Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions
Authors: Duligur Ibeling
Thomas Icard
Venue: NeurIPS 2023 poster (2023)
Keywords: potential outcomes framework
structural causal model
causal inference
logic
probability
graphical causal models
causal abstraction
causal machine learning

Abstract: 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 ameliorative 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.

Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions
Duligur Theling
Thomas Icard Department of Philosophy Stanford University icard@stanf ord edu
Department of Computer Science Stanford University dul aulligu@stanfordd edu
Abstract
The aim of this paper 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 RCM to be representable by an SCM. A key result then shows that every implied by the SCM framework ereeres RCM. Finally, we illustrate the power of this conciliatory perspective by pinpointing important role for SCM principles in classic applications of RCMs; conversely, we offer a characterization of the algebraic constraints implied by a graph, helping substantiate further comparisons between the two frameworks.
those that violate algebraic principles abstraction of some representable
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- formaliiing the effect of one variable another-as 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 remarkable body of theoretical and applied work especially in social and biomedical sciences (see [4] for a review).
A second approach, associated with Pearl 22 and collaborators (see [27] for textbook treatment; see also (37)), focuses instead on assumptions that can be encoded qualitatively, or more specifically, graphically, arising from a fundamental object known calculus is one of the crowning achievements of this approach, and it has been shown derivationally complete with respect
structural causal model (SCM). The do-
wide range of canonical estimation and inference problems 35 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-
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
others grounded in the SCM approach and typically involving graphs (e.g.. 3 38
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 the other" 27 p. 98]. On the other hand, some authors suggest that the two are only equivalent in 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" [2] 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 4 514]; others argue that work in the RCM framework provides "transparent definitions of causal effects and encourages the analyst to consider individual-level heterogeneity as first principle" 23 p. 91] as well as "guidance
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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to researchers and policy makers for practical implementation" [14] p. 1131]. While obviously important, our goal is not to address these disputes about what theoretical primitives are most "natural" or "useful" for practitioners applied researchers. Rather, the aim of the present contribution is to offer number of new technical results that together shed light on a more basic question, namely, how precisely the RCM and SCM frameworks relate at theoretical level. For example, are the two merely notational variants, or does one tactictly enforce
aeseeettsiioo 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., 77. Our second goal is to help clarify the sense in which they may fail. Drawing from recent literature on causal abstraction (e.g.. [33 51)
broadened cover both SCMs and RCMs-we 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. [ buttresses this intuition, showing that every RCM is constructive abstraction of representable RCM (hence satisfying composition and reversibility). We furthermore remark 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, "proba- bility of counterfactual." In the second half of the paper we introduce language for reasoning about such probabilities, which in turn facilities further comparison. With respect 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 D. which also helps illuminate which assumptions are logically required in the derivation. Meanwhile, we offer 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-worls intervention graphs (SWIGs), a framework drawing from both perspectives, can be construed as a completeness result for the same language (Thm. D
Homework-neutrer formal
Taken together, our results are largely conciliatory the same spirit as other important conciliatory work in this context; see, c.g., [30 36 showinng 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.
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 $L..3
1.1 Preliminaries
Common to both frameworks will be set V of endogenous variables. Concerning notation:
Notation. The signature (or range) of variable V is denoted Val(V). Where S is a set of variables, let Val(S) = Xses Val(S), the product of the family of sets Val(S) indexed by S S. Elements of Val(S) represent joint valuations of the variables S. Given an indexed family of sets {S and elements s, E Sg, let { denote the indexed family of elements whose object associated with the index is sg, for all B. The symbol C indicates any subset (or set inclusion) and does not imply strict subset (or proper inclusion). For B' B write म.' X S - X S3 for the projection map sending each {Sa]øeB + {s y eB; abbreviate g = [[e). for B E B. Thus if E Val(S) is a joint valuation of variables S and E S a single variable, then = (s) E Val(S) is a value for S. If S' C S then (s) E Val((') is joint valuation of S', namely the projection of s to S'. Upper-case letters like S conventionally represent those sets of variables that the corresponding lower-case letters are valuations of, Val(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 [2]. It codifies experimental outcomes across individuals drawn from distribution.
2

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These are potential outcomes over variable set V. defined as expressions of the form Yx for an outcome Y E V and an intervention or treatment x E XX) for some X C V. and interpreted as the value observed for Y in controlled experiment where each X E X is held fixed to x (x).
Definition 1. A Rubin causal model (RCM) is a tuple R - (U, V, D, B. P) where U/ is a finite set of wnits or individuals, V is finite set of endogenous variables, D is set of potential outcomes over V. 8 is a set of potential response functions, to be defined shortly, and P : U - [0, 1] is a probability distribution on U. A potential response for Yx E D is a function fy= 4 - Val(Y). For each Xx= E D we require that 3 = {fy,) JYED contain exactly one such function M
RCMs are often specified in tabular form as in, e.g., Fig. 1 below. We adopt the notation (i) or Yx (t) = y as a shorthand for fy, (i) = y: for R as Def. write R. (x(()) iff Yx E 2 and fyx (i) = y. This means that in the above controlled experiment, outcome E VYl(Y) is observed for individual i. Each Yx can be thought of as a variable with range Val(X*) = Val(Y). We call the set Val(D) of joint valuations of these variables counterfactuals. A set of potential responses 3 then maps units to counterfactuals, 3 : l + Val(D), by defining B(1) = {rv, (0))y.e and:
Definition 2. Where R. is as in Def. the counterfactual distribution P/P VVa((D) - [0, 1] induced by R is the pushforward 2 3. (P) of P u - [0, 1] under 3 U + Val(S).
The reason we call Pd a counterfactual distribution (and Val(D) counterfactuals) is because such joint probabilities over multiple potential outcomes appear in the usual ratio definition of counter- factual probabilities. For instance, P( (v = P(yz: 3/ //tp/P((/e P(y/z)) gives the probability that a unit who was withheld treatment and did not recover would have recovered if assigned treatment. But PR also answers via marginalization all questions (whenever defined by R) about "interventional" probabilities like P(xx), as well as purely "observational" probabilities such as P(x); see, e.g., E.
Some authors submit that "probability will mean nothing more nor less than proportion of units" [ p. 945], thereby assuming a uniform distribution on a finite population 2/ (cf. also M.). Of course, in the infinite population size limit we recover all RCMs as Def. T (see Prop. A.I).
While practitioners do not typically consider potential outcomes Yx when Y E X, instead maintaining strict dichotomy between cause and effect variables (e.g.. [1] 12],, it is natural to impose the following requirement (known as effectiveness) whenever such potential outcomes are defined. An
intervention is always assumed be a successful intervention: whenever Y E X,
Effectiveness.
xy)))
(1)
for every u E U. In fact, practice in the RCM framework reflects this same assumption, in the sense that violations of it are taken to signify poor choice of variables. As classic example, the possibility of non-compliance in an experimental trial motivates the introduction of instrumental variables, and specifically separation between, e.g., treatment and assignment treatment (cf. Ex. 3. Crucially, we recover effectiveness with respect to the latter. We will assume any RCM to meet unless otherwise specified; let Retf be the class of such RCMs
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 M (U, V.V. FF,F P(U)) where U is finite set of exogenous variables, V is a finite set of endogenous variables, F is a family of structural
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 Halpem 29) the equivalent class of causal constraint models (CCMs) introduced by Blom et al. [6] Rather than mapping each potential outcome Yx to a value in Val(Y), ). GSEMs map each (allowable) intervention x to a (possibly empty) ser of values for all the variables, that is, to elements of the powerset ((Val(V))
RCMs, by contrast, allow that, e.g., Yx may be defined for all u. while Z, is undefined because Zx & D. The two are thus incomparable in expressive power.
"That is. P (o) P (fo))) for any o E Val(D).
'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
3

