A Hierarchy of Graphical Models for Counterfactual Inferences
Authors: Hongshuo Yang
Elias Bareinboim
Venue: NeurIPS 2025 poster ()
Keywords: Graphical Models
Causal Inference
do-calculus
counterfactual-calculus
Bayesian Networks
Causal Bayesian Networks
Counterfactual Bayesian Networks
Primary area: probabilistic_methods

Abstract: Graphical models have been widely used as parsimonious encoders of assumptions of the underlying causal system and provide a basis for causal inferences. Models encoding stronger constraints tend to require higher expressive power, which are also harder, and sometimes impossible to empirically falsify. In this paper, we introduce two new collections of distributions that include counterfactual quan- tities which are experimentally accessible under counterfactual randomizations. Correspondingly, we define two new classes of graphical models for encoding empirically testable constraints in these distributions. We further present a sound and complete calculus, based on counterfactual calculus, which licenses inferences in these two new models with rules that are within the empirically falsifiable bound- ary. Finally, we formulate a hierarchy over several graphical models based on the constraints they encode and study the fundamental trade-off between the expressive power and empirical falsifiability of different models across the hierarchy.

A Hierarchy of Graphical Models for Counterfactual Inferences
Hongshuo Yang
Elias Bareinboim
Causal Artificial Intelligence Lab
Columbia University
hy27128columbia.cem edu
Abstract
Graphical models have been widely used as parsimonious encoders of assumptions of the underlying causal system and provide a basis for causal inferences. Models encoding stronger constraints tend to require higher expressive power, which are also harder, and sometimes impossible to empirically falsify. In this paper, we introduce two new collections of distributions that include counterfactual quan- tities which are experimentally accessible under counterfactual randomizations. Correspondingly, we define two classes of graphical models for encoding empirically testable constraints in these distributions. We further present sound and complete calculus, based on counterfactual calculus, which licenses inferences in these two new models with rules that are within the empirically falsifiable bound- ary. Finally, we formulate a hierarchy over several graphical models based on the constraints they encode and study the fundamental off between the expressive power and empirical falsifiability of different models across the hierarchy.
Introduction
Causal information is fundamental across a wide range of scientific disciplines and human decision- making, and is increasingly recognized necessary ingredient for advancing AI and machine learning in enhancing robustness, interpretability. and generalizability [16] I]. The Pearl Causal Hierarchy (PCH) organizes such information into three layers: the observational, the interventional, and the counterfactual, corresponding roughly to the ordinary human capabilities of seeing, doing, and imagining [16] 3. Each layer is formalized through a distinct symbolic language and encodes causal quantities with increasingly expressive semantics. For example, consider system with two observed variables, X (Ireatment, e.g., diet) and Y (outcome, e.g., BMJ). Layer (L,) includes observational distributions, like Pyy[z), which represents the probability of observing BMI among those who naturally follow diet æ. Layer (C2) contains interventional distributions, like Ply]do(>) which represents the probability of having BMI among those who were externally assigned to diet x. Layer (Cx) comprises counterfactual distributions, like P(yz [2'), which represents the probability of having BMI if the diet had been set to x among those who would naturally follow diet x'.
When the true causal mechanism underpinning phenomenon of interest formally represented by a Structural Causal Model (SCM) - is known, all layers of the PCH are immediately computable. Unfortunately, it is rare for SCMs be known at this level of precision in real-world scenarios. This limitation gives rise to the field of causal inference, which seeks to understand the conditions under which valid inferences can be made given access limited features and data from the causal system. The inferential process can be illustrated through the causal inference engine [I] Sec. 1.3.4], as illustrated in Fig. [ The engine takes three inputs: {(l) Query, (2) Data, (3) Model), each reflecting different aspect of the underlying SCM. The Query specifies the causal quantity of interest, the Data consists of data gathered through interactions with the environment like random
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Query Q
Yes 2 f=
PCH
Causal Inference Engine
SCM *
Observed Data D
3. Graphical Model
No
(Causal Assumptions A)
Unobserved SCM and the causal inference engine. The engine takes input query, model, and datasets, and returns whether the query is computable from the assumptions and data.
Figure I:
samplings or randomized experiments, and the Model encodes assumptions about the SCM. common language for articulating such assumptions is provided by graphical models, particularly and their variants, which encode constraints describing how different quantities within the PCH relate to one another. For example, Pearl's 's celebrated treatise Probabilistic Reasoning in Intelligent Systems developped comprehensive account of Bayesian Networks (BN) as encoders of conditional independencies
causal diagrams 14 15 3
that is, Li equality constraints within an observational distribution, such as P(y x) = P(y) [14]. In contrast, Causal Bayesian Networks (CBN) encode equality constraints across distributions in both Li and L2, like P(y]do(z)) = P(y(æ: 15 2 3
Counterfactual Bayesian Networks (CTFBN) further extend this framework to encode constraints across C3 distributions, like P(yz, x) P(y.a æ) [T] Sec. 13.2].
For graphical model be sufficient for supporting inference on query, there must be match in expressiveness between the model's ' constraints and the query, as illustrated in Fig. _ This requirement aligns with Nancy Cartwright's famous motto "no causes in, no causes out" S as mathematically formalized by the Causal Hierarchy Theorem (CHT): to perform inferences on quantity layer i, one needs knowledge from layer or above 3 Corollary 1]. For instance, given an Le query, BN encoding only Li constraints is insufficient, while CBN encoding both Li and Le constraints both sufficient and necessary for inference. CTFBN encoding La constraints, while sufficient for the target query in L2, imposes assumptions that are stronger than necessary 66 I
While models that encode constraints higher in the PCH support inferences about more expressive queries, it is also generally preferable to avoid un- necessary assumptions for a given query. This notion of parsimony is grounded by the concept of empiri- cal falsificution in the sciences. As Popper emphasized, system scien- tific only if it is refutable by empiri- cal tests (18). At the same time, the degree of falsifiability varies across domains. some fields, opportuni- ties for direct refutation are abundant; in others, such as cognitive science and AI, they may be scarcer, motivat- ing the introduction of additional as- sumptions that render counterfactuals articulable. This does not weaken the principle falsifiability, but reflects a continuum of empirical accessibil- ity across scientific disciplines. For graphical models, this continuum be- comes concrete question of empiri- cal accessibility: can the assumptions encoded in a model subjected to empirical testing with available data? Concretely, the falsifiability of an assumption in graphical model depends on the feasibility of drawing samples from its underlying distributions (also known realizability, 19). Among the three layers of the PCH, it is generally understood that data from
X f(U
SCM
Y - U,
Query
PCU,UD
CTFBN
PLY, Y POLIY.x>
Countetarauu
=
d(Y)))(()
Intervattonaal Obsonationa
CBN
BN
POY
Model
52
.2
Figure 2: Expressive power of queries and graphical models along the PCH. The model's constraints should be least as expressive as the query for the causal inference engine to work. Layer 3 is partitioned into two sub-regions: the green region represents L3 distributions that cannot be accessed any experiments, while the blue region represents those that can, at least in principle, be sampled via experiments.

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L1 and Lg distributions are, at least in principle, attainable via random sampling and randomized controlled trials [8. In contrast, L3 encodes counterfactual knowledge traditionally considered be- yond the reach of physical experimentation. For example, the probability of necessity and sufficiency (PNS), P( ). an L3 quantity that cannot be obtained via any randomized experiments. Yet, recent work by Bareinboim, Forney and Pearl have revealed that an L3 quantity known as the effect of the treatment on the treated (ETT), P( la'), can be sampled through a new experimental procedure called counterfactual randomization [4). Subsequent work further refined and characterized the set of L3 distributions that are realizable in principle [9]. These advances reveal that L3 is not monolithic but instead contains distributions with varying degrees of empirical accessibility. This heterogeneity naturally raises the question of what assumptions are sufficient and necessary to support counterfactual inference
question now address directly.
The empirical heterogeneity of La distributions naturally raises a central question: what assumptions are sufficient and necessary to support counterfactual inference? In this paper, we addresses that question by analyzing the region between L2 and L33 in the PCH, illustrated in the orange zone of Fig. 2 We introduce formal languages, models, and inferential machinery for two families of realizable distributions that extend beyond the Fisherian interventional world yet remain empirically accessible. In doing so, we give precise mathematical form to Cartwright's httss principle: only when the assumptions built into a model ("causes in") are adequate can the corresponding counterfactual queries ("causes out") be answered. Our main contributions are follows:
We introduce symbolic languages and valuation semantics for two new collections of distributions, each entail quantities that become experimentally accessible by a distinct implementation of counterfactual randomization. We then define two new classes of graphical models, CBN2.25 and CBN2.5, that encode constraints within these distributions which are amenable to empirical testing. We prove that counterfactual calculus with graphical checks form a sound and complete inferential machinery for CBN2.25 and CBN2.5.
(1) Graphical Models & Inferential Machinery:
We formally define a hierarchical structure for graphical models based on constraints they encode and analyze this structure from two angles: (a) expressive power and (b) empirical falsifiability. We show that models higher in the hierarchy encode stronger assumptions that permit more expressive queries, but are increasingly harder to falsify.
(2) Hierarchy of Graphical Models:
Notations. We denote variables by capital letters, X, and values by small letters, x. Bold letters, X represent a set of variables and x a set of values. The domain of X is denoted by Val (X). Two values x and z are consistent if they share common values for X n Z. We denote by x Z the value of X X\ Z consistent with x and by x n Z the subset of x corresponding to variables in Z. We assume the domain of every variable is finite. W. denotes an arbitrary counterfactual event, and V(W.) {W E VIW, E W. G[W] denotes a vertex-induced subgraph over W. We use kinship notation for graphical relationships: parents (Pa), children (Ch), descendants (De), ancestors (An).
Definitions and Background. We use Structural Causal Models (SCM) as the underlying semantical framework (15). An SCM M is a 4-tuple (V, U, P(u))), where U is set of exogenous (latent) variables, distributed according to P(u): V is a set of endogenous (observable) variables; FF is a set of functions such that for each V, V. ft maps from set of exogenous variables U, U and set of endogenous variables Pa, S V to the VV)(V, 3. An SCM M induces a causal diagram G over V where directed edges reflect functional arguments and bidirected edges reflect shared correlated latent confounders. We assume the model has no cyclic dependencies among variables. Two variables belong to the same c-component if they are connected by path made entirely of bidirected edges.
Intervention do(x) in an SCM M creates a submodel Mx, where functions generating X are replaced with constant values x. The functions in My are denoted as Fx. Given set of variables Y C V. the solution for Y in MM defines a potential outcome denoted as Y, (u). || denotes the exclusion n Z and Gx is G with all incoming
operator such that | = Yz with Z X An(Y)sg) edges into X removed. An SCM M also induces all quantities within the Pearl Causal Hierarchy (PCH): for any Y, Z, ..., X, W V, L1 (Observational): P-M(y) E., 1|Y(u) = P(); Lz (Interventional): P. (yx) = Σ, 1(Yx(u) = P(u): La (Counterfactual): (_M ..., kow) Σ. (u) y, (w) = z) P(u). We denote the collection of all L, distributions as PC,, the collection of all La distributions as p and the collection of all L3 distributions as CCa.
Equalities or inequalities between polynomials over L. terms represent special marks an SCM imprints on its distributions, called invariance constraints. A graphical model (also known
3

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compatibility relation) is pair P), where G is a graph and P is a collection of distributions over
V. The missing edges in represent certain invariance constraints within P. Some examples of graphical models corresponding to three layers of the PCH are Bayesian Network (BN) [14. Causal
Bayesian Network (CBN) 3 and Counterfactual Bayesian Network (CTFBN, (I).
The counterfactual randomization action (CTT-RAND(X 7 ()) procedure to fix the value of X as input functions generating C Ch(X) using a randomising device having support over V Val(X). for unit i, where Ch(X) stands for variables taking X as an argument in their functions. A feasible action set describes all experimental actions allowed in a system. The maximal feasible action set contains all sampling, intervention and CTF-RAN actions over all variables and gives the agent the most granular experimental capabilities. More detailed background definitions and examples are provided in Appendix A for reference.
4
19 is an experimental
2 CBN2.25 and CBN2.5: Graphical Models for Realizable Constraints
In this section, we provide fine-grained analysis of the counterfactual layer (Cg) by circumscribing subsets of distributions that are realizable given a feasible action set. We assume all actions required sample from any L2 distribution are available, together with certain counterfactual randomization capabilities. Specifically, we define two collections of realizable distributions, each determined by different degree of flexibility how counterfactual randomization propagates downstream variables (Sec. 2. We then introduce the corresponding graphical models encoding constraints in these distribution sets (Sec. 2.2 followed by the inferential machinery for each model (Sec. 2.3
2.1 Formal Languages for Realizable Counterfactual Distributions
Before introducing the two layers of language, we first provide two definitions of interventional sets that help distinguish between them.
Definition 1 (Interventional Variable Set). Given a set of random variables V, an interventional variable set is subset of V which an intervention is performed.
Definition (Interventional Value Set). Given a set of random variables V, an interventional value set, x E Val (X), is a specific assignment of values to an interventional variable set X S V.
For each interventional variable set X, there may exist multiple corresponding interventional value sets x drawn from (X) For example, given a binary variable X, its corresponding interventional variable set is {X). while the interventional value set can either be {X =0} {X 1).
The first collection of realizable distributions defined under the assumption that each RAND on X fixes single value of across all its children. | The symbolic representation and valuation of distributions this collection, given an SCM, are provided below.
Definition 3 (Layer 2.25 (L2.25)). An SCM M M(U, V, F, P(u)) induces a family of joint distributions over V, indexed by each interventional value set x. For each X, Y S V, Val(X):
PM
A VEY(X
Vitall = Uis
VIRgANA = 04
Σ1
Vinni (u)= = Dis
A
Vilugsy (u) = P(u).
VIEiX
V,
such that (i) X; C. and U;x;= and (ii) for any v; E x and all V, E Y, if E An(V,) in Mxlv, then U/ E Xx. The collection of all such distributions is denoted P6225.
Condition (i) of Def. ensures that only assignments from the intervention value set x appear in the subscripts, and that each value in x appears least once. This prevents redundancy in representing the same distribution under different intervention value sets, e.g., when x C x'. Condition (ii) requires all descendants of an intervened variable X to share the same value x, unless the path from X to a descendant is blocked by another variable in the intervention set. These two conditions reflect the limited flexibility allowed under the restricted counterfactual randomization action.
For both La 25 and L2.5. s. we assume that counterfactual randomization is allowed for all variables in V.

