Val}(\mathbf{U}_V' \cup \mathbf{P}\mathbf{a}_V') \to \mathrm{Val}(V)$ . If  $V \notin \mathbf{X}$  define  $\mathbf{U}_V' = \mathbf{U}_V$ ,  $\mathbf{P}\mathbf{a}_V' = \mathbf{P}\mathbf{a}_V$ , and  $f_V' = f_V$ . If  $V \in \mathbf{X}$  define  $\mathbf{U}_V' = \mathbf{P}\mathbf{a}_V' = \emptyset$  and  $f_V'$  as a constant function mapping to  $\pi_V(\mathbf{x})$ .
Letting  $\mathcal{M}$  be the SCM of Def. 3, for  $\mathbf{v} \in \mathrm{Val}(\mathbf{V})$  and  $\mathbf{u} \in \mathrm{Val}(\mathbf{U})$  write  $\mathcal{M}, \mathbf{u} \models \mathbf{v}$  if we have  $f_{V}\left(\pi_{\mathbf{U}_{V} \cup \mathbf{P}\mathbf{a}_{V}}(\mathbf{v})\right) = \pi_{V}(\mathbf{v})$  for every  $V$ . Let  $\mathfrak{M}_{\mathrm{uniq}}$  be the class of all SCMs  $\mathcal{M}$  such that, for any  $\mathbf{u}$  and intervention  $\mathbf{x}$  there is a unique solution  $\mathbf{v}$  such that  $\mathcal{M}_{\mathbf{x}}, \mathbf{u} \models \mathbf{v}$ . In this case we define the potential outcome  $Y_{\mathbf{x}}(\mathbf{u})$  as  $\pi_{Y}(\mathbf{v})$ . Thus any  $\mathcal{M} \in \mathfrak{M}_{\mathrm{uniq}}$  defines a potential outcome for every  $Y_{\mathbf{x}}$ , giving a natural function  $p^{\mathcal{M}}: \mathrm{Val}(\mathbf{U}) \to \mathrm{Val}(\{Y_{\mathbf{x}}\}_{\text{all } Y_{\mathbf{x}}})$  via these outcomes, and:
Definition 5. The counterfactual distribution  $P_{\mathrm{cf}}^{\mathcal{M}}:\operatorname {Val}(\{Y_{\mathbf{x}}\}_{\mathrm{all}Y,\mathbf{x}})\to [0,1]$  induced by  $\mathcal{M}\in$ $\mathfrak{M}_{\mathrm{uniq}}$  is the pushforward  $p_*^\mathcal{M}(P)$  of  $P:\operatorname {Val}(\mathbf{U})\to [0,1]$  under  $p^{\mathcal{M}}:\operatorname {Val}(\mathbf{U})\to \operatorname {Val}(\{Y_{\mathbf{x}}\}_{\mathrm{all}Y,\mathbf{x}})$
Thus, SCMs in  $\mathfrak{M}_{\mathrm{unic}}$  canonically define counterfactual distributions for all possible potential outcomes via manipulation of functional dependencies. Importantly, Def. 5 provides a bridge to RCMs, as both produce counterfactual distributions (recall Def. 2). As long as the counterfactual probabilities are assumed to mean the same thing—i.e., as long as they highlight the same targets for empirical and theoretical investigation—we can then compare the ranges of assumptions and inference patterns that each framework can encode about them. We thus assume that all our SCMs belong to this class  $\mathfrak{M}_{\mathrm{unic}}$ .