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functions, to be defined shortly, and P Ual(U) + [0, is probability distribution on (joint valuations of) U.
structural function for V E V is function fv Val(Uv U Pay) + VVl(V), where Uy C U, Pay C VV (V). For every V E V we require that J = {fv)v have exactly one such function; the entire collection F thus forms an exogenous-t--endogenooo mapping.
Interventions come derived notion, replacing structural functions with constant functions 22
Definition 4 (Intervention be the SCM of Def. fV Val(U,, U Pay) - Val(V). If & X define U Uv. Pay Pay, and f' = fv. If and fi as a constant function mapping to (x).
SCMs). Let Val(X) for some X C V be intervention and M 3 Then define a manipulated model MMx = (U, V. (V,J,,((U) where each
V E X define U. Pay-
Letting M be the SCM of Def. 3 for v E Val(V) and u E Val(U) write M, u F v if we have fv (TUUUUPay (v)) = TV (v) for every V. Let Manin be the class of all SCMs MM such that, for any and intervention there is unique solution such that Mx: u F In this case we define the potential outcome (u)) as Ty (v). Thus any M E Maiq defines a potential outcome for every Yx- giving a natural function pM : Val(U) + Val((Yx )anl,)) via these outcomes, and:
Definition 5.
The counterfactual distribution P.M. Val([X x - [0. 1] induced by M E Manig is the pushforward (P) of P: Val(U) + [o, 1] under pM Val(U) + Val([X))al) y,x).
Thus, SCMs in Munig canonically define counterfactual distributions for all possible potential outcomes via manipulation of functional dependencies. Importantly, Def. provides a bridge to RCMs, as both produce counterfactual distributions (recall Def. [ As long as the counterfactual probabilities are assumed to mean the same for empirical and theoretical investigation -e 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 Muniq.
as long as they highlight the same targets
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 possible interventions (though this assumption is not universal; see, e.g., K3 5).. For RCMs it is generally assumed that there can be "no causation without manipulation" " IT 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 Let R E Rell and M Nuniq. We say that M represents R if its counterfactual distribution Pet marginalizes down to PR on the potential outcomes defined by R (the in Def. I.
We say R is representable if it represented by least some .M Munin
Thus M represents R if they are counterfactually equivalent with respect to the outcomes defined by R. Toward a characterization of representability, consider two properties of an RCM (
Definition 7. The following Boolean formulas encode assumptions about potential outcomes:
Composition.
(w) = y A Z (u) = - Zwy =
Reversibility.
Ywz (u) y A Zwy = = Yw y.
(3)
Say R E Ren satisfies composition and reversibility, respectively, when the respective statements hold for every unit u of R. whenever all the appropriate potential outcomes are defined.
We understand lower-case values like y, 2, 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 E Val(Y), w E Val(W), E Val(Z). This same usage is repeated, e.g., in
While reversibility seems not to have arisen in the potential outcomes literature, instances of compo- sition have appeared explicitly (e.g., Holland 12] 968) and have been used implicitly in concrete
'With regard representability, our assumption that R E Rall is without loss since Def. easily implies that the potential outcomes induced by any SCM must satisfy effectiveness