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Y
Y
Y
T
Y
X
X
x
Z
Z
Z
Z
(a) Causal Diagram G
(b) L2
(c) C2.25
(d) L2.5
Figure 3: Differences in how intervention on X affects downstream variables in L2, C2 23 and C2.-
The second collection of distributions relaxes this restriction by allowing each child of X to receive potentially different randomized value. This more flexible form of counterfactual randomization expands the class of realizable distributions beyond those in the first collection.
Definition 4 (Layer 2.5 (L2.s)), An SCM M = VU, P(u)) induces family of probability distributions over V indexed by each interventional variable set X. For each Y, Y.X C.
PMM(
A ViEYX
Vipel Ui
A
Vingneo 24
VEEUX, V.
(2)
-E 1
A VINN (u) 2/,
A
Vanarva (u) u. P(u)
VEYYX
ViEYUX, wi Vinx
X; x; X; and for any V; and any B E X XaP((V) and for all V; V YY: if, X; and E An)))) in Mxy. then x, O =x, O B. The collection of all such distributions denoted by Pl2s.
such that (i) X; C X (X), and U,
Def. 3] and Def. 4 serve as templates for enumerating distributions in the two layers. Their key distinction lies indexing: L2.25 distributions indexed by specific interventional value sets x E (X), whereas L2.5 distributions are indexed by interventional variable sets X. The more specific indexing in L2.25 imposes stronger restrictions on the expressiveness of its distributions, as reflected in the corresponding conditions, As in Def. 3 condition (i) of Def. 4 requires each intervention variable to appear least once in the subscript. but relaxes the former by allowing multiple value assignments for X. Condition (ii) is likewise weakened, enforcing value consistency only at the level of X's children rather than from X itself.
Example 1 (SCM inducing 22.25/ L2.5). Consider the SCM M = (U = U,, = (X, P(ui), where F {X (2(U)); Uz IL Uy. The distribution P(X, Y., Zz), indexed by the interventional value set {X = x} belongs C2.25. It satisfies conditions in Defi by having consistent subscripts x across all children of X. i.e., Y and Z. In contrast, the distribution P(X,Y, Y., Zz.') does not belong to L2.25 because it contains conflicting value assignments for X, making unindexable any specific interventional value set. However, it does belong to L2.5 since the conditions in Def. 4 fallow different value assignments for the same variable in the intervention variable set. This difference between the two layers illustrated in Fig. Bec and (d). Finally, the L3 distribution P(Yz, Y) lies outside both languages, as it includes the same variable under two different submodels, which not permitted in L2.25 or L2.5-
((X,U,); + 1,(XUU)) and U2 II
The evaluation processes for distributions in these two new layers are illustrated in Fig. with interventional distributions (C2) shown on the left and full counterfactual distributions (Lx) the right. In L2.25 25 and L2.5. a variable in Y is always evaluated within one submodel: each intervened variable in V, X is evaluated in its is evaluated in Mx; according to the value of it receives. The submodels in L2.25 and L2.5 are further constrained by the conditions in Def. 3 and Def. 4 respectively. Comparing across layers in the PCH, L2 25 and L2.5 are more expressive than C2. since L2 evaluates all variables in Y within a single submodel, while C2.25 and L2.5 allow joint evaluation across multiple submodels (i.e.. counterfactual worlds). At the same time, they are less expressive than the full L3, which imposes no such restrictions on which submodels may be joined.
submodel M Mx. while each non-intervened variable
2.2 Graphical Models
With these new collections of distributions defined, we now introduce two graphical models that encode the corresponding constraints and compatibility relations.
5

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i
(ii) C2.25/ C2.5
iiii
Interventional
Counterfactual
(a)
External state
P(U)
P(U)
P(U)
(b) Transformation
Fx
Fx.
F XxA
Fx
F w
(c) Induced Distribution
x(Yx)
P(Arix Vioul Ax Aod
P(xx.-... . Zww)
satisfying conditions in Def.
Figure 4: Given SCM's initial state (i.e., population) (a), we show the different functional transformations (b) and the corresponding induced distribution (c) of each layer of the hierarchy. (i)
represents the transformation FF. from the natural state of the system (P(U) world (i.e., with modified mechanisms Fx), (ii) to multiple counterfactual worlds representing C2.25/ C2. and (ii) multiple counterfactual worlds with no constraints on the worlds joint.
interventional
Definition 5 (CBN2.25 Semi Markovian). Given a causal diagram G. and let P be the collection of all C2.25 distributions over V. G is a CBN2.25 for Plass if the following hold:
(i) [Independence Restrictions] For a fixed intervention value set v in Val(V) and subset of variables W C V. Let W. be the set of counterfactuals of the form W Waa,, with pa,, taking values in v, Cl...., Ci the c-components of G.V(W.) .)], and Ciay..,, Ci, the corresponding partition over W.. Then P(W.) factorizes as:
P( A
W
pa
II A 5=1
Wpo.)
(3)
Wp EW.
Wpa. EC,,
(ii) Exclusion Restrictions] For every variable Y E V "wit parents Pay, forevery set Z C V\
(Y)), and any counterfactual set W. such that P(Ypa,,.2 W.) E P2235
P(Ypa,,,, W.) P(Ypa,, W.)
(ii) (Consistency Restrictions] For every variable Y E V with parents Pay. X C Pa for every set
Z C (Y)), and any counterfactual set W. such that P(Yxa y, X, = x, W.) E p22.3;
P(Y, = =, y, =x, W.) P(Yx= y. x, W.)
(5)
This definition closely resembles CTFBN M. sharing the same types of constraints but restricted to distributions circumscribed by 22.55. Condition (i) requires variables not sharing latent confounders be jointly independent once their parents are fixed by intervention. Condition (ii) states that once the parents of a variable Y are fixed, no further intervention can affect its value, regardless of any other observation. Finally, condition (iii) connects observations and interventions: if a parent X of Y is observed to be x while both and are under the same intervention do(2 Z = 2). this is equivalent to intervening Y by do(Z = X z). Importantly, the next proposition establishes that a causal diagram G induced by an SCM M is CBN2.25 for the L2.25 distribution generated by M.
Theorem 1 (C2 25 connnnncciion CBN2.25). The causal diagram induced by the SCM M is CBN2.25 for 2 the collection of all L 23 distributions induced by M.
Example 2 (CBN2.25). Given the SCM in Example 7 the fork, the pair (G, (G,2C is CBN2.25, where G denotes the causal diagram in Fig. 3 ()) and C2.25 satisfies the following constraints:
(i) [Independence Restrictions]
P(X))XXXX)
(6)
(ii) [Exclusion Restrictions]
P(X, x,, W.) = P(X = Z, W.) C (2,0)
(7)
P(Y,: = y, W.) ) P(Y, = y, W.)
(8)
P(Zny = 2, W.) = PPZz = =2, W.)
(9)

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(iii) [Local Consistencyl
P(Y = y, 1, = - P(Y - V, = x)
(10)
P(Y, = y, X; =I, W.) P(Y., -, = I, W.)
(11)
P(Z 2, X = =x, ...) P(Z, = =2, X = I, W.) P(Zy X, II, W.) -P(Z - 2, Xy = W.) Here, W. can be any set of counterfactual variables such that P()) p
(12)
(13)
Similarly, a graphical model for C2.5 can be defined by imposing the same types of constraints on distributions restricted to C2.5. In this case, the causal diagram & induced by an SCM M is also CBN2.5 for the C2.3 distributions generated by M. Details on CBN2.5 are given in Appendix B
2.3 Inferential Machinery
can be observed that the listed constraints in the definitions of CBN2.25 and CBN2.5 are local: that is, they involve counterfactual variables with their parents. These local constraints serve the building blocks to derive more global statements involving variables that may be far apart in the system.
Example 3 (Local to Global Constraints). Consider the 25 from Example and annd the distributions P(ye, a) and P(y, x). One may ask how these two distributions related, like, whether P(y,, x)- Pyy x). This re- lation cannot be read off directly from the model, since it does not appear (Example 2). However, it can be derived by composing several local constraints, as shown the right.
P(Uz, = P(yz, K2))
(Eq.
7
(14)
P(yazy
(Eq.
(15)
P(ya,, F:2)
(Eq.18)
(16)
local constraint in the basis
P(ya, x.)
(Fq.17)
(17)
P(y,
(Eq.(10)
(18)
The inferential machinery associated with graphical model facilitates the process of composing the local constraints and determining whether query can be expressed as a function of the available data. For L1 assumptions. the standard machinery for the probabilistic constraints encoded in BNs is d-separation [14]. For CBNs (C2), Pearl's celebrated do-calculus serves this role [15), while for CTFBNs (Cz). the corresponding tool the etf-calculus [6]. As discussed earlier, the key distinction between CBN2.25/CBN2.5 and CTFBN lies in the distributions to which the constraints apply. Building on etf-calculus, we develop an inferential machinery for CBN2.25 and CBN2.5 by restricting the rules to distributions in their respective layers. Definition 6 (Counterfactual Calculus (etf-calculus) for CBN2.25(CBB22.
Let G be a CBN2.25
(CBN2.5) for 2 (Pf25), then 22225 (PC2.) satisfies the Counterfactual-Cate Calauluu rules accord- ing G. Namely. for any disjoint sets X, Y, Z, W, T,R V the following three rules hold:
Rule (Consistency Rule Observvation/ntteevettto Exchange)
Ply,.x;XT., P(yT., Wx)
(19)
Rule 2
(Independence Rule- Adding/Removing Counterfactual Observations)
P w-) P(yzlw.) (Y, IL X.) in GA
(20)
Rule 3 (Exclusion Rule- Adding/Removing Interventions)
P(yxa, w.) P(yz, w.) ) if (X An(Y) 0) in Oz where G is the AMWN G Y, U. and all P(.) in the rules belong to aaa. (PCas).
(21)
The three rules of the calculus can be viewed as global counterparts to the three conditions in the definitions of CBN2.25 and CBN2.5. It is important to note that the etf-caleulus rules must, for each model, be restricted to distributions in the appropriate layer. For example, the calculus for CBN2.25 is limited to distributions in PC2.25, while for CBN2.5 it is limited to PC2.3, To enforce this, we introduce a graphical check to verify that all P(-) appearing in the rules correspond to valid distributions in the given model. In C2.5. it checks the counterfactual ancestor set, whereas in C2.25 it must also examine the descendants of ancestors, since the more restrictive CTF-RAND imposes stronger consistency requirements across downstream variables sharing the same intervened parents.
Definition and algorithm for Ancestral Multi- World Network (AMWN) are given in Appendix
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Definition 7 (Counterfactual Reachability Set), Given a graph G and a potential outcome Yx. the counterfactual reachability set of Yx denoted CR)(Y consists of each ||Wx|| such that W E (An(Y) U (De(V)= VV E xx] X, and (| such that W E (An(Y) U (De(V)= [Ex) X. X. For a set W., CRS(W-) .) is defined as the union of the CRS of each potential outcome in the set, with the following merging rule: if (W Wpaaae C W have CRS sets containing
counterfactual variables { (Rxx)): i over the same variable R. then (Rix.) are merged into a single variable [| whenever [[W (quiin) = Wiip for all i.
Lemma A distribution Q = P(W.) induced by any SCM compatible with a given graph G belongs to: (a) C2.25 if and only if CC.((W. satisfies: (i) it does not contain any pair of potential outcomes Ws. We of the same variable W under different regimes (s & t); and (ii) it does not contain any pair of potential outcomes Ra, We with inconsistent subscripts. i.e., n T + + tnsss n.S. (b) C2.s if and only if An(W>) does not contain any pair of potential outcomes Wa, We of the same variable W under different regimes (s + t).
Example 4 (CRS Check Not in L2.25). Consider the causal diagram in Fig. Baa and whether P(Zu, Y.v) belongs to L2.25 induced by the corresponding SCM. The reachability set is CRS Za,, Ya²) (X, Za,, Yz, Zeg', y Since the joint counterfactual (Z ) oppears in this set with 7. under different regimes, Lemma implies that P(Zz, Yz.) is not an L2.25 distribution.
With Lemma D ensuring that the relevant distributions lie within the corre- sponding layers, we can apply the ctffcalculus in CBN2.25 and CBN2.5. Theorem 2 (Soundness and Completeness for CBN2.25/CBN2.5 Identifiabil- ity). An C2.25 or L2.5 quantity Q is identifiable from a given set of observa- tional and interventional distributions and causal diagram G if and only if there exists a sequence of applications of the rules of the cif-calculus for CBBN225/CBN2. logether with the probability axioms restricted to L2.5. that reduces Q to a function of the available distributions.
Y Figure 5: AMWN GA(G, (Y,, X}) of G in (a).
X
Example (Effect of the Treatment on the Treated). Consider the causal diagram G in Fig. Fa) and the effect of treatment on the treated (ETT), defined as Q = P(yz I x'), with observational distribution P(v) as input. Q can be derived using the ctf-calculus rules as follows:
P(y 1x') = P(yz)
(Rule 2: Y, 1L X in GA (Y,, X)) Fig.
5
(22)
P(vx(æ))
(Rule 2: Y, Il X in G^ X)) Fig. 5
(23)
=
P(ylæ)
(Rule 1: Consistency).
(24)
Steps Eq. (22) and (3) follow from Lemma) 1
since CRS(X,Y,) = {X, Zz, Yz) is in C2.25
Hierarchy of Graphical Models
In this section, we develop refined view of the PCH by incorporating the two new graphical models that allow counterfactual inference between L2 and the full L3. We then illustrate how models in the hierarchy differ in the queries they support and in the falsifiability of the assumptions they encode. First, note that the two new collections of distributions can be positioned naturally within the PCH.
Theorem 3 (PCH*, or Augmented PCH). Given an SCM M and its induced collections of ob- servational (PLI) ), interventional (PC2), ) L2.25 (PL225,, L255 (PLas), and counterfactual (PLa) distributions:
pC Cpcs Cp222s CP22s CPs,.
(25)
The illustration in Fig. provides global view of the components involved in the analysis. The SCM .M" sits at the top of the generative process and induces both the PCH distributions on the left and the causal diagram on the right. This augments the original PCH by explicitly incorporating the intermediate layers 242.2 and pa.. The connection between each collection of distributions (left) and its associated graphical model (right) defines the corresponding compatibility relations. Building on this hierarchy of distributions, we turn to the constraints encoded by each graphical model. Given a causal diagram G, the constraints it encodes arise from the interpretation of its missing edges. As we move higher in the hierarchy of graphical models, the missing edges correspond to increasingly stronger constraints on the distributions, and, by implication, to a finer partitioning of the space of SCMs. Ω. This progression is illustrated in the example below.
8

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Example 6 (Constraints from Missing Edges). Consider the causal diagram in Fig. ga). The constraints encoded by the missing directed edge from Z Y across different layers are (P(-) denotes the power set):
BN:
P(Y [=
= P (X)
(26)
CBN:
P(Yss) = P(Yx)
(27)
CBN2.25:
P(Y.Z) W.) - P( W.), VW. E 2,))
(28)
CBN2.5:
P(Y,,) W.) = P W.), VW. E P([X, (.)x2,)))((XX, 2x
(29)
CTFBN:
P(Ys., W.) = P(Y,, W.), VW.
(30)
Moving from BN to CBN augments the model with L2 constraints, and moving further to CBN2.25, CBN2.5, and CTFBN introduces La constraints. Among the counterfactual models, higher layers allow increasingly flexible forms of W., corresponding to stronger assumptions.
The missing bidirected edge encodes independence constraints at different layers:
CBN:
P(Z,) P(Z X z), P(Y,) P(Y X
(31)
CBN2.25:
P = P(Z,) P(Y,) P(X)
(32)
CBN2.5:
P(Zz, Yz, X)= P(Z)) P(Y P(X)
(33)
CTFBN:
P
A
Zu, A
Y2, X
P
Ze
P
Yo
P(X)
WE Vai()
x'E Va)())
Val(X) I'E Val(X)
(34)
As we move up the hierarchy, independence constraints involve richer sets of counterfactual variables, reflecting the stronger assumptions imposed. CBN encodes the parent do/see constraints restricted to 222 in contrast, CBN2.25 25 introduces counterfactual constraints beyond C2. and together with consistency conditions. these imply the parent do/see restrictions of CBN. CBN2.5 uses the same constraint forms CBN2.25 but permits more flexible subscripts in joint counterfactuals. At the top. CTFBN allows the most expressive independence constraints, spanning broad joint counterfactual distributions and implying those in CBN2.5 via marginalization.
In fact, the constraints encoded by graphical models higher in the hierarchy always subsume those of the models lower in the hierar- chy. This monotonicity property defines the hierarchy of graphical models, as illustrated in Fig.
SCM M *
Induced Distribations
Functional and U Dependencies
Causal Diagram
p
p
2
2
Theorem 4 (Hierarchy of Graph- ical Models, PCH*). Given a causal diagram G. the set of con- straints it encodes when inter- preted as graphical model al layer i is always subset of the constraints it encodes when in- terpreted at a higher layer j. for all < j.
CTFEN
CBN2.5
Graphical
CBN22255
Constraints
CBN
Figure Pearl Causal Hierarchy (PCH*) and hierarchy of graphi- cal models induced by SCM
As discussed earlier in the con- text of Fig. [ the causal infer- ence engine operates by match- ing a query with a model that en- codes sufficient and ideally only necessary - set of assumptions. This matching logic can be understood from two complementary perspectives.
The first concerns the expressive power of model: the extent to which its assumptions are sufficient to support valid inference for given query. When the expressiveness of the query exceeds that the model's ss assumptions, the causal inference engine lacks the necessary ingredients to proceed. For