1.1.3 Representation of RCMs by SCMs
A contentious methodological question is whether all (endogenous) variables should be potential candidates for intervention. Following the literature we have supposed that SCMs allow all possible interventions (though this assumption is not universal; see, e.g., [33, 5]). For RCMs it is generally assumed that there can be "no causation without manipulation" [11, 16], and thus that only some interventions should be allowed. While methodologically important, this is theoretically inessential. We can construe SCMs as possible representations of RCMs in the following sense:
Definition 6. Let  $\mathcal{R} \in \Re_{\mathrm{eff}}$  and  $\mathcal{M} \in \mathfrak{M}_{\mathrm{uniq}}$ . We say that  $\mathcal{M}$  represents  $\mathcal{R}$  if its counterfactual distribution  $P_{\mathrm{cf}}^{\mathcal{M}}$  marginalizes down to  $P_{\mathrm{cf}}^{\mathcal{R}}$  on the potential outcomes defined by  $\mathcal{R}$  (the set  $\mathfrak{D}$  in Def. 1). We say  $\mathcal{R}$  is representable if it is represented by at least some  $\mathcal{M} \in \mathfrak{M}_{\mathrm{uniq}}$ .<sup>4</sup>
Thus  $\mathcal{M}$  represents  $\mathcal{R}$  if they are counterfactually equivalent with respect to the outcomes defined by  $\mathcal{R}$ . Toward a characterization of representability, consider two properties of an RCM [9]:
Definition 7. The following Boolean formulas encode assumptions about potential outcomes:
$$
\text {C o m p o s i t i o n .} \quad Y _ {\mathbf {w}} (u) = y \wedge Z _ {\mathbf {w}} (u) = z \rightarrow Z _ {\mathbf {w} y} (u) = z \tag {2}
$$
$$
\text {R e v e r s i b i l i t y .} \quad Y _ {\mathbf {w} z} (u) = y \wedge Z _ {\mathbf {w} y} (u) = z \rightarrow Y _ {\mathbf {w}} (u) = y. \tag {3}
$$
Say  $\mathcal{R} \in \Re_{\mathrm{eff}}$  satisfies composition and reversibility, respectively, when the respective statements hold for every unit  $u$  of  $\mathcal{R}$ , whenever all the appropriate potential outcomes are defined.
We understand lower-case values like  $y$ ,  $z$ ,  $\mathbf{w}$ , when not bound as dummy indices or otherwise, to be schematic variables carrying tacit universal quantifiers. Thus (2), (3) must hold for all possible  $y \in \operatorname{Val}(Y)$ ,  $\mathbf{w} \in \operatorname{Val}(\mathbf{W})$ ,  $z \in \operatorname{Val}(Z)$ . This same usage is repeated, e.g., in (9).
While reversibility seems not to have arisen in the potential outcomes literature, instances of composition have appeared explicitly (e.g., Holland 12, p. 968) and have been used implicitly in concrete
<sup>4</sup>With regard to representability, our assumption that  $\mathcal{R} \in \Re_{\mathrm{eff}}$  is without loss since Def. 4 easily implies that the potential outcomes induced by any SCM must satisfy effectiveness (1).
derivations (see Ex. 3 below). Note that the well-known principle of consistency [7, 27] is merely the instance of composition for  $\mathbf{W} = \varnothing$ . For  $\mathcal{R}, \mathcal{R}' \in \Re_{\mathrm{eff}}$  that share the same units  $\mathcal{U}$  and endogenous variables  $\mathbf{V}$  but have respective potential outcome sets  $\mathfrak{D}, \mathfrak{D}'$  and potential response sets  $\mathfrak{F}, \mathfrak{F}'$ , if  $\mathfrak{D} \subset \mathfrak{D}'$  and  $\mathfrak{F} \subset \mathfrak{F}'$  we say that  $\mathcal{R}'$  extends or is an extension of  $\mathcal{R}$  and  $\mathcal{R}$  is a submodel of  $\mathcal{R}'$ . Call  $\mathcal{R}$  full if it has no proper extension. Then:
Proposition 1 (SCM Representation). RCM  $\mathcal{R}$  is representable iff  $\mathcal{R}$  extends to some full  $\mathcal{R}'$  that satisfies composition and reversibility.
Note that for an RCM  $\mathcal{R}$  to be representable it is necessary (though not sufficient, in light of the models presented in Fig. 1 below) that  $\mathcal{R}$  itself witness no composition or reversibility failures. Prop. 1 thus clarifies a sense in which RCMs are more general than SCMs, not just by allowing only a subset of allowable interventions, but also by imposing fewer requirements on how potential outcomes relate to one another. However, assuming composition, reversibility, and fullness, the two define the same classes of counterfactual distributions, despite the superficial differences in their definitions. In that case the two are, e.g., equivalent for interpreting the probabilistic logical language of  $\S 2$ . We submit that some version of this result makes sense of statements in the literature, e.g., from Pearl [27], that the twain are essentially equivalent frameworks from a theoretical perspective.
1.2 Causal Abstraction
The goal of this section is to clarify the source of putative failures of principles like composition. We suggest that it is helpful to view these issues through the lens of causal abstraction (the definitions in this section are adapted from [33, 5]). Abstraction has mostly been studied in the context of SCMs; our definitions apply equally to SCMs and RCMs via counterfactual distributions.