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derivations (see Ex. below). Note that the well-known principle of consistency 7 27 is merely the instance of composition for W 2. For R, R' E Rot that share the same units U and endogenous variables V but have respective potential outcome sets D, D' and potential response sets 3, S', if D C D' and B C 3' we say that R' extends or an extension of R and R is a submodel of R'. Call R.lu if it has no proper extension. Then:
Proposition 1 (SCM Representation), RCM R is representable iff R extends to some full R' that satisfies composition and reversibility.
Note that for an RCM R. to be representable is necessary (though not sufficient, in light of the models presented in Fig. below) that R. itself witness no composition or reversibility failures. Prop. | thus clarifies a sense in which RCMs are more general than SCMs, not just by allowing only subset of allowable interventions, but also by imposing fewer requirements 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 $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 [3] 5)) Abstraction has mostly been studied in the context of SCMs; our definitions apply equally to SCMs and RCMs via counterfactual distributions.
In causal abstraction, one has a set VT. of low-level (or concrete, or micro-) variables representing a finegrained description and set VH of high-level (or abstract, or macro-) variables representing a coarser- coarser-grained description of the same scenario. The correspondence between the two descriptions given by a partial translation map : VV) + Val(VH). Translations extend canonically to maps of partial valuations (e.g., interventions) T Uxcv. Val(X) + Uxcvn Val(X) by setting
T(XL) = xH iff (=xx (xL) = xxn (xH).
We overload 5 once more so as to cover counterfactuals, defining as follows yet another partial :: Val(DL) - VaI(On) for any sets Dr. Du of potential outcomes over VL and Vи respectively, Index an element of Val(Dr.)
{(yf.);; Jisism. where xi. + x for any + j and yf. for each i, and an element of Val(Dn) likewise {(rH)sg hissn-
Val((Y VL Yx, E DL)) Define 7 (((yi))xx hsssmm) ((r))) Jissn if 7(x : 15 m)) (x1 1 S j S n) and r(y1.) - yÿ for any pair xf, Xi where T(xi,) = x/1
Definition 8. With counterfactual translation in hand, we define abstraction relation between probabilistic causal models. The model H abstracts L over the aligned variables (written H -, L) if the translation pushes the latter's counterfactual distribution to the former's, s, that is, P (PG).
stricter and typically more useful notion is that of constructive abstraction (e.g.. (. These arise from translations that can be generated variable-wise, and thus correspond to coherent "clustering" of variables:
Definition 9. Translation 7 Val(VL) - Val(VH) is constructive if there is partition II of VL with non-overlapping cells {IIv \veVHu(1)) each Hl C VL, where Hy is non-empty for all V * I, and collection (vve each of which is a partial surjective map TV : Val((Ty) - Val(V), such that T(V1.) = {Tv (mn, (VL)) vev, for any V1. E Val(VL).
simple abstraction, ubiquitous in the literature (see, e.g., 16 $1.6.2] and 77), is that of variable treatment levels. Here legeer-level value corresponds to some collection of lower-level specifications that might represent the potency or dosage of the drug administered. the time of administration, etc.: for example, a distinction of whether one took 300, 400, 500, or 600 mg of aspirin is made at the low level, but at the high level, there only the binary distinction between having taken aspirin and not. Formally, a treatment variable T is only binary with values c, ir (control, treatment resp.) at the high level but takes on many values c, trt, ..... tr at the low level. The abstraction made by omitting the fine-gre -neined details; symbolically, one forms new model by eliding the superscripts, collapsing
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all tr' into tr. So long as for any outcomes we have (u) = Yiri (u), the model thus formed will be a constructive probabilistic abstraction of the low-level model.
The next result provides some useful properties of constructive abstraction.
Proposition 2. Suppose H < L with constructive. Then H is effective if & effective. Also, for any submodel "t' of H there is submodel L' of L such that H 3, L'
Thus our general class of effective RCMs closes under constructive translation. The next example shows that this is not the case for the narrower class of representable models.
= Example 1. Let X,Y, Y' be variables with Val(X) {0, 1, and Val(X') = Val(Y")
Val(Y)= {0, 1}. Consider the RCM R, defined over Vt. = {_,, as a conjunction of POs:
X = LAYY= IAAX= 0 XY= =1
(4)
for a single unit (suppressed above for clarity). Consider a second RCM Ry over VH = (X', Y'): ===
X' = AAY x = 0 o 1.
(5)
Note that Rh X, Rr. where is constructive abstraction with Ix (X), Hy (Y) and T(X = 0) = 0, 7(X = 1) = -(X 2) = 1, -(Y = y) = y. Now Ru violates both composition and reversibility, while Rr, is representable.
A second observation is that the analogue of the claim about submodels in Prop. 2 does not hold for extensions:
Example 2. Consider enlarging (4) with the potential outcome Yx=1 1. Then there is no high-level abstraction under т that defines the outcome x since Yx=2- = A Yx=1 = translates to
- = 0 MAYX x1==== =1
The main result of this section is that the phenomenon exhibited by Ex. accounts for all repre- sentability failures:
Theorem 1 (Abstract Representation). Let R. be an RCM. Then there is a representable Rr. and constructive translation r such that 3, RL.
It worth remarking the connection between well-known twofold condition called the Stable Unit Treatment Value Assumption (SUTVA [16] $1.6)) and causal abstraction. The first part of SUTVA is the assumption that "the potential outcomes for any unit do not vary with the treatments assigned to other units"; this is already presumed by our definition of causal model, which does not admit interventions within our framework as an application of abstraction). The second part is that "for each unit, there no different forms or versions of each treatment level, which lead different potential outcomes." " Note that this assumption can be seen as guaranteeing the viability of the variable treatment levels abstraction, it is simply a restatement of the condition we already identified that the outcomes Y,,- (u) for any unit and treatment level tr' must all agree 5
multiple units (however, see Rmk. Rmk. for way model this condition
2 Inference
The raison d'être of a causal inference framework is to provide language for encoding causal assumptions and showing when and why conclusions follow from available data and appropriate assumptions. In this section, to provide further granularity on the comparison between RCMs and SCMs, we introduce neutral formal language that is naturally interpreted relative to both of these models. The language systematizes reasoning about the probabilities of counterfactuals, Fixing set D of potential outcome pairs, we define formal language L in two stages:
Definition 10.
The base language Cress is given by all Boolean combinations of statements Yx = alternatively written V/. for all Yx E D, y E Val(Y). Meanwhile, L defined as the set of Boolean combinations of inequalities t1 > t2, where t,, t2 are generated as sums, products, and additive inverses of probability terms PE)), where E Loase-
'Imbens and Rubin (16 also mention ways of fulfilling this condition requiring changes to the causal model definition. In the supplement (Rmk. A.2) we show these alternative models can be represented within our framework.