---

example, a BN encodes only L1 constraints and is therefore blind to La structure, making it unable to evaluate queries such as P(y do(z)). Similarly, CBN, which encodes only L2 constraints, cannot support inference for La queries like P(Y., X). In contrast, when model's assumptions are expressive enough to support the query, we say that the query and the model are matched. A CTFBN, which sits at the top of the hierarchy in terms of expressive power, can in principle match the most demanding queries in the P
The second perspective concerns the empirical falsifibbiiyy of a model: whether the assumptions it encodes are not only sufficient but also necessary for the query at hand. As one ascends the hierarchy, models impose increasingly stronger counterfactual constraints, many of which cannot be directly falsified with the given data collections, whether observational (PC1), ), interventional (P*)), or those realizable via counterfactual randomization (P P If a model encodes assump- tions beyond what is strictly required, these may become empirically untestable and ontologically burdensome. Accordingly, the preferred model for given query is the most parsimonious one that still supports valid inference, striking a balance between expressive power and empirical falsifiability.
Example 7 (Natural Direct Effect (NDE)), Consider 9 in Fig. 7 The nat- ural direct effect from X to Y, DD at (y) P(ye 2.) P(yz). Ap- plying unnesting, the first term becomes Σ. P(yz'z kx), which is ID if and only if NDE is ID. Let Q be P(ya's 2x), which an L2.25 query this case. Q can be identified in the CBN2.5 associated with G via aff-calculus as P(yx'z Zz. >=)==(()) P(2(z) A CTFBN, which encodes stronger con- straints than CBN2.5, can also identify Q, but it brings in unnecessary as- sumptions such as P(Z.g, Zeg; X)= = >>)=P( P(Zz, P which cannot be empirically falsified with current data collection methods. In contrast, CBN is not expressive enough to represent Q at all. This example highlights the parsimony dimension: although both CBN2.5 and CTFBN can in principle support the query, the more parsimonious CBN2.5 is preferable since it avoids unnecessary, unfalsifiable commitments.
Z
Figure 7: Causal di- agram for NDE
z)
This example highlights the trade-off between the expres- siveness of queries and the parsimony of models. The optimal match occurs when the assumptions in the model are both sufficient and necessary for the intended inference (illustrated in Table D. The introduction of CBN2.25 and CBN2.5 refines the necessity boundaries in L3, ensuring that queries in C2.25 and C2.5 can be matched with more parsimonious and empirically testable models. In short, higher models in the hierarchy gain inferential power by encoding constraints over increasingly expressive distri- butions. but at the expense of falsifiability. is therefore crucial for researchers to choose models that strike an appropriate balance between expressive power and empirical testability for the task at hand.
Q Layer
GM
Suff.
Nec.
L2.5
CBN
L2.3
CBN2.5
L2.5
CTFBN
X
Table 1: Examples of Matching between Graphical Models and Queries. 'Suff.': Sufficient and 'Nec.' := Necessary
4 Conclusions
In this paper, we introduced two new classes of graphical models, CBN2.25 and CBN2.5, which encode constraints over distinct collections of distributions realizable under counterfactual random- ization. We showed that these models are naturally induced by SCMs (Thm. [ and established sound and complete inferential machinery for them (Thm. 2 We then placed the new distribution classes within the PCH (Thm. 3) and proved that graphical models over the PCH form hierarchy
(Thm. 4. Finally, we highlighted the trade-off between expressive power and empirical falsifiability across the hierarchy. Taken together, these results refine the landscape of graphical models and provide a more nuanced map of the space between La and L3. They also offer practical guidance for selecting models that balance inferential power with empirical testability, while accounting for the societal risks and ethical considerations inherent to each application domain. We further hope that these results serve as theoretical foundation for future work on broader empirical evaluation and computational cost analysis of counterfactual graphical models across diverse domains.
"This does not immediately imply that the query is identifiable, only that it can be represented in principle.
10

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Acknowledgments and Disclosure of Funding
This research is supported in part by the NSF, ONR, AFOSR, DoE, Amazon, Morgan, and The Alfred P. Sloan Foundation. We thank Arvind Raghavan for their thoughtful comments.
References
[1] Elias Barcinboim. Causal Artificial Intelligence: A Roadmap for Building Causally Intelligent Systems. http://causalaibboo..ooo 2025.
[2] Elias Bareinhoim, Carlos Brito, and Judea Pearl. Local Characterizations of Causal Bayesian Networks. In Lecture Notes in Artificial Intelligence. Springer, 2012.
[3] Elias Bareinboim, Juan D Correa, Duligur lbeling. and Thomas Icard. On Pearl's Hierarchy and the Foundations of Causal Inference. In Probabilistic and Causal Inference: The Works of Judea Pearl (ACM, Special Turing Series), 2022.
[4] Elias Bareinboim, Andrew Fomeyy and Judea Pearl. Bandits with Unobserved Confounders: A Causal Approach. In Proceedings of the 28th Annual Conference Neural Information Processing Systems, 2015.
[5] Nancy Cartwright. Nature's Capacities and Their Measurement. Clarendon Press, Oxford, 1989.
[6] Juan D Correa and Elias Bareinboim. Counterfactual Graphical Models: Constraints and Inference. In Proceedings of the 42nd International Conference on Machine Learning, 2025.
[7] Juan D Correa, Sanghack Lee, Elias Bareinboim. Nested Counterfactual Identification from Arbitrary Surrogate Experiments. In Advances in Neural Information Processing Syxtems, volume 34, 2021.
[8] Ronald Fisher. The Design of Experiments. Oliver and Boyd, 1935.
[9] Yimin Huang and Marco Valtorta. On the completeness of identifiability algorithm for semi Markovian models, Annals of Mathematics and Artificial Intelligence, 54(4)3363-408, 2008
[10] Amin Jaber, Murat Kocaoglu, Karthikeyan Shanmugam, and Elias Bareinboim. Causal discovery from soft interventions with unknown targets: Characterization and learning. Advances in neural information processing systems, 33:9551 9561, 2020.
[11] Murat Kocaoglu, Amin Jaber, Karthikeyan Shanmugam, and Elias Barcinboim. Characterization and learning of causal graphs with latent variables from soft interventions. Advances Neural Information Processing Systems, 32, 2019.
[12] Murat Kocaoglu, Karthikeyan Shanmugam, and Elias Bareinboim. Experimental Design for Learning Causal Graphs with Latent Variables. Advances in Neural Information Processing Systems 30, 2017.
[13] Adam Li, Amin Jaber, and Elias Barcinboim. Causal discovery from observational and interventional data across multiple environments. Advances Neural Information Processing Systems, 36:16942 16956, 2023.
[14] Judea Pearl. Probubilistic Reasoning in Intelligent Systems. Morgan Kaufmann Publishers, 1988.
[15] Judea Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, second edition, 2009.
[16] Judea Pearl and Dana Mackenzie. The Book of Why. Basic Books, 2018.
[17] Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017.
[18] Karl Popper. The Logic of Scientific Discovery. Routledge, 2002.
[19] Arvind Raghavan and Elias Barcinboim. Counterfactual Realizability. In Proceedings of the 13rd International Conference Learning Representations, 2025.
[20] P Spirtes, C N Glymour, and R Scheines. Causation, Prediction, and Search. MIT Press, Cambridge, MA. 2nd edition, 2000.
[21] Jin Tian and Judea Pearl. A General identification condition for causal effects. In Proceedings of the 18th AAAI Conference Artificial Intelligence, 2002.

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[22] Julius von Kügelgen, Michel Besserve. Liang Wendong, Luigi Gresele, Armin Kekić, Elias Bareinboim. David Blei, and Bernhard Schölkopf. Nonparametrie identifiability of causal representations from unknown interventions. Advances Neural Information Processing Systems, 36, 2024.
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Appendices
Contents
Background and Definitions
A.I SCMs and Graphical Models
21
A.2
L:: Bayesian Networks
27
A.3 Ly: Causal Bayesian Networks
27
A.4
L3: Counterfactual Bayesian Networks
30
A.5
Counterfactual Randomization
31
Details on Layers 2.25 and 2.5
33
Nested Counterfactuals
33
B.2
Examples for 22.25 and 22.5
................................
33
B3
Details on CBN2.5
B.
44
Independence Constraints and AMWN
35
Proofs
36
Supporting Lemmas
36
2
Proofs for Main Theorems
39
Frequently Asked Questions
42
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A Background and Definitions
A.1 SCMs and Graphical Models
In this section, we review key concepts and definitions that are fundamental to this work.
We adopt Structural Causal Models (SCMs) the baseline generative framework, following the presentation in [3] The discussion there is more detailed and may be consulted if additional context on these foundational notions is needed.
Definition 8 (Structural Casual Model (SCM) (U, V, F. P(U)), where
3 A structural causal model M is a 4-tuple
U is set of background variables, also called exogenous variables, that are determined by factors outside the model;
V is set (V1, V2....: Vn) of variables, called endogenous, that are determined by other variables in the model - that is, variables in U UUV;
F is a set of functions {f1. f2....,fr such that each ft is mapping.frot from (the respective domains of) Ug Pa; to VG where U. C. U, Pa; S. V V,, and the entire set F forms mapping from U to V. That is, for 1,.... n, euch Ji E J is such that
0x < fi(pa, u:),
(35)
i.e., it assigns value to V; that depends on (the values of) select set of variables in UUV: and
* P(U) is probability functtooo defined over the domain of U.
Intervention in an SCM can be viewed as a modification of the model by changing the mechanism of the intervened variables, while keeping all other components of the SCM intact.
Definition 9 (Submodel- - "Interventional SCM" [55).. Let M be a structural causal model, a set of variables in V, and a particular realization of X. A submodel Mx of M is the causal model
Mx (U, V F P(U)),
(36)
where
Fx (fi V. X
(37)
The impact of the intervention on an outcome variable Y is commonly called the potential outcome: Definition 10 (Potential Outcomes (15). Let and Y be two sets of variables V, and u be unit. The potential outcome (u))) is defined as the solution for Y of the set of equations Fx with respect to SCM M (or, Y MM (n)). That is, (u MM,(u) (u).
An SCM induces observational, interventional, and counterfactual distributions over the endogenous variables, which form three layers known as the Pearl Causal Hierarchy (PCH).
Definition 11 (Pearl Causal Hierarchy (PCH) (3) ). An SCM M = (U, V, F.P(()) induces three layers of probability distributions that form the Pearl Causal Hierarchy, For any Y. Z..... W S V, the three layers of distributions are given by:
L1 (Observational):
p- = [
(38)
u
L2 (Interventional):
pM((x) Σ x)
(39)
L3 (Counterfactual):
p. (y Zw) Σ 1| = y.... Zw(u) = P())
(40)
21

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ii)
iii Interventional
iiii Counterfactual
Observational
(a) External state
P(U)
P(U)
P(U)
(b) Transformation
F
Fx
F
Fw
(c) Induced Distribution
P(Y)
P(Yx)
P(Y x:
Zew!
Figure 8: Given SCM's initial state (i.e., population) (a), show the different functional transformations (b) and the corresponding induced distribution (c) of each layer of the hierarchy, (i) represents the transformation (i.e., F) from the natural state of the system (P(U)) to an observational world, (ii) to an interventional world (i.e., with modified mechanisms Fx), and (iii) to multiple counterfactual worlds (i.e., with multiple modified mechanisms).
The collection of all Li (Observational) is denoted as p the collection all La (Interventional) is denoted as pLn, and the collection of all L3 (Counterfactual) is denoted as P
PCH specifies both the symbolic representation and the valuation of each probabilistic quantity given an underlying SCM. the SCM is fully specified, every quantity in any layer the PCH can be computed directly via Def. IT (Fig 8). In the causal inference engine (Fig. T. this correspondence is depicted by the arrow from M to the PCH.
In practice, however, only partial knowledge of the SCM is available, so only a subset of the PCH can be observed. For example, the observational distribution may be available (Fig. [1] item (2)), while the interventional distribution remains unobserved and must be queried (item (1)). Each causal inference task therefore rests on assumptions about the structural "marks" left by the SCM on its distributions. These assumptions take the form of invariance constraints, defined as follows:
Definition 12 (Invariance Constraint). Given an SCM M an invariance constraint is an equality or inequality between polynomials over L. terms of the PCH.
A common example is conditional independence in the observational distribution. For instance, Ply ω) = P(y) encodes that X is probabilistically independent Y.
Invariance constraints can be seen coarsening the PCH: they abstract away from specific numerical values and instead capture relationships among distributions. included, the granularity of knowledge about the underlying SCM increases. To avoid enumerating constraints individually, we exploit graphical models, which encode them systematically and parsimo- niously by linking invariance constraints to graph topology (e.g.. relations among parents, neighbors, and ancestors). The natural first step in this process is to construct a causal diagram from given SCM, since the diagram directly captures the topological relations that determine which invariances hold.
more invariance constraints are
Definition 13 (Causal Diagram 33.. Consider an SCM M VU, Then G is causal diagram of M if constructed as follows:
(I) add vertex for every endogenous variable in the set V
(2) add an edge V, + V, for every VG, V, E V if appears as an argument of
(3) Add bidirected edge V; t----> Vi for every V., V2 V
(a) the corresponding functions Jú, Jj share some common U U argument, or (b) the corresponding U,, Uj E U correlated.
The causal diagram G can be viewed as a non-parametric coarsening of the SCM M: it preserves the structural signatures (i.e., the arguments of the functions and dependency relationships among exogenous variables) while abstracting away from their specific parametrization. In Fig. T this depicted by the arrow from M to G.
22

---

X
Z
X
Z
Y
(a) Fork
(b) Chain
Figure 9: Two causal diagrams encoding knowledge about the causal mechanisms governing three observable variables X, Z and Y.
Graphical Model
Physical Sampling Procedure
Inferential Machinery
Related
Layer
Distribution
Lit.
L1
BN
P(V)
random sampling randomized controlled
d-separation
14
L2
CBN
P(V do(x)), Vx
do-calculus
US
3
trials (RCT)
L3
CTFBN P(W.) VW.
counterfactual randomization
eti-calculus
6
Table 2: Sample graphical models and their corresponding distributions, physical sampling procedures, and inferential machinery for each layer of the PCH.
Pairing a causal diagram with the of invariance constraints it encodes over collection of distributions defines graphical model, also referred to as compatibility relation. For any SCM, its induced causal diagram & together with the corresponding PCH distributions naturally form such relation. In this way, the invariance constraints encoded in G serve as surrogates for the empirical content of the underlying SCM, summarizing knowledge about the different layers of distributions induces.
Definition 14 (Graphical Model). Let V denote a set of endogenous variables. graphical model is pair (G, P), where
1. G FV, is graph whose nodes correspond to the variables in V, and
2. P is a collection of probability distributions defined over V,
such that the absence of an edge (Vi, Vj) E F. encodes one or more invariance constraints that are satisfied by distributions in
Depending on the assumptions made on different layers of the PCH, a different graphical model can be defined. As alluded earlier, some examples of models corresponding the three layers of the PCH are Bayesian Network (BN) 14), Causal Bayesian Network (CBN) [ and Counterfactual Bayesian Network (CTFBN) II Sec. 13.2]. These graphical models are powerful tools for encoding assumptions to perform causal inference tasks such as identification (Fig. []. with each model accompanied by its inferential machinery like d-separation for BNs, do-calculus for CBNs, and cff-calculus for CTFBNs 14 15] 6).
As ascend the PCH, the corresponding graphical models encode invariance constraints increasingly richer sets of distributions. These enlarging sets of constraints naturally induce a hierarchy: models higher in the hierarchy support powerful inferences but depend stronger assumptions, which are correspondingly harder to verify empirically.
Example 8 (SCM and Graphical Models). Consider the SCM M = -U= (U2, U,), (x, Z. F, where
X + U2
==
Z XOU
(41)
Y X a U
P(u): Ug Bernouli((02)) Bernoull((..),, Uy ~ Bernoulli(0.3)
(42)
The endogenous variables V represent, respectively, treatment X (e.g., diet) and outcomes Z and Y (e.g. patient's BMI and cholesterol level). The exogenous variables Un. Us, and U, represent other variables outside the model that affect X. Z. and Y, respectively.
23

---

5
8
Ux
0
0
U
0
U
0
0
0
X
Z
0
Y
0
7x=0
0
0
Yx=0
0
0
Zx=1
Yx=1
Xz=0
Yz-0
Xz=1
0
Yz=1
Xy=
Zy-o
Xy=1
Zy=1
X.=
X
Xz=1Y=
0
Xz=1Y==1
=x=
Zx=0,Y=1
0
7x=1,y=0
Zx=1Y=1
0
Yx=0,z=0
0
0
Yx=0,z=1
0
0
Yx=1,2=0
0
2,=
P(u)
0.336
0.144
0.224
0.096
0.084
0.036
0.056
0.024
Table Mapping events the space of to potential outcomes in the context of Example
Given the SCM, all quantities from the PCH can be computed following Def. by mapping from each U u to all potential outcomes derived from M, as shown in Table 3
En distributions will
only involve the observed variables X. Z, and Y. and Eg distributions will have additional access to the all potential outcome variables like Zz and Yz,, individually. L3 includes joint distributions over all potential outcomes in the table, capturing the full range of counterfactual dependencies.
However, in practice, the SCM is often not observed with such details, and we analyze the invariance constraints that hold in the distributions it induces. For example, it can be calculated from Table that
P(x) = P(r,), ((tr,),Var)) V(z v) P(yz, x) P(y x). ((z,g)
(43)
(44)
These invariance constraints can be represented using the causal diagram shown in Fig. aa). When this causal diagram is interpreted as the graphical model for different layers of the PCH, it encodes different constraints according to the definitions of models:
C1 BN:
2(z.g,2)=P(z)) P())))
(45)
L2 CBN:
24