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The language L is the most expressive in a sequence of three languages introduced in [1 4 to formalize the "causal hierarchy" [27] By restricting probability terms purely "observational" or "interventional" quantities, it is possible to study the inferential limitations of data and assumptions at lower levels of this hierarchy. For present purposes, L. naturally encodes prominent reasoning patterns in RCMs and in SCMs. Its semantics are straightforward in any M or R that includes all outcomes D: we generate a mapping of each polynomial term t + [t] E RR recursively with the crucial case being to map P(e) to the probability calculable by marginalization of pef Pef M. and then evaluate the atom 6 2 t2 true iff [t]] 2 [t2]] recursing to define a semantics for all of C. Over the class of all (recursive, possibly infinite) SCMs, L has been axiomatized [13] by set of principles called AXs. and the complexity of its satisfiability problem has been shown complete for the class EI [24). The class of simple probability distributions over the atoms of Lese is axiomatized by principles known as AX1 13 which we will abbreviate AX.
2.1 Potential Outcomes Assumptions
Reasoning about potential outcomes is often encoded in what we call the base language Ltsse. augmented with (typically implicit universal) quantifiers over units. For instance, the well known
monotonicity (or "no defiers" who do the opposite of their prescription) assumption [1] 16 says
Vu.X,- (u) = Xx+ (w) 1,
(6)
where X and Z are binary variables respectively meaning the treatment (actually taken) and the treatment prescribed, with 2+, = abbreviating % = 1, 7 = 0 respectively. We will use the same abbreviation for other binary variables, so that the above condition can be written succinctly as - + _ We also adopt this interpretation of X, 2 for the rest of 2. We now explain how this and other causal assumptions in the potential outcomes framework can be encoded in
Definition 11. Let & be a well-formed, closed predicate formula in prenex normal form with single quantifier over a variable {u} and matrix in Liuee: the u can alternately be included in the atoms, e.g..
by writing Yx()) y. Define its encoding T({) L as follows:
(P(-T(()) 0, ç Vu.G, where Ç Lrase
T($)= (((T()) 0, ξ Bu.G, where ÇE E Lrase
Note that Ç is quantifier-free in both cases. Thus, c.g., the encoding of (6) is + +x 0.
Where S is a set of Ltuse assumptions let R(S) be the class of RCMs whose potential outcomes obey every assumption in S, thus obeying Vu.o where u ranges over units for any o E S. Also let
T(S) = (T(Vu.o) JoEs be the encoding of S via Def. 11 Then we have the following:
Theorem 2. AX + TSS) is sound and complete for R(S).
Corollary 1. RCM = AX T is sound and complete for Ref.
One consequence of this completeness result is that purely propositional and predicate logic reasoning about potential outcomes can be interweaved with probabilistic reasoning, as in Ex. 3 below. Another consequence a complete axiomatization of SCMs (which can be seen as probabilistic lift of [T)::
Corollary 2. Let C, Rev be universal statements of (2) and (3) respectively. Then SCM RCM T(C) T(Rev) is sound and complete for Maniq (where every outcome included in L
Example 3. A seminal result from [15] I is that it is possible estimate the Average Treatment Effect among the population of units who comply with their treatment assignment, a quantity known as the Local Average Treatment Effect (LATE): E( which we assume binary purely for simplicity, and without loss generality. Thm. implies that this can be verified in our calculus, by appeal to two key assumptions [15 1 monotonicity (6) and
Y. læt A x ), with Y the outcome,
Exclusion restriction (ER).
Vu.ye-. (w) < (n),,(())
The original discovery was that these principles guarantee that LATE ITT, the average "causal effects of assignment on treatment received and on the outcome of interest" Α. or symbolically:
--T,,//TT, /ITT,, where the latter
ITT2 = E(X,+ - xx-) = P((tt- A ^rz-)) P( Axt))
ITT E E( Y=-,x.-) =P Xx+ AVT xx-) -P xx+ Art xx
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R(E)
u
P(u)
X51 (u)
Xg
(u)
Y
(1)
Y
(u)
Y (u)
Y (u)
u0
3/1
1
0
1
0
0
11
1/1 -
0
0
1
1
U2
1
0
0
0
0
U3
E 1
0
1
1
Figure family of RCMs (R(e) ((le)Jusee Josesy/ such that R(s) is representable iff = (though (2 and (3) are met for any unit). Note that all members of this family meet (6) and 7 (the latter
guaranteed since columns (u), -==_=(u)) give the potential outcome for any 2). Also, any experimentally testable quantities .onntitte-iiclluiinng... in particular, ITT, and ITT, agree across the family, with IT
ITT, 1/2. However the assumption in question (8) holds in R(E) iff = 0, and LATE = 1/2+5.
so that LATE ITT.
ITT, only when this holds, and it is not estimable in general.
where in ITT;, interventions like X= set X at the unit level to the value that it would take under the intervention setting Z to 2; thus, e-g.. we have ((y..x.) = Pyy... A x ) + P( A x Crucially, these two quantities can be estimated, e.g., through randomized experiments [16 Ch. 23].
However, over our most general class Reff of RCMs, these assumptions are not in fact sufficient to identify LATE. Fig. [illstraates a family of RCMs that satisfy (6) and (7). but disagree on LATE. An additional principle, which Angrist et al. [ offer as a matter of notation, we dub:
Outcome decomposition.
Vu.yuu (u) ++ Vz+ (.)..(u)..
(8)
It can then be shown that, taken together, 66) (7). and (8) do indeed logically entail LATE
I see Prop. B.I in the technical appendix for the derivation.
There has been much discussion of monotonicity and exclusion restrictions (which are closely related to graphical assumptions; see (2.2wwelow) but what might justify outcome decomposition (8)? One intuition might be that it somehow follows from the exclusion restriction (7): if the effect of on Y the same no matter the value of Z, then it would seem that omitting in the intervention should have no impact on the effect of X on Y. Of course, the example in Fig. shows that this is too quick.
It turns out that (8) does follow from 7 if restrict attention to representable RCMs. In fact, (8) is derivable from (7) and the principle of composition (2) in the calculus AX, so long as we can reason along the way about the potential response Ze. By composition, for any and y we have Ue A _ + V2 a and Yı A 2Ø + V/s .a- and by ER 7 the latter gives yu A 2 - Yz! .z- As Zz binary, we have 2 V zæ. and thus by propositional reasoning, V2 > V/+ .z- The other direction Yzt - V2 follows from the same argument by contraposition, Y too is binary.
Thus, while the full power of composition is not invoked, it is natural read this example and much of the literature as implicitly assuming something like representability (thus implying composition). Another source of support for this is that under representability one can show (see Prop. B.2 that ITT, = E(Y - -Y--), ), an even simpler and manifestly identifiable expression for this average effect.
2.2 Graphical Assumptions
As we saw above (Prop. D. SCMs can be understood as representations of suitable RCMs. As such, they also suggest further sources of assumptions for deriving causal inferences. In particular, qualitative patterns of functional dependence introduce the possibility of graphical methods:
Definition 12. Let M (U, V. {fr lvev, P) be an SCM. Then define the causal diagram (MM) of M as a graph over nodes V, with mixed directed edges - and bidirected arcs +---- For any V, V' E V, there is directed edge V - V' if V Pay, and there is a bidirected edge V Ve----+V* Uv, Uv C are correlated under (including Uy n Uv + Ø).
Letting (G) be the set of SCMs with diagram G. we have MG) C M provided the directed edges in G form dag (see Lem. A.3). We thus assume this acyclicity any G. When interpreting over an SCM, we include every possible potential outcome in L. Just as we earlier encoded assumptions about the potential outcomes of an RCM into L, we may do the same for SCMs regarding their
graphs. A first observation is that Def. translates axiom C6 of [10 to ProbRec of (13), thus
8

---

rederiving the system AX₃ for the class of all acyclic SCMs, i.e. Ug (g()), from the latter. We now encode the content of (the assumption of having) a particular diagram 9 into L. Let Paş = {V' V V + V E }) be the directed parents in graph 9 of a vertex V. We encode by way of two schemas, encapsulating what some (28] have called "the two fundamental laws of causal inference":
Definition 13. Let the exclusion restriction schema ER be the L principle Ya + Up. for all variables Y E V and sets of variables A O Paf. where y E Val(Y), a E Val(A), p T (a). Let the counterfactual independence schema cf-sep" be, for all pairs of variable sets (Yihsisn: (Yihsisw C. V such that there are no Y,, Y; for which Y, = Y2 or Y,+ Y,+=---+Y( in g,
cf-sep". P[ A (wi)p: A A (h/p = Pl A (..)p.) P A W/)pss (9)
15i50
IS/S
1sis
15/Sm'
where E Val(X)), J/} E Val(y)), P. E Val(Pa),) ). E Val(Pa for each Y, Yj. Then the
translation of G is the combination of axioms T(G) T(ER)) + cf-sep".
Note that while Ex. 3 in no way relies on graphs, if we accept a G where Z A Y, then ER yields Y/ + Yız + y without further ado. Importantly, however, T is not valid over Mg) for any g containing the edge Z + X, revealing extra-graphin ttra-ggraphiaa provenance. On the other hand, cf-sep is inexpressible infrennttl,,,,ally, the two approaches are incomparable.
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 & (see. e.g.. 139 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 L-and thus including inequality constraints as well- but making certain restrictions on G, we answer this question in the affirmative:
Theorem 3. For any acyelic diagram G, axioms T(g) SCM are sound for L over MG), and also complete if the bidirected ares in 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. is stronger (see Cor. [.I.) This completeness result implies that for such G, any known graphical conclusions including do-calculus, identifiability results, and bounds-cor can be rederived in our calculus, e.g.:
Example 4 (Verma constraints). We derive the Verma constraint [40 39
over the graph 9 of Fig.
that Σw P(y | 2, P(W
zz) does not depend functionally
Ply, 2, w. 20 P (w, x))
TIC)
P(yawars Zywaz Wyzer; yew) P(Way Fw))
P(z, w. (x)
P(Zwz Wass Fzw P(x)
T(ERS)
P(y Zwo, Wz, 2. ((W,. x))
P P(yz, w. ) POPP P(x) = P(yz).
P(Zw, Way P(
P(Z))P(W)PPPP
To summarize, for wide class of cases, Thm. tells that the two principles encoded in T(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-Worls 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 wider class of expressions by combining graphical and potential outcome notations (see, e.g., (0)). 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 D be a dag over V and let x be an intervention. Let An
D = {V' V V
V in D) be the directed ancestors of a variable V in D. Then the SWIG Dy has nodes labeled
Lee., the transitive closure of the converse of the parent sets Paf. Note that V E And and Paf C Anf.

---

X
U
W
W
W
w
Zuo
.ut
Mixed Diagram G
Directed Graph D
SWIG D Dw=
Figure 2: (a) Graph G for Ex. Note that we have ER F Vaww ++ ys Zyuz + Rwo Wyzw ++ Wx.
Tyaw + 2, 2w 2, Zwz + Zwo etc., while pairs where cf-sep applicable include {W,Y }
{X, ]} and {2). [W, X (and any pairwise subsets thereof). These instances are used in Ex. 4
(b) Directed graph D, obtained from & by making the latent exogenous variable U explicit.
(c) SWIG obtained from D for the intervention W wt, with fixed node w+ (TO) implies
((w, Vutt, -w+ PP()) (w, u) (Nutt Pw+, u)) since every path between Wa and either Zw+ or
Yu+ goes through Up. where the directions of the arrows do not collide.
(V (x) V E
V E X), where A = (Any n X) \ (VV, with nodes in the first set this union termed random, and those in the second termed fixed. This SWIG has edges
(xx(x) > >Vp E X,9 has edge UV V & X, has edge V - V')
See Fig. De for an example of this construction. Note that edges in the first 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 sw-sep">, for any of random nodes { ((X))..
< i & 1) and {(G))p; : I m) that are d-separated (Def. B.1 given {(Zk) 1 s k n] in the SWIG Dx:
sw-sep Dx P A
(x:),. A (W5)p;
PL A Isksn
(ZN)pg]
1SK5
1515
=P A (z:)p, (2k)pg) P
A 1
(g))e a (zx)pg) (10)
ISiS
15k5n
1SA50
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 R be a full RCM. Then is a FFRCISTG over every instance of T(ER*)
and sw-sep holds in its counterfactual distribution. Let 3(D) be the class of FFRCISTGs over D.
Proposition 3. Suppose the SCM M E (D) represents the full RCM R. Then R E ((D). L
Given that Def. alaacay defines 3(D) in terms of C-principles, while 33 have shown the soundness direction, the following straightforward:
Theorem 4. RCM + T(ERP) + U. Sw is sound and complete over B(D).
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]. [4], [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 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. to encompass all possible causal diagrams (not just those in which the bidirected ares form disjoint union of complete graphs). Does the theorem hold with no further principles, or do additional algebraic constraints arise? This important open question 39 & B1 is crucial step toward complete thcoretical synthesis of the two frameworks.
10

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Acknowledgments
This work was partially supported by seed grant from the Stanford Institute for Human-Centers Artificial Intelligence. We also thank Elias Bareinboim and Guido Imbens for helpful conversations.
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12