---

All Global
Local Constraints
Constraints
P(Y do(X)
PLY d,(X),
P(Y do(X)))
P(Z do(X))
P(Y (X)
MY do(Z))
P(Z X
POY (Z)
Figure 10: Constraints listed in the definition of a graphical model serves as a local basis that implies all constraints encoded in the model. Blue lines represent of local invariance constraints that can be composed to imply the global constraint represented by the red line.
p(), y. P(s))(222))((111) P(y.2)do(z)) P(y))o(()))(
(46)
(47)
(8)
P(z]do(a)) P(x), va C {,,0} P(=]do(æ, v)) P(zldo(x))
(48)
(49)
Ply/do(z, =)) P(yldo(a))
(50)
(ill)
P(z[do(x)) P(z|z)
(51)
P(z|do(z, v)) P(zldo(y), x)
(52)
P(yldo(x) P(y|z)
(53)
Plyldo(x, 2)) P(yldo(=),
(54)
La CTFBN:
P(X, Za, Zo Ya, Yor P
(55)
(i)
P(rn: w. Ple, w.), Va S (z,1)
(56)
P(Zay: W. = P(ez, W.)
(57)
P(yaz; W.) P(ye, W.)
(58)
(iii
P(z,2, x., W.) = P(žzs D. W.)
(59)
P(ay, Ty. W.) = P(Zoy, Keys W.)
(60)
P(y, x. w.) = P(yas Ir, W.)
(61)
P(ye, Te. W.) P(yazy Lz. W.)
(62)
Interestingly, the constraints explicitly listed the definitions of graphical models represent only subset of all the constraints implied by the model. These explicitly stated constraints are typically
25

---

SCM
U,
F= Z - x U,
Y X U,
Plu): U. Bern(22,, U, Berm(U.4), U, (ern(O.3)
PCH
Def. 11
Def. 13
2 P(X 1)= 0.2 1|
2; P(X (do(y))) 0.2
23
Cansal Diagram
Pi 1|X
+
1|do 1))
P(Kx- 1|x
X
POY= |X 0.7
P(Y 1odo0X 1J) - 0.7
PIZz 1.7'
y
Def. 12
Def. 23,
Invarlance Constraints
Graphical Models
Plylc) Pellyi- P(clo)
Pix Pis) do(in)
Po,lx)= ((r)
Local Basis
P(c.a) P(a,x)
BN
CBN
CTFBN
P(yldo()) P
P(y,))= P(,,)
Global Closure
Figure 11: Illustration of how an SCM induces the PCH, invariance constraints, causal diagram, and graphical models, following Example 8
local, involving variables and their immediate parents. Importantly, these local constraints form a "basis" from which all other (global) constraints in the model can be spanned, as illustrated in Fig. 10 The process of composing local constraints to derive global ones underpins the operation of the causal inference engine.
For example, consider the constraint:
P(y]do(z)) P(y)
(63)
This is constraint that does not involve the parents and does not appear in the local basis, since X which is in the parents Y Y does not appear in the expression. Still, it can be derived by combining various local constraints shown next:
Ply]do(z))
Σ (
(64)
Ply)do(z:))
(Eq 54
(65)
Σ P(zldo()
(Egg
50
(66)
((sd(o())))((3))
(Eq
(67)
P()))))(x)
(Eq 48)
(68)
P(y)
(69)
The connections among these different moving parts the SCM, the PCH, the causal diagram, the invariance constraints and the graphical model are illustrated in Fig. 77 The constraints in each model determines its inferential power. Given the L1 constraints, the only inference that and Z are independent conditional on X in the observational distributions. However, with the L2 constraints, the causal effect from the treatment to the outcome be inferred, and in this case it coincides with their observational correlation (i.e. P(y]do(z)) = P(y)x)). If we are able to Las object, say CTFBN, the local constraints can be leveraged to
be drawn is
interpret the causal diagram infer that the effect of the treatment on the treated (ETT). To witness, the ETT also equal to the conditional distribution, P(yzkz') P(y]z))
26

---

Z
X
Y
X
X
(a)
(b)
(c)
Figure 12: SCM Induced DAG or Causal Diagrams
In the following sections, we give the definitions and examples for graphical models introduced in previous works [14 15 I.)
A.2 L1: Bayesian Networks
The first graphical model encodes invariance constraints in the observational distributions. Firstly, we formally define how to construct a graph from an SCM.
Definition 15 (Confounded Component of an SCM (3). Given an SCM M = -U, P(u)). ler Ui Uf. ..... Uf C U be disjoint maximal subsets of the exogenous variables in M such that P(u) I'_ P(U)). Then, we say that V Vy are the same confounded component (for short, C-component) of M if KuU n U, & 0, Ug Sinn U * 0H > 0, that is, if ft and ff have both latent urguments some common U
Definition 16 ISCM-induced DAG B Consider an SCM M = (U,,F,,((u), Then G is DAG induced by M it:
has a vertex for every endogenous variable in the set V
has an edge V₁ + for every V, E V V,, appears as an argument of fr
there exists an order over the functions in F such that for every pair Vi, V, in the same C- component of M such that Si < Jú, the edge V; - Vj and the edges Va Vj, Vv E Pa; are in G.
Definition 17 (Markov Relative to [4). A probability distribution P(V) over set of observed variables is said to be Markov relative to graph if:
P(V) II P(s/paa)
(70)
where Pa, = [V, VIOY + VV) 9}.
Definition 18 (Bayesian Network [14). A directed acyelic graph (DAG) 9 is a Bayesian Network for probability distribution P over the variables in V if P is Markov relative to G.
Example -SCM-induced BN). Consider the SCM M = (U U2, V
{Z, X, Y) P(u)) where
P(u))
Z + Uz
F
X + ZVU,
(71)
Y + 7 U,
P(u): U2 (ernoulli(O.5), U, Bernoulli(0.)) U, (ernoulli
(72)
Its SCM-induced DAG is shown in Fig. 12 a) and its induced observational distribution P(v) satisfies:
P(v) P(z)
(73)
for all x, u, in Val(X) Vol(Y) Val(Z). The DAG in Fig. 12 BN for P(v).
A.3 Lg: Causal Bayesian Networks
The second graphical model encodes invariance constraints in the interventional distributions. Definition 19 (CBN Markovian B). Let P. be the collection of all interventional distributions do())) X V, (X), including the null intervention, P(V), where is the set of observed variables. directed acyclic graph G is called Causal Bayesian Network for P if:
27

---

1. [Markov] d(V)) is Markov relative to 9;
2. [Missing-link] For every V, E V, Vi & X such that there is no arrow from X to Vi in G: P(y:ldo(pa;)
P(vs|do(pa.)
(74)
3. Parents dolsee) For every Vi E V. X
P(c:/do(pa:) do(x)) = P(us paa, do(x))
(75)
Example 10 (SCM-induced CBN Markovian). Consider the SCM from Example 9 Its induced
causal diagram is shown in Fig. 12 a) and its induced set of interventional distributions P. satisfy:
1. Markov]
P(v) P()))(2))))())))
(76)
P(v]do(x)) = P( ((zz))) P(v]do(y)) P(= do ((2))
(77)
(78)
P(v]do(2)) = P(zz)((()))(()dd((())
(79)
2. MMissing-link)
P(æ|doly, 2)) P(xldo(z))
(80)
P(yldo(z, 2)) = P(y]do(2))
(81)
P(z]do(a)) P(z), Va S {x,y)
(82)
3. Parents do/see)
P(=]do(=)) P(z|2)
(83)
P(æ]do(y, =)) = 2(x|2, do(y))
(84)
Ply]do(2)) P(vl2)
(85)
P(yldo(z,2) 2)) = P(ylz, do(x))
(86)
The causal diagram in Fig. a2 a) is CBN Markovian for P..
Similar the confounded components in an SCM (Def. 15), there is also corresponding set of confounded components in the causal diagram induced.
Definition 20 (Confounded Component (2I). Let C1 Ck. be a partition over the set of variables V, where C; is said to be confounded component (for short, C-component) of G for every V,, V3 E C₁ there exists a path made entirely of bidirected edges between V₁ and V, in and C, is maximal.
Definition 21 (Augmented Parents). Let < be topological order over the variables V1, Vn in G. let (V;) be the subgraph of G consists only of variables in Vi.... Va, and let C(V,) be the C-component of V; in (VV).. The augmented parents of V,, denoted Pat is the union of parents of all variables in C(V.) that comes before V. in topological order:
Pat = UJIV,ET, Paj Paj\(Vs) M(V)
(87)
where T; = (X E C(V.) : X V.).
We use Gx to denote the mutilated graph with all incoming edges to X removed from G. The augmented parent of V, in 9x is denoted Pax+.
Example 11 (Augmented Parents). Consider the SCM M = (U = (U,,U,, V F, where
U X t ZVU
F
(88)
t X
P(u) U2 (er5oulli(0.5), U ~ Bernoulli(0.5)
(89)
28

---

The causal diagram G it induces is shown in Fig. b). The respective augmented parents of X, Y, Z in G are:
Pa = {
(90)
Pa = {7}
(91)
Pay [X,
(92)
If we consider the induced subgraph G(Y, 2) where there are no edges at all, it is the same graph as 9x. In this graph, nodes Y and form their own e-components respectively, so their augmented parents are both empty:
Pa = 0 Pag =
(93)
(94)
Definition 22 (Semi-Markov Relative to (3. probability P(V) said to be semi-Markov relative to graph G if for any lopological order < of G:
P(V)= II P(uilpat)
(95)
Definition 23 (CBN Semi-Markovis BD. Ler P. be the collection of all interventional distributions P(V do(x))) X V,x VX)(X), including the null intervention, P(V), where V is the set of observed variables. A directed acyclic graph G is called Causal Bayesian Network for P. if. considering Patt all compatible topological orders over V:
1. [Semi-Markov] P(VIdo(x) is semi-Markov relative to 9;
2. [Missing directed-link) For every V, E V.X. W C VV(Pa UXU (V)):
P(vs)do(x), paot dd(ww) P(u:)do(x), po
(96)
3. [Missing bidirected- For every V; E VXX, let Pa be partitioned into two sets of confounded and unconfounded parents, Paς and Pa Gxi:
P(us pal, do(pa?)) = P(u: P(ux)d((x), pat, pa))
(97)
Example 12 (SCM-induced CBN Memi-Markovian). Consider the SCM from Example 77 Ils
induced causal diagram is shown in Fig. 72bb b) and its induced set of interventional distributions P. satisfy:
1. [Semi-Markov]
P(v) P(=) P(s3))()))))))
(98)
P(v]do(x)) = P( (()))))( P(v]do(y) P( P(al
(99)
(100)
P(v]do(=)) = P(a))do()))P((})xdd
(101)
2. [Missing directed-link)
P(a do(y)) = P(x|z)
(102)
P(s)do(s), do(y)) P(xldo(2))
(103)
P(yldo(z), do(z)) = P(y]do(z))
(104)
P(z[do(a)) P(z), Va C {z.v}
(105)
3. [issing bidirected-link-
P(z]do(2)) = P(x|=)
(106)
P(zldo(y,2)) P(z|2, do(g)) P(ylæ, do(z)) = P(y(,,
(107)
(108)
The causal diagram in Fig. 12 b) is CBN Semi-Markovian for P..
29

---

A.4 L3: Counterfactual Bayesian Networks
we climb further up the PCH, we get another graphical model that encodes structural constraints in the counterfactual distributions.
Definition 24 (CTFBN Markovian [1 Def. 13.2.1J). A directed acyclic graph & is Counterfactual
Bayesian Network for P if:
1. Independence Restrictions] Let W. be a set of counterfactuals of the form W Wpa., pa.. then P(W. ,) factorizes as
P(
W pew.
II
P(
A
W paw
(109)
yow EWW
VEV(W.)
W puu (WEV(W.)
2. Exxlusion Restrictions] For every variable Y V with parents Pay. for every set Z (Pa, U (Y)), and any counterfactual set W. we have
pa 2: W.) P(Y pa, W.)
(110)
3. [Local Consistencyl For every variable Y E V with parents Pag. let X C Pay then for every set S VV and any counterfactual set W s. we have
P(Y₂ y. xx
P(Yxa x,,,,
(111)
Example 13 (SCM-induced CTFBN Markovian). Consider the SCM from Example Is induced causal diagram is shown in Fig. 12 and its induced set of counterfactual distributions P satisfy:
1. IIdepeedeecce Restrictions]
P(z, L' U2" U.. (t
(112)
2. [Exclusion Restrictions)
P(ayes w.) = P(εe, w.)
(113)
Plyxe wa) = P(ye, w.)
(114)
P(zn, W.) = P(z, W.), Va (x,y)
(115)
3. Local Consistency]
P(x, 2) P(Iz, 2)
(116)
P(ay, [y) P(Iyz: 2n)
(117)
P(y,z) P(Uz,
(118)
P(ye, 2x) P(yze Zo.
(119)
The causal diagram in Fig. a2 is CTFBN Markovian for Ps-
Definition 25 (CTFBN Semi-Man Markovian [ Def. 13.2.2)). A directed acyclic graph & is a Counter- factual Bayesian Network for P # if:
1. [Independence Restrictions] Let W. be set of counterfactuals of the form W WP the c-components of GVV -)), and Cr....., Ct, the corresponding partition over W.. Then P(W.) factorizes as
pa.. Ci.... C
P( A
W Wpan
P( - f=1 Wynn EC2,
A
W Wpe.)
(120)
2. Exclusion Restrictions] For every variable Y V with parents Pay. for every Z C (Pa, (Y))), and any counterfactual set W., we have
P(Ypa, 2: W.) P(Ypa, W.)
(121)
30

---

3. [Local Consistencyl For every variable Y E V with parents Pag. let X _ Pay. then for every set Z
U ( and any counterfactual set W 0. we have
P(Y, y, X2 x, W.) P(Yxa = y, W.)
(122)
Example 14 (SCM-induce) CTFBN Semi-Marte Markkoian)).. Consider the SCM from Example [1 Its induced causal diagram is shown Fig. b) and its induced set of counterfactual distributions P satisfy:
1. "Independence Restrictions]
P(z,Tz, Yur v2..) P(=) 2'2,, Yur Va.")
(123)
2. [Exclusion Restrictions
P(ayz, W.) = P(xz, w.)
(124)
P(yer wa) = P(yle W+)
(125)
Piza, w.) = P(z, W.). ) Va S fx.y)
(126)
3. [Local Consistency]
P(x,2) P(Iz, 2)
(127)
P(ay 2n) P(IyE: 2ny)
(128)
P(y,x) ((
(129)
P(ya, T2) P(Wazz w2)
(130)
The causal diagram in Fig. is CTFBN Semi-Markovian for P#
A.5 Counterfactual Randomization
An agent may sometimes interact with a system interest through experiments. thereby collecting data from different layers of the PCH. Counterfactual randomization is an experimental procedure that enables an agent to observe the value of a variable before intervention takes effect 44. For instance, doctor may be able to determine patient's ss natural choice of drug prior to randomly assigning treatment clinical trial. This extension of experimental capability formalized in the following definition of a new type of physical action that an agent may be able to perform in an environment.
Definition 26 (Counterfactual (ctf-) Randomization (Def. 2.3 19)). For variable X and some particular unit in the target population of the environment, the operation
CTF-RAND(X + C)
(131)
denotes fixing the value of X as an input to the mechanisms generating C C Ch(X) for this particular unit, where Ch(X) is the set variables whose mechanisms take X as argument.
The value of X is assigned randomly with support over Domain(X),
The essential differences between Fisherian randomization and CTF-RAND(X + C)(1 are:
1. CTF-RAND does not erase unit i's natural decision x
2. While Fisherian randomization affects all children of X, CTE-RAND only affects the chosen subset C S Ch(X), leaving Ch(X) \ C.C untouched.
Importantly, CTF-RAND can only be enacted under certain structural conditions. These include environments where one can measure a unit's natural decision while simultaneously randomizing its actual decision [4) settings where counterfactual mediators allow altering how subset of children perceive the value of [19]. either case, ctf-randomizztion enables multiple randomizations on the same variable X for single unit i. Further, CTF- RAND must always be applied with respect to a graphical child variable; is not possible to bypass a child and directly alter the perception of a descendant.
"This definition discusses a unit speciipe experimental procedure, it takes physical perspective on how an agent interacts with the units a system. Another way understand this difference is that the unit's natural inclination is taken into account.
31

---

Action Sets for Realizability
POY, Y,) P(Y,ly'x)
Fucutefictaal
CTF-RANDIX Cy V.VV Ch())
2Q)
2
CTF-RANDIX ChN))
=(x1Ix=x)
W
Intervenional
P(Y|do(X))
2
RANDOX) V Y
READIV) VVEV NVV SELECT
P(
Figure 13: Hierarchy of action sets to realize distributions in different layers
Example 15 (CTF-RAND). Consider the SCM from Example Counterfactual randomization on X allows an agent to observe the natural value of X, say x', while simultaneously assigning a specific value x input to its children Z and Y. This is illustrated graphically in Fig. (b), and result, the L3 distribution P(X- = x', -xxxxxx Yx=a) becomes experimentally accessible (i.e., realizable).
By including the counterfactual randomization action into our experimental toolkit, we obtain the action set that gives the agent the most granular experimental capabilities.
Definition 27 (Maximal Feasible Action Set (SCM) (1)). Given an SCM M VU, P(u)..
The maximal feasible action set (MM is the set of all actions the agent can perform in M with the most g granular interventional capabilities:
(i) SELECT()) randomly choosing, without replacement, a unit i from the target population, to observe in the system;
(ii) (E VV V: measuring the way in which causal mechanism fv F has physically affected unit by observing its realised feature V(O;;
(ii) RAND(X)O VX E V: erasing and replacing i's natural mechanism fx for decision variable X with an enforced value drawn from randomising device having support over Domain(X):
(iv) CTE-RAND(X + C)(0), VX, VC E Ch(X): fixing the value of X as an input to the mechanisms generating O Ch(X) using a randomising device having support over Domain(X) for unit i, where Ch(X) stands for the set of variables that take X as an argument in their mechanisms.
SELECT with READ correspond to random sampling. When SELECT and READ are permitted over all units and variables, all distributions L, are realizable. Adding RAND to the action set gives the agent the ability to perform randomized experiments, When SELECT, READ and RAND are permitted over all units and variables, all distributions in C2 are realizable. With CTE-RAND, some distributions in L3 also become realizable. These distributions are the ones that lie within P and 2 If we can perform all actions from the maximal feasible action in an environment, we are able to draw samples from any distributions in C2.55 For L3 distributions that lie beyond P there is currently no known experimental procedure to sample from them. From the definitions of action sets, we observe hierarchical structure in the feasible actions agent can perform to access distributions at different layers, as illustrated in Fig. [3] This hierarchy of action sets match the hierarchical structure of distributions from the perspective of realizability.
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B Details on Layers 2.25 and 2.5
B.1 Nested Counterfactuals
The counterfactual variables in the symbolic representation of 22.25 and C2.5 are all the form Yx. where the subseript indicates that intervention do(X x) has been performed in the system. There is another type of counterfactual variables which represents interventions like do(X x2), where the variable X is set to behave as another counterfactual variable, say Xz. A random variable Y in such a system is represented with a counterfactual of the form Yx, which is called nested counterfactual.
All nested counterfactuals can be unnested via the Counterfactual Unnesting (CUT) process below and be transformed into non-nested ones.
Corollary (Counterfactual Unnesting (CUT) b)). Let E, V, and let z be a set of values for Z. Then,the nested counterfactual P(YT.x, y) can be written with one less level of nesting as:
P(Yr.x,=1)) E(PTY
X2 x)
(132)
Nested counterfactuals may also belong to L2.25 and L2.,, provided that their unnested form, derived via the Counterfactual Unnesting Theorem, contains only distributions admissible within the corresponding layer.
Lemma (Nested Counterfactuals in L2 15.2//Co L2.)). nested counterfac- tual belongs to C2.25/L2. if and only if there exists sequence of appli- cations of the CUT procedure that reduces a function of unnested counterfactuals in 22.25/L2.55
X
Y
Figure 14: Causal Dia- gram: path X - Y rep- resents the Natural Direct Effect (NDE)
Example 16 (Natural Direct Effect (NDE)). Consider the causal diagram in Fig. 14 The natural direct effect from X to Y can be written in counterfactual language as
NDE, x. (y) P(yx P(yz)
(133)
The first term is a nested counterfactual, and we can derive its unnested expression by applying CUT.
Plyz Σ P(yer.2e)
(134)
From this unnested expression, can conclude that it is in C2.5 P(Y Zz) satisfies the conditions in Def. 4 However, it is not in L2,25 due to the conflicting subscript x and xf in the two counterfactual variables joint.
B.2 Examples for C2.25 and L2.5
Def. and Def. can be viewed as the template enumerate distributions in C2.25 and C2.s. The key difference between the two layers is that C2 25 is indexed by specific interventional values, while C2.5 is indexed by interventional variables. This difference illustrated in Example where different value assignments for the interventional variable set X is allowed in L2.5 but not in 22.25. We further illustrate this difference in another example below.
Example 17 (Difference in Indexing between C2.25 and L2.3). Consider the SCM from Example 11 where the variables V (2. Y form a chain X - topologically, Let the interventional variable set be {Y}.
For L it is indexed by specific interventional value. So we need to fix the value assignment of to be y E Vol(Y). Then by Def. Xy, where Z and X share the same subscript, is distribution in L2.25-
For L25, the interventional variable can take any value in its domain unless it is constrainted by Cond. (ii) of Def. f when two variables are descendants of the same child of an intervened value. In this example, Z and X are not in descendants of Y. As result, there no constraint on value assignment Y for Z and X. Taking any y. E Val(Y), P(Zy, Xy, Y) is a distribution in L2.5-
33

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This example shows that the increased flxibbiity of indexing in C2.5 compared to C2.25 allows it to include more distributions.
Cond. (i) of Def. 3 and Def. ensures that all variables in the intervention set must appear at least once as subscript the counterfactuals joint. This avoids any redundant symbolic representation to appear during the enumeration of distributions in the languages, as illustrated in example below.
Example 18 (Cond. (i) of Def. and Def. 4.. Consider the same SCM from Example 77 where the variables V [2, X, Y) form a chain Z + - Y lopologically. Given two interventional variable sets { and {Y).
The empty interventional set gives the distribution P(Z,X, XY), where all subscripts are empty. This is consistent with our understanding that empty intervention equivalent to observation. For the interventional variable set {Y}, if Cond. (i) is not imposed, P(Z,X,Y Y) would also be compatible with the symbolic representation for distributions in these layers. This means that the same distribution is repetitively enumerated under different interventional variable sets. To avoid this redundancy, we impose Cond. (i) to require the union of all subscripts to cover the interventional variable set. In other words, y must appear as a subscript in at least one of the counterfactuals joint. As a result, the enumeration would not produce P(Z, X, Y), but rather, produces distributions like P(Zy, X.Y), P(Z,Xp Y) or P(Zy, Xy, Y) for L2.25, and also P(Zyy Xyy, Y) for L2.55
Cond. (ii) of Def. B and Def. reffects how counterfactual randomization enforces consistent values over downstream variables. For C2.25, counterfactual randomization variable X is restrained such that all children of X shares the same value z. As a result, all descendants of X share the same value x. In contrast, counterfactual randomization in L2.5 allow each child of X to interpret X differently. Yet, given that counterfactual randomization cannot bypass child to affect descendants directly, still imposes consistent value constraint over the descendants of X. This constraint starts at the children of X, instead of X itself.
Example 19 (Cond. (ii) of Def. and Def. 4.. Consider the causal diagram in Fig. Fa and the intervention on X. In L2, the submodel fixes = x and we obtain the distribution P(Y, Z ())( In C2.25, all downstream variables of X must include in its subscript, i.e., Y,, Zz. At the same time, counterfactual randomization allows us to join the natural value of X with the other counterfactual variables and obtain the distribution P(X, Y Zz). In L2.25, the downstrean variable consistency is only enforced at the child level. In this example, different subscripts of X for Y and are allowed and we obtain the distribution P(X, Yz, )z The difference belween the three layers are illustrated graphically in Fig. 3
B.3 Details on CBN2.5
In this section, we give the detailed definition and theorem for CBN2.5.
Definition 28 (CBN2.5 Semi- Markoviann... Given a mixed graph & and let P be the collection of all L2s distributions. G is a Causal Bayesian Network 2.5 for p if:
[Independence Restrictions] Let W. be a set of counterfactuals of the form W pa with distinct W. Ci...., Ci the c-components of GVV(W=)) =)). and Ci...... Ct. the corresponding partition over W. such that P(W.) E P a Then P(W (.)=))actrrizes as
P( A Wynng EW.
W Wpa.>
II P( A j=1
W Wpe..
(135)
Wyo EC2.
2. [Exclusion Restrictions] For every variable Y E V with parents Pay, for every set Z C (Pa, U (YY)), and any counterfactual set W. such that P(Ypa, W.) E PC2ss we have
P(Y paya,,.2. 2: W.) = P(Y pa., W.)
P(Ypa,
(136)
3. LLocal Consistency) For every variable Y E V with parents Pay. let X C Pay. then for every Z C U (Y))), and any counterfactual set W. such that P(Yxx. y, y,
X. W.) ) E PC2s, we have
P(Y, xx x, W.)
P(Yxa v, =x W.)
(137)
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Theorem 5 -o.ss-onnnetion - CBN2.5 (Markovian and Semi--arrovvann), The Causal diagram
induced by the SCM M following the constructive procedure Def. 13 CBN2.5 for P the collection of all L2.5 distributions induced by M.
Example 20 (CBN2.5). Consider the SCM from Example 7
The Causal Diagram is induces shown in Fig. aa and the collection of realizable distributions plas it induces salisfies the following constraints:
1. [Independence Restrictions]
P(, = P(X))(X)1)))
(138)
2. [Exclusin Restrictions)
P(a. 2, W.) P(X I, W.), {2,3}
(139)
P(Y:: y. W.) P(Y2 = y. W.)
(140)
P(Zzy 2, W.) P(Zx = W.)
(141)
3. [ocall Consistency]
P(Y = y, X = =x) = P(Y, = X = x)
(142)
P(Y, = y, X, z) P(Y 50 = y, = x)
(143)
P(Z = 2, X x) = P(Z, = xxx=
(144)
P(Zy Xy r) P(Zyr x, =
(145)
B.4 Independence Constraints and AMWN
The independence rule in etf-calculus requires the construction another graphical object, known as the Ancestral Multi-World Network (AMWN) 16. We reproduce the algorithm for AMWN construction and the theorem stating soundness.
Algorithm AMWN-CONSTRUCTIG, W.)
Input: Causal Diagram and set of counterfactual variables W.
Output: (A(..), )) the AMWN constructed from 9 and W. 1:
Initialise g' by adding variables in An(W., together with the directed arrows witnessing the ancestrality
2: for each node V E V appearing more than once in G' do
3: Add node Uv and edge Uv Vx for every instance V of V.
4: end for
5: for each bidirected V ------ W where V and W are in G' do
6: Add a node Uvw and edges from it to Vx and Wx for every instance Vx of V or Wx of W
in G".
7: end for return G'.
Theorem 6 (L3 Independence Constraints Counterfactual d-separation). (Theorem in ( Consider a causal diagram G and a collection of counterfactual distributions, Phs, induced by the SCM associated with G. For counterfactual variables X ; Y,, Z,
Xell 1 ||Y|l| [Zz,)).. = [I]Xell IL ||Ye|l [Z-11)ps
(146)
In words, xXee and || are d-separated given [Z.]l in the diagram GA Yr Z.), then || || and [|Y|ll are independent given | [Z,|| |l in every distribution PCs compatible with the causal diagram G.
When adapting ctf-caleulus CBN2.25 and CBN2.5, there is an extra step to ensure that the distributions belong to the corresponding layers. This can be added as an extra step before Step 1 of Alg. to check that:
CBN2,25: CRS(W.) satisfies Lemmal
. CBN2.5: An( An(W.) satisfies Lemma |
The same check applies to the other two rules of ctf-calculus too.
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C Proofs
C.1
Supporting Lemmas
Lemma C.1 (Casual Diagram of Submodel), Given an SCM M and causal diagram G. the causal diagram induced by its submodel Mx is Gx. i.e. 9 with all incoming edges to X &emoved.
Proof. By Def. 9 Mx replaces fa with X <- for all X E X. As result, X have no endogenous or exogenous parents. By the causal diagram construction Def. [3] edges that point to X are added only when X have parents. Thus, there is no edges incoming X. the causal diagram induced by MM. In addition, given that MMx keeps all other components of .M intact, all other edges remain the same. Therefore, the causal diagram induced by MMx is G with all incoming edges to X removed, denoted as 9x.
Corollary 2. Condition (ii) of Def. and Def. can be translated to an equivalent gruphical condition:
C2.25 For any v, E for all V3 Y, if,, E An(V,) in Ixy,, then Us E x,.
D2.5 For any V,, B BeXn Pa(V.). for all V2 E Y, Vf & X, and V; E An(V)) in Jxy. then x, x, B = n
Proof. It follows from Lemma
Lemma C.2. Given a causal diagram 9 over V and a set of counterfactual events W., if P(W.). in p of all SCMs compatible with G, then P(]) W. is also in Pl225 of all SCMs compatible with g.
Proof. If P(W.) is in a of all SCMs compatible with G, it satisfics both conditions of Def. 3 We prove that after applying the exclusion operator to W.. the distribution still satisfies both conditions of Def.
Let the set of potential outcome variables in W. be denoted as { [ Wa }. P(W.) is indexed by the union of subscripts of all Wa,, E W., and we denote this index by & U, t.. The exclusion operator does not add subscripts to the variable, so let the new index set be the union of subscripts of all [[W,, E |W.]l and denote it as t' A U, th. Cond. (i) of Def. 3 still holds.
Given that P(W.) also satisfies Cond. (ii) of Def. 3 and by Cor. 2 it means that whenever there directed path from T E to W; V[W.] in Grow t in the subscript of W, i.e. t.. Applying the exclusion operator on W,, removes variables in t; that does not have a directed edge to W; in G Thus, it does not affect those that satisfy the antecedent of Cond. (ii) of Def. 3 As result, whenever, the antecedent of Cond. (ii) of Def. 3 hollds, 1. still belongs to the subscript of W,. So Cond. (ii) of Def. still holds.
Given that P(] W. Il) satisfies both conditions of Def. is in PC222
Lemma C.3. Given causal diagram G over V and a set of counterfactual events W.. P(W.) in 2 of all SCMs compatible with G, then P([[W. II) is also P of all SCMs compatible with g.
Proof. The proof is very similar to Lemma C.2 with the key point being that the exclusion operator on W., removes variables in t; that does not have a directed edge to W, in GT,- Thus, it does not affect those that satisfy the antecedent of Cond. (ii) of Def. 4
Lemma C.4. Given a causal diagram & over V and a set of counterfactual events W, (Wing) with all subscripts taking consistent values from the same set v E Val(VV, if ( tee = [[Wijoee for all i, then P(W.) x) is in 2 of all SCMs compatible with G.
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Proof. The exclusion operator removes subscripts 2 from W; if there is no directed path from X to W; in -. Thus, the subscripts that remain after exclusion capture precisely the cases in which the antecedent of Coid.(ii) in Definition 3 holds. 1 i ll. the subseript in x; accounts for all instances in Ul * 2;, that are restricted by Cond. (i). Therefore, P(W.) satisfies Def. 3 and belongs to P
Lemma C.5 (ctf-caleulus - do dolcclluuus reduction (Lemma in 6)). etf-calculus subsumes do- calculus.
Lemma C.6 (ctf-calculus 2.25 do-calculus reduction), etf-calculus restricted to 2 subsumes do-calculus.
Proof. This result follows from the proof of Lemma C.5 where all steps in the reduction only involves quantities within 222.25
Given a graphical model with bidirected edges, 9, the set V of observable variables represented as vertex can be partitioned into subsets called c-components [2I] such that two variables belong the same c-component they are connected in g by a path made entirely of bidirected edges.
Definition 29 (Ancestral components (6). Let W. be a set of counterfactual variables, X. C W., and G be causal diagram. Then the ancestral components induced by W., given X., are sets Agn...... that form a partition over AnW made of unions of ancestral sets Anlgx. (w.) We W.. Sets (W and (W(gg.(Waa gether if they are not disjoint or there exists a bidirected arrow in 9 connecting variables in those sets.
are put to-
Lemma C.7 (Ancestral Set Factorization (Lemma in (6)). Let W. be an ancestral set, that is, An (oW = W., and let w. be a vector with value for each variable in W.. Then,
P(W. = w.) P(
Wa = w)
(147)
WoEW.
where each w is taken from w. and paw is determined for each We W, as follows:
(i) the values for variables in Paw O T are the same as in t, and
(ii the values for variables in Pa \T are laken from W. corresponding to the parents of W.-
Lemma C.8 (C-component Factorization (Lemma 4 in ()). Let P(W. =w.) ) be distribution such that each variable W. has the form W Pa llet W1 < W2 < ... be topological order over the variables in G V(W ..), and let Ci,..., Ck be the c-components of the same graph. Define C, (Wpa.. E W. W E C,} and Cj. as the values w, corresponding to Cj,, then P(W. -W, decomposes
P(W. w.) = II P(C,, =C,))
(148)
Lemma C.9 (Ancestral Set in 22 C2.t). P(W.) ) is in 22.25/ L2.5 if and only if the distribution over its ancestral set ([An(W.) is also in 22.25/ C2.5.
Proof. For C2.25. CAS(An(W)) .) = CRS(W.) by Def. 7 and for C2.5. An(An( (W)) An (n) by Def. ??. Thus, W. satisfies Lemma I and only if An(W.) satisfies Lemmal
Lemma C.10 (Ancestral Set Factor in 22.25/ L2.5). Let W. be an ancestral set, that is, An(W.) W., and let w. be vector with value for each variable in W. Then, P(W.) )) is in 2225 L2.5 only if its ancestral set factor P(Aweww Wpa.. = w) is in L L2.5
Proof. If P(W.) )) is L2.25/ L2.5, then there does not exist two variables W. and W, in W. with
inconsistent subscripts. Therefore, the ancestral set factorization will also have distinct W for each
W Wpa... satisfies conditions Def. Def. 4 with consistent values from w, for 22 and with
Pa blocking all directed path from other variables to W.
O
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Algorithm 2 CTFIDU Y,,,,
Input: 9 causal diagram variables V: Y, set of counterfactual variables in V; y. set of values for y; and available distribution specification Z.
Output: P(Y, y+) in terms of available distributions or FAIL if not identifiable from (G, 2)
1: let Y. |-
y.nny.y,,,, (Yx) or Yr E Y. with y. (Y2) 4: if there exists Yx E Y. with two consistent values in y. y.x)) or Y, E Y. with y. = then remove repeated variables from Y. and values y..
2: if there exists Yx Y. with more different values in y. then return 0.
3: end if
5: end if
6: let W. An.)), and let Ci....., Ck. be corresponding ctf- in G[V(W.).
7: for each C, s.t. (C;, C.+) ) is not inconsistent, Z E 2 s.t. C, n Z = @ do
8: let B, be the e-component of Gz such that C, S B,, compute Pv:B, (B.) from Pz (V).
9: if IDENTIFY(C;; B,, Pv, (B,), 9) does not FAIL then let (v,c,(C.) t IDENTIFY(C,; B,, Pv.B, (B,),, (B,).). 11: let P(C C.v) <- (yc,(C, evaluated with values (c,, U Uc.ec. pa.) move to the next C, 13: end if
10:
12:
14: end for
15: if any P(C;. C.+) is inconsistent not identified from Z then return FAIL.
16: end if
17:
return P(Y. = y + Ew. w. P(C, C..).
Lemma C.11 (C-component Factor in C2.25/L2.5) Let P(W.=wi be a distribution such that each variable in W. + has the form W Wa.. with its c-component factorization P(W.- ww.) 11; P(C = C..). Then, P(W.) is in 2 L2. only if its c-component factors P(Cj. = Cj.) are in 2 22.5.
Proof. If P(W.) is in C2.25/L2.5. then has distinet for each counterfactual the set and satisfies Def. Def. This property is not affected by e-component factorization only partitions
W. into subsets connected by bidrected paths. As a result, each P(Cj. = cj.) will also satisfy Def.
3
Def. 4
O
Lemma C.12 (Consistency (Lemma 1 in 6))). Given SCM M and X,Y E V,T.C V, and let be a value in the domain of X. Then,
=) P(P Xr. x),
(149)
where T. represent any combination of counterfactuals based on T.
Lemma C.13 (Exclusion operator (Lemma 2 in 6)).. Let Yx be a counterfactual variable, g causal diagram, and
Ya such that Z = X x Ans (Y) and x x
(150)
Then, Y2 = Yx holds for any model compatible with G. Moreover, this transformation is denoted as [[YY)) := Yg-
Lemma C.14 (Independence in (2.55/22.) Given a CBN2.25/CBN2. Theorem 6 is sound when the AMWN constructed over W. where P(W.) in L2.25/ (2.5
Proof. The soundness follows from soundness of Theorem 6 where the ancestral set factorization constructed over [X, Y,, Z} in the proof also the corresponding layers 25.25/ L2.5 by Lemma C.9lndd and Lemma C.10
38