Reviews:
[{"review_id": "GHVx0xh53x", "reviewer": "Reviewer_vtiC", "summary": "The paper presents a logical framework to represent both the Rubin potential outcomes (PO) approach and the structural causal model (SCM) approach to causality. It shows that under mild assumption (composition and reversibility) every PO model is representable by an SCM. The paper proceeds to show how the underlying logical framework can then be used to elucidate the assumptions necessary for instrumental variable inference (commonly discussed in the PO context), in particular for the derivation of the local average treatment effect (LATE), and similarly, how the logical language permits the derivation of graphical conditions used for identifiability results in the SCM framework.  ", "questions": "1) I would very much like to understand the connection of the present work to SWIGs. This seems to me to be crucial for a paper that provides a unifying approach to PO and SCM frameworks.\n2) Am I missing something about the causal abstraction? A whole page is dedicated to it, but for what is discussed in the actual paper it appears to only ensure that one can appropriately control the state spaces of the variables. \n3) Please explain cf-sep as a generalization of d-separation.", "limitations": "limitations are appropriately discussed", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "-- offers formal connection between potential outcome framework and structural causal model framework which elsewhere is often only stated informally\n-- gives specific examples using the underlying logical language (developed in more detail in the papers cited) that show the need for specific assumptions in the PO framework that have been subject to much discussion\n-- shows with examples how to recover identifiability results in the graphical models framework\n-- offers the prospect of a unifying logical framework for understanding causality  \n\nThese are very interesting results and I really encourage a thorough and clear discussion of how they fit in. ", "weaknesses": "-- Section 1 provides the most interesting contribution by formally connecting the PO and SCM framework, but the paper then quickly moves on to treat questions of causal inference, rather than fully explaining what can and cannot be translated between the two frameworks now. There has been a very substantive discussion in the literature about (a) the connection between PO and SCMs and to what extent it has been addressed by the single world intervention graphs (SWIGs, citation 28) and (b) the differences in the treatment of counterfactuals even within different versions of the structural equation modeling framework. The present paper is missing a more detailed discussion of how the present contributions fit into that context. Obviously, this would require more space and so a journal paper would be more appropriate for this material. As it stands the paper is very dense and provides a mixture of half a discussion of the unification of PO and SCM and then a couple of very nice examples of re-derivations of known results using the logical framework.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their insightful comments!\n\nWe very much appreciate the point about SWIGs and alternative frameworks for reasoning about counterfactuals. In fact, in an earlier draft we had included some remarks about SWIGs in particular, including how they fit into our logical framework. We removed this due to space constraints, but we could use a fraction of the additional alloted page to discuss that connection (still relegating some of the more technical discussion to the appendix).\n\nIn short, we can define FFRCISTGs as a subclass of RCMs and show that SWIGs are in fact sound and complete (in the logical sense) with respect to FFRCISTGs. Specifically, the SWIG framework derives the crucial independences and restrictions of these models within our formal language L. We will explain this connection—also re-emphasizing its importance in the larger reconciliation project between counterfactual and graphical traditions—in a new, brief subsection (2.3). \n\nThere are aspects of the SWIG framework that are, so to speak, extralogical—meaning that they pertain to how easy it is for humans to understand and assess them—but we believe it is a further testament to the framework in the present work that it can help locate these \"quasi-graphical\" models in the broader context of RCMs and SCMs, particularly from an inferential perspective.\n\nIn response to Question 2 about abstraction, we have edited the text to make the role of abstraction maximally clear. The point here is that the state space of the RCM is given, and our Thm. 1 reveals that by moving to some finer state space (given in the proof) it is possible to ensure that principles characteristic of SCMs hold while maintaining consistency between the finer and coarser spaces.\n\nAs for Question 3 about $\\textsf{cf-sep}$, it is a generalization in the sense that $\\textsf{cf-sep}$ and exclusion restrictions ($\\textsf{ER}^{\\mathcal{G}}$) suffice (by none other than our completeness result) to derive *all* conditional independences given by d-separation. On the other hand, every instance of $\\textsf{cf-sep}$ represents *some* conditional independence implied by d-separation (in an appropriate twin network); our result just shows that strictly speaking the others can be derived from these few (given $\\textsf{ER}$). We have elaborated the discussion in our paper to make this relationship clearer."}, {"review_id": "2YNXVXHNjp", "reviewer": "Reviewer_w4ki", "summary": "This paper compares the Rubin causal model (RCM) and structural causal model (SCM) frameworks for causal inference. Specifically, the authors show that RCMs, when encoded with the composition and reversibility properties, represent the same space of counterfactual distributions as SCMs. Moreover, they show that all RCMs are constructive abstractions of some RCM that is representable by an SCM. Finally, they characterize the axioms and assumptions that are sound and complete for counterfactual inferences in both frameworks.", "questions": "1. The preliminaries focus on the space of effective RCMs $R_{\\text{eff}}$ and the space of SCMs $M_{\\text{uniq}}$. Are there SCMs that are not in $M_{\\text{uniq}}$? Is this uniqueness property a characterization of effectiveness in the SCM framework?\n\n\n2. Can you clarify the relationship between Thm. 1 and Prop. 2? I think both results are interesting, but they seem to have conflicting messages. Prop. 2 seems to be saying that $\\mathcal{H}$ is representable if it is an abstraction of a representable RCM, while Thm. 1 seems to be saying that all RCMs are abstractions of a representable RCM.\n\n\n3. Is Eq. 6 reversed? It seems like the $z^+$ and $z^-$ should be swapped.\n", "limitations": "Limitations are clearly stated.", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 2, "strengths": "1. The paper offers a refreshing and productive perspective of the two causal inference frameworks that compares them on their axiomatic properties and abilities to encode causal assumptions and perform inferences, as opposed to more controversial philosophical aspects.\n\n\n2. The work is logically grounded given the definitions and axioms and provides concrete connections between the frameworks. The logic is explained clearly through examples.\n\n\n3. I found the usage of causal abstractions in the paper unexpected and very interesting, especially since the original sources were developed under the SCM framework. Causal abstractions provide a powerful framework for describing causal properties across different sets of variables. Prop. 2 demonstrates that certain properties such as effectiveness are preserved across abstractions, allowing for a more flexible way of interpreting RCMs.\n\n\n4. Causal assumptions are an important aspect of causal inference research, but they are typically studied with respect to a fixed framework. This paper provides an interesting approach to encoding the assumptions in a manner that allows comparisons between the RCM and SCM frameworks.\n", "weaknesses": "1. It is not clear what is the ultimate takeaway of this work, as most of the paper details mathematical connections but not much about their implications. For example, Sec. 1.1 and Prop. 1 claim that SCMs and RCMs are equivalent frameworks in theory, but how should a reader change the way they approach causal inference given this knowledge?\n\n\n2. The flow between different sections is somewhat strange. It is not clear exactly what the abstractions of Sec. 1.2 are adding to, say, the results of Prop 1. Sec. 2 also seems to be mostly unrelated to the previous section. Perhaps the transitions between the sections could be reworked to fix this.\n\n\n3. As mentioned in the conclusion, Thm. 3 only applies to graphs for which the components connected through bidirected edges are complete cliques, meaning that the theory is currently incomplete for SCMs. This did not impact my score in this review, but this is definitely an interesting direction of future work. I am curious if the authors have any counterexamples of cases with other graphs where there are additional inequality constraints not captured by the given axioms.