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Algorithm 3 CTFID(Y Y.: X., 2,
Input: 9 causal diagram over variables V; Y. X, set of counterfactual variables in V; yx, x, set of values for Y. and X.; and available distribution specification Z.
Output: P(Y, y+ X. = x.) in terms of available distributions or FAIL if non-ID from Z).
1: Let A1,, A2..... be the ancestral components of Y. UX given X..
2: Let D. be the union of the ancestral components containing variable in Y, and d, the
corresponding set of values.
3: let Q < CTFIDU(UD,ED p d.;
4: return Ed. My..). 2/E Q.
C.2 Proofs for Main Theorems
Theorem (L2.25- Connection - SCCMCCB2.25), The Causal diagram G induced by the SCM M following the constructive procedure in Def. 13 is CBN2.25 for P4225, the collection of all 22.25 distributions induced by M.
Proof. Let M be SCM, p222s the L distributions it induces and & its causal diagram. We prove that (g. pl2225) is a CBN2.25, by showing that the 3 conditions defined in Def. ?? holds in p22.5 according G.
(Independence Restrictions) Given a potential response of the form W Wa... its value only depends on the exogenous variables U.. which appear arguments in fw. Let W. be the set of counterfactuals of the form Wp. with pa... taking consistent values from v E Val( P(W.) falls in 22.25 as satisfy conditions of Def. 3 Let Ci...., C₁ be the c-components of G[V(W.)) ]. and C1., ....... Ct+ the corresponding partition over W.. Then the set of exogenous variables U(W.) can be partitioned U) U(C,,) where U(C,,) and U(C,,) are disjoint for all j 1, "..... 1, + j. due to the absence of bidirected paths between variables in C; and and variables Cj. Then by Def.
(Exclusion restrictions) Given a potential response of the form Yps,,z. its value only depends on the exogenous variables U, which appear as arguments in fy as pa, are fixed. Thus, Ypa, (u). Then by Def. for any counterfactual set W. such that P(Y pa, y, W. =..)
=
P(Ypa,, = y, W. w.. = Σ 1(Y 1(Y Da, y, W-(u) = w.)P(u)
(151)
Σ 1(Y pa, (u) = = W. P.)
(152)
P(Ypa, = W. w.)
(153)
which proves the exclusion restrictions are satisfied.
(Consistency restrictions) Given u E Val(U) such that Yg = y, (z) xx W.(n) = w., for
some Y X. Pay. S V U (Y)), R Pay (X U Z), we have
Yz(u) fy (z n pa,, (u). R₂ (u), u(U,))
(154)
fr(2 n pay, x, R2 (u), ((U)))
(155)
_u)
(156)
Then by Def. 3 for any counterfactual set W, such that P(Y, = y. X, = W. -..) E P2225
P(Yz y. X2 = x, W. = =w.) = Σ 1(Yx(u) = (u) x, W. = =w.)P(u)
(157)
u
1 = y, X, (u) x, W.(u) w. P())
(158)
P(Yxe = y, xx W. -..)
(159)
which proves the consistency restrictions are satisfied.
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Definition 30 (Counterfactual Reachability Set). Given a graph G and a potential outcome Yx. the counterfactual reachability set of Xx denoted CRS(Yx), consists of each I| Il s.s. W (An(Y) U (De(V): : AV E x])XXX and [| xwwll s.t. W E (An(Y) U (De(V) VV E xX).XX For set .. CRS(W.) )) is defined to be the union of the CRS of each potential outcome in the set, such that for any set of variables (Wsggss S W. with their CRS set having counterfactual
variables {Rix))s hx over the same variable R. (Rx.) merged into one variable [[ if
[|W [
[1pm0 W all i.
Lemma 1. A distribution Q = P(W.) is in the C2.25/ L25 distributions induced by any SCM
compatible with given graph G if and only if the set CRS(W..) .) satisfies (i) and (ii) / An(W.) .) satisfies (i): (i) Does not contuin any pair of potential outcomes Ws. We of the same variable W under different regimes where + t; (ii) W. does not contain any pair of potential outcomes Ra, Wt. with inconsistent subscripts where n T t.
Proof. Consistent values across the variables are enforced by (ii). Each CRS set corresponding to a potential outcome Y. includes all variables that must remain consistent with Y. under the regime When taking the union of CRS sets over multiple potential outcomes, and if the union does not contain any pair of potential outcomes Wg. We for the same variable W under different regimes, then two cases arise:
(a) All CRS sets are disjoint with respect to the variables from which their potential outcomes are derived. This implies that the ancestral and descendant sets of these variables are also disjoint, so there is no directed path crossing the CRS sets in way that would trigger the antecedent of Cond. (ii) in Definition
(b) Any overlapping CRS sets must involve counterfactuals over the same variable, which are merged as [W;, ipu,,..] = a for all i. This condition implies that the variables underlying these merged CRS sets are consistent, by Lemma C.4
Therefore, P(W.) satisfies conditions in Def. 3 and belongs to p4222
The graphical check for C2.5 is proved in Corollary 3.7 of [9
Theorem 2 (Soundness and Completeness for CBN2.25/ Identifiability). An 2225//2..5 quantity Q is identifiable from a given set of observational and interventional distributions and CBN2.25/CBN7 if and only if there exists a sequence of applications of the rules of ctf-calculus for CBN2. 25/CBN2.5 and the probubility axioms restrained within L2.25/L2.5 5 that reduces Q into function of the available distributions.
Proof. The soundness of the calculus for L2 22.55/22.5 follows from the soundness of the etf-calculus rules. The soundness of the etf-caleulus rules in turn follows from Lemma C.12 for Rule Lemma C.14 for Rule 2 and Lemma C.I3 for Rule 3.
To prove that it is complete, we rely on the completeness of the CTFID algorithm reproduced as Algo. and Algo. 2 77. Specifically, we show that if the query is in L2.5,/ all steps of the CTFID algorithm can be justified by the rules of etf-calculus for CBN2.25/CBN2.5 and the probability axioms restrained within L2.25/L2.5-
Line 1 and 2 of Algo. 3 are justfied by Lemma C.9 and Lemma C.10 if the input query P(Y, y. X. = x.) is in L2.5, then the ancestral set factorization P(UD,eD, Dpa, = d) over D. = An(Y., X.) and d, E Val(D.) consistent with y,,x, x, is also in 22.25/ Thus the probability axioms underlying the marginalization step have all quantities within the corresponding layers.
Line 1 of Algo. is justified by rule 3 of the etf-calculus and Lemma C.2 and Lemma where both D. and ||D.| are in the corresponding layers. Line 2 to 3 are justihie by quantities in L2.25/ having consistent values. Line 4 to 5 follow from probability axiom to remove redundant variables. From line to 14, the algorithm identifies the factors based on c-componentes using IDENTIFY [21] which soundness can be justified with do-calculus ( which in turn is subsumed by ctf-calculus 2.25 by Lemma C.6
line 17, the algorithm returns the result as a product that is justified by Lemma
C.I1
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Therefore, given query in L2.55/C2.5. CTFID is both sound and complete to determine if it is identifiable from the available data without any intermediate step having quantities outside the layer.
n
Theorem 3 (PCH*), Given an SCM M and its induced collections of observational (P interventioaa(((aa) L2.25 (PL22), L2.5 (Pl2s), and counterfactual (PCs) distributions: p C pla C p2ass C ppas C pls.
Proof. With PCH already established and proved for Cr.C2 and C3 [3. we prove that (1) p C
plan, (2) 2 Ç p and (3) plzs C pCs,
It is easy to show that P C P42.25, because each distribution in P can be derived from marginalization of a distribution in pl223;
P(Y y/do(X XEYnX ViEYXX
P(
ViN - U
WEENX, Vinx Vipeaa v4) (160)
where the subscripts for all variables take the whole set x. Clearly, is in P as the consistent subscripts satisfy conditions of Def.
It is also easy to see that P22. C CCs because pCs contains all possible joint distributions over all counterfactual variables, whereas p4a. imposes additional constraints over the joint of counterfactual variables.
To prove that 2 C CCass we show that if a distribution satisfies Def. it also satisfies Def. First, note that the key difference between Def. and Def. A in the two conditions. Thus, only
4
need to prove that a distribution of the form P P(Aneeix |xx| = 2/ war./vveenx... -vv.vnx V Vix
w:) satisfying the two conditions in Def. must also satisfy the two conditions in Def.
For Cond. (i), both languages require the subscripts to cover the whole space of X. However, Def. is stronger by restricting the value assignments to the set x, while Def. 4 allows x; to take different values from Val( Val(..). Thus, if Cond. (i) of Def. 3 holds, Cond. (i) of Def. iimmediately holds.
For Cond. (ii) and by Cor. 2 the antecedent in Def. 4 cchekks there a directed path from B E X to Vi E Ch(B) to V, in Gx.- If such a path exists, we denote by p. There two possibilities: (a) p in G C (b) p is not in Cx,7. For (a). Cond. (ii) of Def. will enforce b to appear in the subscript of both V; and Vj. For (b), it implies that there exists a variable X E X)X, that lies on p between Vú and Vy. We focus on the subpath p' of p directed from X Vg. If X is An())) in Gx., then X must be in X, by Cond. (ii) of Def. 3 which leads to contradiction. If X is not in An(V;) in Gx then there exists another X' E X that lies on p' between X and Vj. We can apply the same logic to shorten p until there is no more variable in X XX that fulfills the same condition. When this terminal condition is hit, the final subpath enforces the variable in X X on the path to be in the subscript of Vj. The same contradiction is achieved. As a result, there cannot be any variable X X X that lies on between Vi and V,. Therefore, whenever the antecedent of Cond. (ii) of Def. is triggered, Cond. (ii) of Def. also holds to enforce consistent subscripts between Vi and Vj-
This proves that all distributions in 22222 are also in Pf2s, or equivalently PC2.5 C PC2.5.
Theorem 4 (Hierarchy of Graphical Models, PCH"). Given a causal diagram 9, the set of constraints encodes when is interpreted as a graphical model on layer i subset of the constraints encodes when it is interpreted as graphical model layer j. when j.
Proof. The constraints encoded by BN are included as Cond. (i) of the corresponding CBN, making the containment relationship is straightforward. The hierarchical relationship among the constraints encoded by CBN2.25, CBN2.5, and CTFBN is also straightforward, as they share the same structural form while progressively increasing the flexibility of distributions allowed at each level in the model hierarchy. The containment relationship between CBN and CBN2.25 follows from the fact that do-calculus subsumed by the etf-calculus 2.25 (Lemma (.6). and that the constraints defined in CBN imply all rules of do-caleulus, while those in CBN2.25 imply all rules of etf-calculus 2.25.
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Graphical Model
Meaning of Missing Directed Edge P(u pa,, nd,) P(u. pa, P(Vipac:a) ) = P(uipa))
Meaning of Missing Bidirected Edge
L1: BN
P(us (o(x), pa,, do(pa,))
L2: CBN
P(us (ol])d(((),,),)))))))))) paf, paa)
Piotpo "Jom; = P()/pm. )P Upngy
P( W.) = P(uipan: We); with P(Uipa;. W.) E P2222
22. CBN2.25
with V; + V2
and pa, and pa, taking consistent values
P(Uipara W.) P(uipas: W.),
P Plvcpa U U/ppa P(ulppa P(U)pa))) P( with V; + V, P(u Opnn ) P( )P(Ujpa)) with pa, 7 pa, if = V,
L2.5: CBN2.5
with P(Uipo W.) E P22s P(Uipapx W.) P(Uipa, W.). for any W.
C3: CTFBN
Table 4: Summary of how missing edges are interpreted in graphical models at different layers
Since the constraints encoded by graphical models are encoded by the missing edges in G. we can alternatively establish the hierarchy by comparing how different models interpret these missing edges, as summarized in Table 4 For missing directed edges, the constraint forms are consistent across layers. but higher layers allow increasing flexibility in the sets w. that can be jointly conditioned on. Similarly, for missing bidirected edges, the independence constraints in CBN2.25s, CBN2.5s, and CTFBNs share common structure, with each successive model relaxing the limitations on how these independencies are expressed:
Independence constraints in CBN2.25s only apply to distributions over distinct variables that share consistent parent values.
Independence constraints in CBN2.5s extend to distributions over distinct variables, allowing their parents' values to vary freely.
Independence constraints in CTFBNs apply to distributions over any variables, including those of the form P(Wpa.. W Wpac)
as long as pa o paw..
Theorem 5 Seass-Connection CBN2.5 (Markovian and Semi- Cemi-Markovian)). The Causal diagram G induced by the SCM M following the constructive procedure in Def. is is CBN2.5 for PCa.s, the collection of all f2.5 distributions induced by M.
Proof. The proof is similar to the proof for Theorem I with the independence restrictions expanded to allow inconsistent parent values, and the exclusion and consistency restrictions expanded to join more W. such that the distributions are within L2.5 instead of 22.25-
D Frequently Asked Questions
Q1. Where the causal diagram coming from? Is it reasonable to expect the data scientist to create one?
Answer. First, the assumption of the causal diagram is made out of necessity. The causal diagram is a well-known flexible data structure that is used throughout the literature to encode a qualitative description of the generating model, which is often much easier to obtain than the actual mechanisms of the underlying SCM 15 20 17]. The goal of this paper is not to decide which set of assumptions is the best but rather provide tools to perform the inferences once the assumptions have already been made, as well as understanding the trade-off between assumptions and the guarantees provided by the method.
Second, the true underlying causal diagrams cannot be learned only from the observational distribution in general. More specifically, there almost surely exist situations that M1 and M2 induce the same observational distribution but are compatible with different causal diagrams (see 3 Sec. 1.3] for details). With higher layer distributions (such as distributions from L2), possible to recover a more informative equivalence class of diagrams that encode additional constraints present in the input layer [2]
[1] 10 13 22
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Q2. What is graphical model and how can it help us in causal inference?
Answer, A graphical model is a modeling tool that allows one to represent a compatibility relationship between a causal diagram G and a collection of distributions P. Specifically, encodes how the topological structure of the diagram can be interpreted to impose constraints the associated distributions. For instance, when restricting attention to Li distributions (i.e., purely observational), Bayesian Networks (BNs) are the most prominent graphical models to encode conditional independence constraints of the observational distribution [14]. As we climb up the PCH and include more distributions into the collection, more constraints start to emerge. To encode the richer set of causal constraints in C2 distributions (i.e., interventional), the Causal Bayesian Network (CBN) was introduced [3 More recently, CTFBN is introduced to encode the compatibility relationship between the causal diagram and L3 distributions (i.e.. counterfactual) M. The models defined in this work further refine the space of C3 distributions by restricting to constraints that are, at least in principle, empirically falsifiable. In a nutshell, a graphical model should not be viewed merely as causal diagram, but rather as formal specification of the compatibility relationship between a pair (g. P)- An example of a CBN is illustrated in Fig. 15| where missing edges in the causal diagram represent invariance constraints in the distributions.
The causal diagram in the graphical model offers compact representation for constraints in the associated distributions. These constraints are fundamental to causal inference, as they constitute one of the three core inputs the causal inference engine (Fig. D As discussed ealier, the main task in causal inference is to determine whether query from higher layer of the PCH can be identified as a function of observed data from lower layers. For example. the task may be to identify causal effect P(y]do(z) when only the observational data P(v) is available. According to the Causal Hierarchy Theorem (CHT), these layers are strictly distinct, and it is impossible to ascend to higher layer without additional assumptions about that layer [ Thm. 1]. The constraints encoded by graphical models serve precisely this role- they encode the assumptions about higher layers that enable to bridge the gap and make such inferences possible. Given the CBN in Fig. [5] the invariance constraint ((Y|X) allows us to identify the L2 query (())) as P(y|z), which only involves observational distributions. Question below will provide further details on the inferential process by explaining how the local constraints defined in graphical model can be composed to derive additional constraints implied by the model.
P(Y (())
Q3. Why do we need to introduce new layers to the PCH, besides the existing ones?
Answer. The original three layers of the PCH, capturing observational, interventional, and counterfactual distributions, provide natural partition among distinct capabiliitess in causal reasoning. Layers 1 and correspond to well-understood physical procedures: random sampling for observational distributions and random experimentation for interventional distributions. contrast, Layer 3 consists of purely counterfactual quantities, that are traditionally considered detached from empirical data collection in principle. In addition, while Layers 1 and 2 are well-structured and homogeneous (cach quantity within a layer having similar interpretation), Layer 3 is more heterogeneous and contains quantities that represent different aspects of the underlying data- gentratinn process.
More recently. Bareinboim, Forney and Pearl introduced a new experimental procedure, counterfactual randomization, that allowed one to sample directly from an La distribution [4]. This work was further extended in [19]. The introduction of counterfactual randomization reveals finer structure within Layer 3, distinguishing between counterfactual distributions that are empirically accessible and those that are not. This fine-g fine-graining of Layer 3 is illustrated in Fig. 16 Notably, these new families of distributions have attractive properties, including well-defined symbolic languages as well as a closed set of inferential rules, as shown in this work. This new view opened up natural way of partitioning L3. In this work, we studied the interplay between graphical models that inherent these features of the PCH and have the property of empirical falsifiability.
To answer the question, the new layers introduced in the refined PCH may not be necessary for all researchers. The original PCH already represents a major milestone in formalizing the logic of causal inference. Still, for some researchers, the refinement and further parti- tioning of Layer can offer valuable insights. particular, it allows for a more precise understanding of the trade-off between empirical falsifiability and the inferential power of
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Graphical Model
Casual Diagram
Distributions
Space
Space
2 Distributiins
Dittiiuttto
P(X Y)
P(Y do(X))
X Y
P(Y X)
P(Y)
P(X do()))
P())
Figure 15: A CBN is pair (g,P), Blue lines represent invariant constraints in P, which are represented by features from G: missing directed edge from Y to X corresponds to the invariance constraint P(X dY(Y)) )) = P(X) and missing bidirected edge between X and Y corresponds to the invariance constraint P(Y)do(X)) X)
p41
p
P22.55
P
p4
P(U)
P(X, Y do()))
P(Y) P(X,Y)
P(Y., 2,r)
P(Y]do(=))
P(Y,, x)
Layer 2.25
Layer 2.5
Layer (Observational)
Layer2 (Interventional)
SCM (unknown)
Layer 3 (Counterfactual)
Figure 16: Pearl Causal Hierarchy (PCH") induced by an unknown SCM M. Layers and are realizable, and Layer 3 is partially realizable, The realizable portion of Layer 3 are further refined into two new layers: 2.25 and 2.5.
graphical models, and provides tighter feedback loop between theoretical assumptions and experimental capabilities.
Q4. What is the difference between layers 2.25 and 2.5?
Answer. The main difference between L2.25 and L2.5 lies in the type of counterfactual randomization allowed. For 22.25, a counterfactual randomization applied to a variable X assigns the same value across all its children and descendants. As result, distributions in this layer cannot contain pairs of potential outcomes Ws. R₂ with conflicting subscripts where x E t and of x', In contrast, the counterfactual randomization action variable L2.5 is more flexible and allows each outgoing edge from X take a different value. This flexibility leads to the possibility of some distributions in the layer include potential outcomes with different subscripts. This difference is graphically illustrated in Fig. However, all descendants of each child of X must still share the same value of 2, unless all directed paths from to the descendant are blocked by other intervened variables. This restriction stems from the rules of counterfactual randomization, which prohibit an intervention to bypass a child and directly affect a descendant's perception of X. In summary, the constraint consistent subseript begins at the intervened variable X in L2.25, but shifts to the children of in L2.5- These differences are reflected in the relaxed conditions that define the symbolic language of L2.5. relative those of L2.25-
Q5. Are all distributions within Layers 2.5 realizable?
Answer. Theoretically. all distributions in C2.3 are realizable every action the maximal feasible action set is permitted. That is, in principle, an agent could draw samples from any distribution this layer through experimental procedures. However, whether a distribution is realizable in practice depends on the physical constraints of the system. If certain actions such as counterfactual randomization on specific variables are not feasible, then some distributions in C2.5 will not be realizable in real-world settings [19].
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Query Layer
Graphical Model Sufficient
Necessary
L1
BN
L1
CBN
L2
BN
X
C2
CBN
/
C2
CBN2.25
2..55
CBN
X
2,
CBN2.25
C2
CBN2.5
22.5
CBN2.25
X
/
22.5
CBN2.5
22.5
CTFBN
L3
CBN2.5
X
C3
CTFBN
Table 5: Examples of Matching between Graphical Models and Queries. Rows highlighted in green represent a match between the model and the query such that the assumptions in the model are both sufficient and necessary for making inference about the query.
The same principle applies to other layers of the PCH. For example, all distributions in C2 are realizable in principle, assuming the agent can freely intervene on all variables. However, practical constraints - such as cost, ethics, or technological barriers may render some interventions infeasible, thereby restricting the subset of L2 distributions that can be realized.
Given causal diagram and specification of the allowed actions, one can determine whether given set of distributions is realizable [99]. Viewed this way, the full collection of distributions in L2.5 be interpreted as the theoretical boundary of what is empirically accessible through physical experimentation.
Q6. How does the hierarchical structure defined over graphical models provide useful information on the models?
Answer. The hierarchical structure over graphical models offers a clear picture of the differences in the strength of assumptions encoded by each model. In causal inference specifically, the strength of the assumptions determines what queries the model may in principle support speccfsccclilll.. whether the causal inference engine can proceed and provide useful insights about the query. For instance, an L2 query P (z)) cannot be answered by a BN, which only encods C1 constraints that does not have the power bridge the gap between the two layers. This limitation is formally captured by the Causal Hierarchy Theorem (CHT), which states that to answer questions at one layer, one needs assumptions at the same layer or even higher. This understanding allows practitioners to select models from the hierarchy with sufficient inferential power for the query at hand.
On the other hand, the hierarchy also provides guidance in the opposite direction - helping identify when model might be stronger than necessary. For instance, while any model at or above CBN in the hierarchy can answer an L2 query P(y]do(=)) using model that makes counterfactual assumptions (e.g., a CBN2.5) would be unnecessarily strong and harder to falsify. Therefore, knowing the hierarchy of graphical models also allows practitioners to avoid choosing models that make extra assumptions not required in the target inferential task.
Putting these observations together, Table summarizes when a model sufficient and/or necessary for queries from each layer of the PCH. In short, the hierarchy serves as a practical guide for selecting models that are both sufficient and necessary maximizing inferential power while minimizing unfalsifiable assumptions.
Q7. What is the differenne between the hierarchical structure of languages and graphical models? Answer. The hierarchical structure of the languages (i.e., the PCH) defines how different families of distributions are related specifically, each layer's distributions form subset of those in the layer above. In parallel, the hierarchy of graphical models reflects how constraints on these distributions are encoded through the topological properties of the
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causal diagram. Each graphical model at layer encodes constraints over the corresponding family of distributions in layer of the PCH. Therefore, the hierarchy of the languages directly informs the hierarchy of graphical models.
However, since a graphical model is defined as a compatibility relationship between a pair (G, P). the expressiveness of the topological features in G also plays critical role. As we move up the hierarchy, the causal diagrams must support richer or more expressive interpretations of missing edges to capture the increasingly complex constraints required by higher-layer distributions. Both hierarchies are illustrated in Fig. [6] where square boxes depict the hierarchy over distributions, and round boxes represent the hierarchy over the constraints encoded by graphical models.
Q8. Why should data scientist care about the trade-off between expressive power and empirical falsifiability of the graphical models?
Answer. In any modeling task, it is generally desirable to construct model that accurately reflects the underlying generative process while also supporting future inferential tasks. Achieving stronger inferential power often requires incorporating stronger assumptions into the model. However, these assumptions can make the model more prone to errors that does not match with reality. Empirical Falsifiability acts as form of regularization, enabling the data scientist to identify, falsify and possibly correct wrong assumptions using empirical evidence. As a result, the model can yield more reliable and trustworthy causal conclusions. The importance of falsifiability echoes Karl Popper's philosophy, which argues that scientific theories must be testable and refutable setting science apart from pseudoscience [18]. Thus, understanding where each graphical model falls the spectrum of expressive power versus empirical falsifiability is essential for practitioners who align with Popper's 's principle.
Q9. What are the differences between local constraints and global constraints?
Answer: As discussed earlier when we introduce the inferential machinery for CBN2.25/CBN2.5, local constraints refer to those that are defined over distributions involv- ing a variable and its parents, and they are the constraints that are explicitly stated in the definitions of graphical models. For example, the local constraints in a BN are the conditional independencies of the form P(u:|pa, nd,) = [(v.] (r,),, where pa, denotes the parents and nd, the non non-descendants of V,, Given a BN over the chain diagram X -> Z Y, the local constraints include P(w] x) = P(w] z) and P(y| ur, 2, x) = P(ylw).
Global constraints, on the other hand, involve arbitrary subsets of variables, possibly far apart in the causal diagram. These constraints are not explicitly listed in the model's definition but can be derived by composing local constraints. For example, given the same BN over the chain above, a global constraint is P(yl x) P()(2), where the direct parent of Y. namely W, is no longer explicitly conditioned on.
This distinction highlights the role of local constraints as basis for implying the full set of global constraints that a graphical model implies, as illustrated in Fig. 10 This relationship is mirrored in the connection between graphical model and its associated inferential calculus: the calculus rules form the closure of all global constraints that logically follow from the local ones encoded in the model.
The process by which local constraints can be composed to yield global constraints was illustrated in Example 3 We revisit this idea with a new example Fig. IO Consider a CBN over the chain diagram X + Z + Y. The local constraints specified in the definition of the CBN are depicted as connecting lines between nodes within the small yellow circle. These local constraints can imply additional constraints not explicitly listed in the definition. One such global constraint is Ply P(yldo(ar) = Pyylar), represented by the red connection line in the figure. This global constraint can be derived by composing "gluing" sequence
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of local invariance constraints, shown as blue connection lines.
P(yldo(z)) = Σ P(yldo(a)),)(((((d((()) (Probabilityy Axiom)
(161)
Σ P(()do((z)))
(Cond. (iii) of Def. 19
(162)
Σ P(y)do(2)) (Cond. (ii) of Def. 19
(163)
Σ P(ul=))(zla))
(Cond. (iii) of Def. 19
(164)
Σ P(2)))))()1x)
(Cond. (1) Def. 19
(165)
a)
(Probability Axiom)
(166)
In summary, although all constraints are explicitly included in the local basis of graph- ical model definition, many are implied through its structure. Since the 1980s, this ability to encode parsimonious, polynomial-sized set of local constraints that implicitly represent an exponential number of global constraints has been an attractive feature contributing to the popularity and usefulness of graphical models in inferential tasks.
Q10. What is the connection between realizability and empirical falsifiability?
Answer. Realizability is a property of distributions, indicating that an agent can draw samples from them through physical experimentation. For example, if an agent can intervene on a variable X and fix P(v do(z)) in layer L2.
value x, it gains access the interventional distribution
In the context of graphical models, empirical falsifiability is property of constraints over empirically falsify a constraint, the agent must have the experimental
these distributions. capabilities draw samples from all distributions involved in the constraint. In other words, the constraint's 's falsifiability requires the realizability of the associated distributions. For instance, testing the constraint Pyy do(x, =)) Ply do(z)) requires the ability to sample from both Pyy do(z, z)) and Pyy do(x)). Whether this feasible depends on the experimental capabilities and limitations of the system in question.
QII. What is the difference between an SCM and Layer 3 distributions or Layer 3 graphical models?
Answer: An SCM is more granular level model with details about the exogenous variables U, which induces the full set of distributions over the endogenous variables V in the PCH, as illustrated in Fig. and Fig. S Specifically, given a distribution over the exogenous variables P(u) and the structural equations that determine each endogenous variable function of its parents (both endogenous and exogenous), we can compute all distributions over the endogenous variables following the formula in the PCH definition (Def. TT). In contrast, the PCH abstracts away from the exogenous variables, treating them as hidden background factors unobserved by the agent. As result, Layer 3 distributions and its corresponding graphical models are defined solely in terms of the endogenous variables.
Given an SCM, it is also possible to evaluate individual level effects when the exogenous state of a specific unit u E Val(U) is known, by solving the set of mechanisms following the topological order of evaluation. Layer 3 distributions and graphical models, on the other hand, offer population-level descriptions of causal relationships, without access to individual-level information.
In nutshell, an SCM provides full access to Layer 3 distributions and graphical models, as it encodes the necessary generative mechanisms. However, the reverse does not hold: Layer 3 distributions graphical models do not determine unique SCM, since an SCM requires additional, often unobservable, information about the exogenous variables and structural mechanisms.
Q12. Given that the constructive procedure for the causal diagram is the same, why do we need, or even have, different layers of graphical models?
Answer. Even though the same causal diagram G shared across many different models, the compatibility relationships it represents differ depending the model. As discussed
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earlier, graphical model is a pair (G. P), where graphical feature in G are interpreted to represent constraints P. As expands to include distributions from higher layers of the PCH, the set of constraints that the graph must represent also becomes richer. As a result, each missing edge is required to encode stronger and more expressive constraints over broader class of distributions. This is illustrated in Example and Table 4
48