\n\n\nI would also appreciate it if the authors could answer some of my questions in the next section.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the constructive comments. \n\nIn response to your very helpful questions:\n\n1. We have chosen to focus on effective RCMs because this assumption is almost always made in practice. Effectiveness is so desirable that (as we have noted) one would go so far as to introduce additional variables should it fail to hold.\nOn the SCM side, we focus on $M_{\\mathrm{uniq}}$ because it is the most general class (up to measure zero) in which the SCM induces a unique counterfactual distribution, thus permitting comparison with that induced by an RCM. We have added a sentence explicating this. There are certainly SCMs that are not in $M_{\\mathrm{uniq}}$. Given the above consideration, they are outside the scope of our paper, but one example is the following: $f_X(Y = y, U = u) = y$, $f_Y(X = x, U = u) = x$. If $X$, $Y$ are both binary then $(X, Y) = (0, 0), (1, 1)$ are both solutions to this SCM under any $u$.\nThere is no relationship per se between the uniqueness property and effectiveness, but because of the interventional semantics effectiveness will hold in every SCM solution (whether these are unique or not), which we have also pointed out in a new footnote.\n\n2. Spurred on by your comment, we have decided to remove the final claim of Prop. 2 (and its mention in Ex. 1) from the main text. We decided it is more important to focus attention on Thm. 1 and avoid the potential confusion you identified. Thm. 1 is central to the paper as a whole, as it identifies a source of apparent incompatibilities between the RCM and SCM frameworks—namely, the level of abstraction.\n\n3. Indeed, $z^+$ and $z^-$ should be swapped in equation (6), and we have done this for the final version. We thank the reviewer for the catch."}, {"review_id": "TJZyhnsPdW", "reviewer": "Reviewer_Ef3C", "summary": "The paper offers a mathematically rigorous comparison of the potential outcomes and structural causal models frameworks for causality. Their comparison goes further than previous work, in part by invoking the idea of an abstraction, and in part by invoking recent  axiomiatizations for probabilistic SCMs. As a result, they offer several new and important connections between the two frameworks, as well as offering novel insights into the frameworks separately. ", "questions": "Main questions:\n\n1: As the authors point out, in the literature on abstractions it is common to add a set of allowed interventions to an SCM, bringing them closer to RCMs. In fact, this makes me wonder: is an SCM that includes a set of allowed interventions, simply an RCM which satisfies effectiveness, composition, and reversibility? Clarifying this would help in understanding the intuitions behind some of the results.\n\n2: Relatedly, an RCM such as R' in Proposition 1 that satisfies these three axioms and has no proper extension, must be one which includes every intervention, and thus would simply be equivalent to an SCM, right? This would also help in interpreting the comment in 182: an RCM generalizes an SCM by not requiring all interventions to be allowed, and by not requiring the same axioms to hold. It might be interesting to compare how GSEMS fit into this picture. (GSEMS are Generalized Structural Equations Models, introduced by Halpern and Peters.)\n\n3: Again a follow-up: L in Proposition 2 would then be equivalent to an SCM, right? It has no proper extension, thus it must include all interventions, and hence by Proposition 1 it satisfies the axioms for an SCM.\n\n4: An obvious question: have you looked into whether new interesting results emerge if you generalize beyond constructive abstractions?\n\nMinor technical questions/clarifications:\n1: When introducing SCMs, why assume that U_V and Pa_V are strict subsets?\n\n2: 228: \"close under constructive abstraction\". \nThis wasn't clear to me, because for effective we go from H to L, whereas for representable we go from L to H, so it sounds like close means something different in each case.\n\n3: monotonicity: Here the variables are given a specific interpretation, right? X is treatment, and Z is prescribed treatment? Or is the idea that this relation holds for all potential outcomes? More generally, my background in PO is limited, and I have no sense of what this condition is capturing. Some intuition would be nice. \n\n4: Theorem 2: I’m confused, here because members of S are part of the base language and thus do not include quantification, so are we assuming an implicit universal quantification? Because that sounds at odds with the possibility of using an existential quantifier in Definition 11.\n\nTypos:\n\n35: a \"a...\n\n44: tactictly\n\n", "limitations": "yes", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "Given the importance of both the RCM and SCM frameworks, and the long-standing debate as to how exactly they relate, these novel results about their relation are extremely valuable. This paper does so much in such little space, that I am confident its results will have an impact on many different issues within the causality literature. I believe this paper is an important milestone when it comes to the mathematical foundations of causal frameworks. ", "weaknesses": "The paper is extremely dense, introducing very sophisticated and complex ideas that are usually given chapter-length expositions in just a few paragraphs, using very compact notation. Therefore it requires both a substantial amount of familiarity with the related literature and a lot of effort from the reader to understand all of it. Ideally there would have been more simple examples such as Example 1 that highlight the main intuitions and illustrate all of the important concepts, making life easier for the reader. I suggest to the authors to write up a much longer journal version of this paper that simply lays out the same content but with a lot more handholding for the reader, both in terms of conceptual clarifications as in concrete examples. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the detailed review!\n\nResponses to questions:\n\n1. Yes, fixing a set of allowed interventions over an SCM yields an RCM with these three properties. However an RCM with these properties does not always come from an SCM (see Ex. 3). We have clarified this on line 181, changing \"possibly not sufficient\" to \"not sufficient, in light of Ex. 3\" and added a footnote to the text of the example.\n\n2. Indeed, such an RCM can be considered equivalent to an SCM.  It will, for example, be identical with regard to interpreting our probabilistic logical language. Note however that RCMs and SCMs (Def. 1, 3) are still defined as very different objects. We have updated the explanation on line 185 (\"However, ...\") to point out this equivalence explicitly.\nWe are also grateful for the suggestion to compare Peters and Halpern's GSEMs, which were also observed to be expressively equivalent to Blom et al.'s \"causal constraint models\" (UAI 2019). These are similarly offered as \"mechanism-free\" generalizations of SCMs. But the focus in that work is on allowing multiple possible outcomes, corresponding, e.g., to multiple equilibria in a dynamical systems model. We've added citations to this important related work. \n\n3. Yes, such an $\\mathcal{L}$ would likewise be equivalent to an SCM.\n\n4. We thank the reviewer for this suggestion, which we had not considered. Generalizing beyond constructive abstractions is an interesting direction for future work, as it may allow one to require, e.g. that $\\mathcal{R}_{\\mathrm{L}}$ be representable by an SCM with a specific graph.\n\nMinor questions:\n\n1. We have changed the notation $\\subset$ to $\\subseteq$. Indeed, this was not intended to indicate a strict subset.\n2. In both directions, the abstraction is from $\\mathcal{L}$ to $\\mathcal{H}$ (finer to coarser); the phrasing was in opposite orders though so we can see that it may have been misleading. Note that we have actually decided to remove the final claim of Prop. 2 (see response to Question 2 of Reviewer w4ki), so in the event, there will no longer be any conflict here.\n3. This principle means that there are no units that do the opposite of what they were prescribed. To make this clearer, on line 283 we have added an explanation of the meaning of $X$, $Z$, which are indeed interpreted as treatment and prescribed treatment respectively, and elaborated on our explanation of monotonicity (\"no defiers,\" line 281).\n4. $S$ is meant to be a set of quantified (over $u$) assumptions in $\\mathcal{L}_{\\text{base}}$, making it consistent with Def. 11, although as you point out this was not explicit. We have clarified this in line 290."}, {"review_id": "obd2sZfMs1", "reviewer": "Reviewer_Qo71", "summary": "In this paper, the authors compare the causal frameworks of the Rubin causal model (RCM) and the structural causal model (SCM) from a logical perspective. This is a pretty theoretical work in which RCMs are connected to SCMs using the notion of *representability* which describes whether an RCM can be represented by an SCM. Utilizing causal abstractions, it is shown that there exists a representable low-level RCM for any RCM. Using a neutral formal language, the authors also model assumption for causal inference which can be applied to both RCMs and SCMs. The authors highlight how related both frameworks are and do not state a general preference towards one or the other.", "questions": " 1. As far as I understand it, the SCM definition only covers hard interventions (Definition 4). How would the extension to soft interventions for SCMs change further results of the paper?\n\n2. I do not understand the sentence in line 354/355. How can equation 6 not be valid if $Z \\rightarrow X$. Do we not assume that $Z \\rightarrow X$ (that the treatment assignment has an effect on whether treatment is applied)? Or is there another mistake in my understanding? Overall, this sentence is not clear to me.\n\n3. While not the scope of this paper, I would be interested in the implications of the differences, which were hinted at. For example, what kind of \"insights that arise when using each that are less transparent when using the other\"? Could a language as in this paper help make insights more transparent in the framework in which they would be seen as not as transparent? And more generally speaking, how do these results help us use causal frameworks \"better\"?", "limitations": "No concerns on this.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper is a good contribution towards a theoretical comparison of RCMs and SCMs, comparing them on a very precise, fundamental level.\n\n2. The examples included in the paper are nice and are helpful for understanding (but also see point 2 in weakness section).\n\n3. The writing is very clear and concise.\n\n4. I like that a neutral language is used, giving no preference to either framework.", "weaknesses": "1. While I acknowledge the page limit for the submission, the paper would have benefited from some additional examples and clarifications for an easier and faster understanding. Maybe the appendix would be a good place for that.\n\n2. There are several parts in the text where the notation could have been explained better and is not easily understandable (see \"Minor Criticism and Comments\").\n\n3. Results about the relationship between RCMs and SCMs included in this paper are not very surprising to me, however, the theoretical framework is still very useful, so this should not be seen as a major weakness.\n\n### Minor Criticism and Comments\n\n- Line 53: It should say here that \"SUTVA\" means \"Stable Unit Treatment Value Assumption\" \n- Line 77: What is $B$?\n- Line 121, also later: \"Ex. 2\" does not link to \"Example 2\"\n- Line 197: What is the inverse of a projection, i. e. $\\pi^{-1}$? Intuitively, it makes sense to me but I think it could be explained more clearly\n- Line 200, 201: $m$ and $n$ are used without explanation\n- Line 203: I understand the idea behind $\\tau^{-1}$ but I think a clear definition would be useful\n- Line 283: What is $Z$? As in what is the meaning of $Z$? Is it the treatment assignment, with $X$ being whether treatment is applied?\n- **Please check** I think the monotonicity assumption (Equation 6) is not defined correctly. It depends on what on how exactly to interpret $X$ and $Z$ but this definition looks not correct to me. I read it as \"If an individual gets assigned treatment and takes it, it follows that this individual would also take treatment if it was not assigned\"\n- Line 304: Please state what \"ITT\" stands for (\"Intention-To-Treat\" I assume)\n- Lines 347, 348: Should it not be $\\mathbf{Pa}_Y^\\mathcal{G}$ instead of $\\mathbf{Pa}_V^\\mathcal{G}$?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their meticulous comments.\n\nWe appreciate the call for more examples and clarifications and have added several new examples, with discussion, to the appendix. Please see also our \"global\" response to all reviewers.\n\nConcerning the minor criticism and comments: \n- Line 53: thanks for the catch. We have moved the full explanation of this abbreviation from line 247 to 53.\n- Line 77: $B$ is an arbitrary indexing set for the family mentioned on line 77. We have now made this explicit.\n- Line 121: we have fixed the link.\n- Line 197: the ${}^{-1}$ refers to the inverse image, a notion defined for mappings in general (the same applies regarding $\\tau^{-1}$ in your comment about line 203). To make this maximally transparent, we have now defined inverse images in Section 1.1 (\"Preliminaries\").\n- Line 200, 201: $m$, $n$ abbreviate the cardinalities $|O_{\\mathrm{L}}|$ and $|O_{\\mathrm{H}}|$ of low- and high-level outcome sets respectively. We have added an explanation.\n- Line 203: see response to comment about Line 197\n- Line 283: yes. To make this clearer, we have added an explanation of the meaning of $X$, $Z$, which are indeed interpreted as treatment and prescribed treatment respectively, and elaborated on our explanation of monotonicity (\"no defiers,\" line 281).\n- \"Please check\": Indeed, $z^+$ and $z^-$ should be swapped in equation (6), and we have done this for the final version.\n- Line 304: we have added an explanation.\n- Line 347, 348: we have fixed these typos.\n\nResponses to questions:\n1. Perhaps surprisingly, it is possible to show that soft interventions are fully reducible to hard interventions within our language $\\mathcal{L}$ (in the sense that soft effects are equivalent to distributions on the results of hard effects). Thus, including such soft interventions would not change our results to the extent that they are couched in $\\mathcal{L}$. If the reviewer feels it would be important to mention this, we would be happy to add a footnote (and short accompanying appendix subsection) clarifying this.\n2. We mean that there is no $\\mathcal{G}$ including the edge $Z \\rightarrow X$ such that $\\mathrm{T}(6)$ is valid over $\\mathfrak{M}(\\mathcal{G})$, i.e., is true in all SCMs with graph $\\mathcal{G}$. Since, as you say, we do indeed assume that $Z \\rightarrow X$, this means that monotonicity (6) is not implied by any acceptable graph. Thus it is not a graphical assumption—its source must be fundamentally extra-graphical, contrasting it with, e.g., the exclusion restrictions. We have inserted an additional sentence making this clear.\n3. We really appreciate this question and think it's an important point for connecting this theoretical work with more practical issues in causal inference. We believe the framework in the paper offers a distinctive—and notably, objective—perspective on the comparison. By framing causal inference in terms of derivability in $\\mathcal{L}$, this can help illuminate inferential relationships among assumptions and possible conclusions. Examples of this are the observations in the paper about what exactly is necessary for the well-known LATE derivation, as well as our graphical completeness result. But we definitely don't want to suggest that the framework is unique in its promise for such clarification. The important body of research on SWIGs and related topics (cf. the discussion with Reviewer vtiC) has many other examples of shifts in transparency afforded by news ways of packaging and formulating existing ideas and results, which can in turn spur new results. We are optimistic that the (meta-)framework in the present work will continue to be an important contributor to this wider endeavor. "}]

Author rebuttal:
We are sincerely grateful to the reviewers for their truly helpful and constructive feedback, and also for their encouraging remarks about the work and its significance. We feel that addressing their constructive suggestions has improved the paper and made it more effective.

In the individual responses below we comment on the specific points raised by each. At a general level, we appreciate the concern from several of the reviewers about density. In addition to some of the particular amendments detailed in the individual responses, we are enthusiastic about using the extra page for the final version to add further explanation of key ideas, notation, and concepts. In addition, as discussed in the response to vtiC, we will use about 1/3 of the extra page to comment on how SWIGs fit into the broader framework, emphasizing the importance of such work for illuminating the many facets and connections between research programs in these two traditions.

We look forward to further discussion and opportunities for clarification, and want to thank the reviewers again for their careful and thoughtful attention to our work.

Meta-review: This paper builds a bridge between the two principal schools of thought regarding causal models: potential outcomes and structural causal models. The reviewers were appreciative about the contributions made to the foundations of causality theory and the excellent quality of writing.
Decision: Accept (poster)