Reviews:
[{"review_id": "0AtWGCw9Vm", "reviewer": "Reviewer_XJnG", "summary": "The paper proposes two novel types of causal queries, labeled $L_{2.25}$ and $L_{2.5}$, that lie between the interventional ($L_2$) and counterfactual ($L_{3}$) layers. Both can be realized using the method of counterfactual randomization. To represent the distributions associated with these intermediate layers, the paper proposes two graphical models based on CBNs with additional constraints. An inference procedure based on the ctf-calculus [Correa & Bareinboim] is developed to identify these queries, and is proven to be both sound and complete. Finally, the paper offers a detailed discussion on the expressivity and falsifiability of different layers within the causal hierarchy.", "questions": "1. Definition 1: is the choice of $\\mathbf{x}_i$ unique? It'll be great to clarify it here.\n2. If I understood correctly, the definitions imply that queries in $L_{2.25}$ disallow conflicting values of treatments and outcomes, and queries in $L_{2.5}$ disallow conflicting values in the outcomes?\n3. I wasn't able to find the reference for counterfactual BN (CTFBN)? Is CTFBN the same as AMWN?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. The paper is well-structured and clearly written overall.\n2. The two new layers in the causal hierarchy are quite novel and theoretically meaningful. Additionally, the paper includes a comprehensive appendix with detailed background and proofs.\n3. I found the examples very effective in aiding readers' understanding of these abstract ideas. Furthermore, the discussion in Section 3 (on expressivity and falsifiability) offers valuable insights from both theoretical and practical standpoints.\n\nWeaknesses:\n1. I believe more real-world examples for Layers 2.25 and 2.5 would help demonstrate their importance. I understand their theoretical significance, but how are they practically meaningful (comparable to L2 and L3)? I know additional details are given in the appendix, but it’s still worth emphasizing the importance (ubiquity) of counterfactual randomization in the main paper.\n2. Theorem 1: Def 10 in the supplementary material does not describe a constructive process. Did you mean Definition 11 here?\n3. Lemma 1: \"set CRS(W*) satisfies (i) and (ii) / An(W*) satisfies (I)\" — I'm confused about this part. What does \"(ii) / An(W*)\" mean here? \n\nMinor typos:\n\n5. footnote 2: \"B\" -> \"Appendix B\"\n6. line 200: \"constrains\" -> \"constraints\"", "was_revised": true, "final_justification": "The paper is technically rigorous and contains important results.", "rebuttal": "We appreciate your time and positive assessment of our work. We provide answers to your questions below. \n\n>*Weakness 1*: I believe more real-world examples for Layers 2.25 and 2.5 would help demonstrate their importance. I understand their theoretical significance, but how are they practically meaningful (comparable to L2 and L3)? I know additional details are given in the appendix, but it’s still worth emphasizing the importance (ubiquity) of counterfactual randomization in the main paper.\n\nYes, we agree on the importance of emphasizing the significance of the counterfactual randomization action and the practical implications of the two layers of distributions that are derived from this action. In both the introduction and Section 3, we highlighted that the counterfactual randomization action “characterizes the set of  $\\mathcal{L}_3$ distributions that are realizable in principle”. To connect with real-world applications, we included Example 5 (ETT) and Example 7 (NDE), which show that several quantities of significant practical interest across the sciences lie within the two new layers. Due to space constraints, more explanations and examples are currently placed in Appendix A and B. That said, we will consider moving more of the intuitive explanations and visual aids (e.g., Figure 10) from the appendix into the main text given the additional one page allowed. We also welcome any suggestions on specific examples or explanations that can make the importance of the layers clearer.\n\n>*Weakness 2*: Theorem 1: Def 10 in the supplementary material does not describe a constructive process. Did you mean Definition 11 here?\n\nYes, it is Definition 11. We apologize for the confusion since the numbering is not fully consistent. In the version submitted with the supplementary material, the numbers are consistent across the main body and appendix.\n\n>*Weakness 3*: Lemma 1: \"set CRS(W*) satisfies (i) and (ii) / An(W*) satisfies (I)\" — I'm confused about this part. What does \"(ii) / An(W*)\" mean here?\n\nIn this lemma, we use the slash symbol “/” to distinguish conditions that apply to the two different layers. Specifically, conditions to the left of “/” apply to $\\mathcal{L}\\_{2.25}$, and those to the right apply to $\\mathcal{L}\\_{2.5}$. In other words, a distribution belongs to $\\mathcal{L}\\_{2.25}$ if “$CRS(\\mathbf{W}_\\*)$ satisfies (i) and (ii)”, whereas it belongs to L2.5 if “$An(\\mathbf{W}\\_*)$ satisfies (i)”. We did a bit of gymnastics to make things fit in the allotted space, but will clarify this notation in the revised version to make it more immediately interpretable. Thank you for the observation.  \n\n>*Question 1*: Definition 1: is the choice of x_i unique? It'll be great to clarify it here.\nWe clarify that the choice of $\\mathbf{x}_i$ is not unique.\n\nNo, the choice of $\\mathbf{x}_i$ is not unique. There can be multiple valid assignments as long as they satisfy the two conditions in Definition 1, i.e., \n\"(i) $\\mathbf{x}_i\\subseteq \\mathbf{x}$ and $\\cup_i \\mathbf{x}_i =\\mathbf{x}$; and \n(ii) For any $v_i\\in \\mathbf{x}$, for all $V_j\\in\\mathbf{Y}$, if $V_i\\in An(V_j)$ in $\\mathcal{M}\\_{\\mathbf{x}\\backslash V_j}$, then $v_i\\in \\mathbf{x}_j$.\"\nThis is illustrated in Example 17 in Appendix B.2. For example, a causal chain $Z\\rightarrow X\\rightarrow Y$, the $\\mathcal{L}\\_{2.25}$ distributions indexed by the interventional value set $y$ include $P (Z_y, X, Y ), P (Z, X_y , Y )$, and $P (Z_y , X_y , Y )$. Although the subscripts on $Z$ and $X$ differ across these distributions, they all satisfy the required conditions. \n\n>*Question 2*: If I understood correctly, the definitions imply that queries in L2.25 disallow conflicting values of treatments and outcomes, and queries in L2.5 disallow conflicting values in the outcomes?\n\nYes, your intuition on ‘disallowing conflicting values of treatment’ is correct. As elaborated in Section 2.1, the distributions in $\\mathcal{L}\\_{2.25}$ come from fixing “a single value of $X$ across all its children”, while the distributions in $\\mathcal{L}\\_{2.5}$ allow “each children of $X$ to take a potentially different value”. \n\nTherefore, it is correct that queries in $\\mathcal{L}\\_{2.25}$ disallow both conflicting values of the treatment variable itself and conflicting subscripts across the outcomes. This is the reason that distributions in $\\mathcal{L}\\_{2.25}$ are indexed by a fixed “interventional value set” to ensure consistent values over the treatments and subscripts.\n\nIn contrast, queries in $\\mathcal{L}\\_{2.5}$ may involve different value assignments for the same treatment variable across different children, which in turn may induce different subscripts on downstream outcomes. As such, $\\mathcal{L}\\_{2.5}$ distributions are indexed by a more relaxed “interventional value set”, allowing variation in the treatment assignments, as long as they satisfy the two conditions in Definition 2.\n\nThis distinction is illustrated in Example 1 and further visualized in Figure 10 of Appendix A.4:\nFigure 10(b) corresponds to $\\mathcal{L}\\_{2.25}$, where all outgoing edges of the treatment variable $X$ receive the same value $x$, yielding the distribution $P(X, Y_x, Z_x)$.\nFigure 10(c) corresponds to $\\mathcal{L}\\_{2.5}$, where different values $x$ and $x'$ are assigned along different outgoing edges, yielding the distribution $P(X, Y_x, Z\\_{x'})$.\n\nFinally, if by “conflicting values in the outcomes” you are referring to the same outcome variable appearing more than once with different subscripts (e.g., $P(Y_x, Y\\_{x'})$), then yes, you are correct that neither layer allows for it. Following Definitions 1 and 2, each variable $V_i$ appears at most once in the symbolic representation of a distribution in these two layers.\n\n>*Question 3*: I wasn't able to find the reference for counterfactual BN (CTFBN)? Is CTFBN the same as AMWN?\n\nThank you, we updated the reference to the correct one. The definition and examples of CTFBN can also be found in Appendix A.4 of the submitted paper. A CTFBN is not the same as AMWN, but rather it is a graphical model, similar in nature to CBN, CBN2.25, and CBN2.5. Specifically, a CTFBN encodes a collection of local constraints over distributions in $\\mathcal{L}_3$ of the PCH. In contrast, AMWN is an algorithm that takes a CTFBN as input and checks whether a global counterfactual independence is implied by the local constraints specified in the CTFBN definition. One analogy to this is the relationship between the causal diagram and Pearl’s celebrated twin network, where the former encodes the local constraints, and the latter is a data structure that allows for the evaluation of global constraints implied by the local ones.   \n\n>*Minor typos*: footnote 2: \"B\" -> \"Appendix B\"; line 200: \"constrains\" -> \"constraints\"\n\nThanks, we will correct these typos in the revised version of the paper."}, {"review_id": "CzmWsPMUJi", "reviewer": "Reviewer_u4aR", "summary": "This is a theoretical paper on the lack of falsifiability in the third, counterfactual layer of Pearl's hierarchy. Two new classes of models to express counterfactual quantities that are experimentally accessible are presented, along with a sound and complete calculus (mostly based on Bareinboim's CTF calculus).", "questions": "As mentioned, the supplementary material contains a long (~40 pages) version of the paper, which is very clear and well-organised. The same cannot be said for the 10-page version, which is obviously more dense, but also less precise in its organisation of the contents. A natural question is whether the authors believe they could make the conference version of their work entirely self-consistent and clearer than what appears to be now.\n\nExample 7 is excellent to advocate the importance of the \"2.5\" layer. Is it possible to do something similar for \"2.25\"?\n\nDoes the calculus derived here make the standard CTF calculus useless?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 4, "originality": 4, "strengths_and_weaknesses": "[+] This paper might be very influential, as it extends the classical Pearlian hierarchy using the key concept of experimental falsifiability\n[+] The counterfactuals considered by this paper include some relevant queries for applications (ex. NDE, which is important for fairness analysis)\n[-] The paper is a shrunk version of a much longer paper (included in the supp-mat). The shrinking process was a bit hurried, which made the clarity of the short version of the paper somewhat poor.", "was_revised": true, "final_justification": "After reading the other reviews and the (satisfactory) rebuttals I confirmed my positive opinion about the paper.", "rebuttal": "Thank you for taking the time to review our work and for your positive assessment. We respond to your questions below. \n\n>*Weakness*: [-] The paper is a shrunk version of a much longer paper (included in the supp-mat). The shrinking process was a bit hurried, which made the clarity of the short version of the paper somewhat poor.\n\n>*Question 1*: As mentioned, the supplementary material contains a long (~40 pages) version of the paper, which is very clear and well-organised. The same cannot be said for the 10-page version, which is obviously more dense, but also less precise in its organisation of the contents. A natural question is whether the authors believe they could make the conference version of their work entirely self-consistent and clearer than what appears to be now.\n\nIndeed, the compression step made the paper a bit dense, even though our goal was to preserve the core technical contributions and the more interesting results in the main body. To answer your question, yes, with the additional page allowed in the camera-ready version, we will revise the current version by incorporating comments from reviewers to improve consistency, structure, and clarity, while maintaining the technical depth of the work. \nWe also welcome specific suggestions regarding content reordering or additional examples that could improve clarity and flow.\n\n>*Question 2*: Example 7 is excellent to advocate the importance of the \"2.5\" layer. Is it possible to do something similar for \"2.25\"?\n\nYes, Example 5 highlights the importance of Layer 2.25. It shows that the effect of treatment on the treated (ETT), $P(Y_x,X)$, a quantity of significant practical interest across the sciences, belongs to $\\mathcal{L}\\_{2.25}$. Moreover, it is identifiable from the observational distribution $P(\\mathbf{V})$ given the causal assumptions encoded in the CBN2.25 associated with the causal diagram in Figure 4(a).\n\n>*Question 3*: Does the calculus derived here make the standard CTF calculus useless?\n\nNo, the calculus for CBN2.25 and CBN2.5 does not make the calculus for CTFBN useless. As discussed in section 2.3 and FAQ Question 9, the constraints listed in the definitions of CBN2.25 and CBN2.5 are local, namely, they involve counterfactual variables with their parents in the subscripts. The corresponding calculus can be viewed as an algorithm to derive all global constraints that are implied by the local constraints listed in the definitions. The same idea applies to CTFBN and its calculus as well.\n\nSince CTFBN encodes a larger set of local constraints than CBN2.25 and CBN2.5, its calculus also allows for the derivation of more global constraints that are implied by these additional assumptions. For example, consider the simple causal chain $X\\rightarrow Z \\rightarrow Y$. The distribution $P(Y\\_x, Y\\_{x’},X)$ lies beyond layer 2.5 because it joins two potential outcomes over $Y$ with different subscripts. As a result, CBN2.25 and CBN2.5 and their corresponding calculus cannot make any statements about this distribution. In contrast, by leveraging the richer local assumptions encoded in CTFBN, the ctf-calculus can derive that $P(Y\\_x, Y\\_{x’},X)=P(Y\\_x, Y\\_{x’})P(X)$.\n\nIn summary, ctf-calculus remains a valuable tool for deriving global constraints implied by stronger assumptions that lie beyond Layer 2.5."}, {"review_id": "aGwlEDXpO7", "reviewer": "Reviewer_mqbs", "summary": "The paper introduces two new classes of causal graphical models, CBN2.25 and CBN2.5, which encode constraints for distributions made experimentally accessible via counterfactual randomization. These models bridge the gap between interventional (L2) and counterfactual (L3) layers in Pearl's Causal Hierarchy (PCH). The authors demonstrate a novel hierarchy of graphical models where higher-level models offer greater expressive power at the cost of reduced empirical falsifiability. They provide theoretical guarantees for their calculus, proving soundness and completeness for inference within these new frameworks.", "questions": "1. How do the computational costs of ctf-calculus in CBN2.25/CBN2.5 compare to Pearl’s do-calculus or ctf-calculus in CTFBN? Could the additional constraints impact scalability?\n2. What concrete applications have been tested with these models, and how do their results compare to prior methods like CTFBN? For instance, in estimating ETT (Example 5), are there cases where CBN2.25/CBN2.5 fail due to insufficient assumptions?\n3. The authors discuss falsifiability but do not address potential societal risks (like biases in counterfactual reasoning). Are there scenarios where CBN2.25/CBN2.5 could amplify ethical concerns compared to prior models?", "limitations": "The authors adequately address limitations by explicitly analyzing trade-offs between model expressiveness and falsifiability (Section 3).", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## Quality\n### Strength\n1. The paper is technically robust. Definitions of distributions (Def 1/2), graphical constraints (Def 3/4), and theorems linking models to SCMs (Thm 1) are precise. Lemma 1 (CRS criterion) provides a critical tool for verifying distribution membership in PCH layers, ensuring theoretical validity.\n2. The authors prove that ctf-calculus is sound and complete for CBN2.25/CBN2.5 (Thm 2), which is foundational for their utility in causal inference. This strengthens the practical relevance of the proposed machinery.\n3. The refinement of L3 into experimentally accessible sublayers (L2.25/L2.5) and formalization of a hierarchy connecting BN, CBN, and CTFBN is conceptually original. Thm 4 rigorously establishes how constraints propagate across layers.\n### Weaknesses:  \n1. The paper does not provide empirical validation or experiments to demonstrate the practical utility of these models. While theoretical completeness is ensured, real-world performance remains unexplored.\n\n---\n## Clarity\n### Strength\n1. The paper follows a logical flow: problem motivation -> formalisms -> hierarchical analysis. Figures 3–6 and tables like Table 1 aid conceptual understanding.\n2. The paper systematically maps new models into PCH (Thm 3) and establishes their hierarchical relationships via constraint nesting (Thm 4). This enhances conceptual clarity for model selection in causal inference tasks.\n### Weaknesses:  \n1. Dense mathematical notation in Def 1/2 and equations for distributions. For example, the distinction between P^L_{2.25} and P^L_{2.5} could benefit from clearer intuitive explanations alongside formal definitions.\n\n---\n## Significance\n### Strength\n1. The work advances the understanding of counterfactual inference by bridging theoretical limitations (non-experimental L3) with empirical feasibility via counterfactual randomization. This is critical for AI systems requiring robust causality foundations, particularly in fields like healthcare or policy analysis.\n2. The hierarchy framework provides a principled tool for model selection: researchers can choose models that balance expressive power and falsifiability per task.\n### Weaknesses:  \n1. The paper focuses almost exclusively on theoretical contributions without illustrating their application to real-world problems. For instance, it lacks examples of how these models improve over existing methods, like CTFBN, in scenarios like estimating ETT or NDE.\n\n---\n## Originality\n### Strength\n1. The formalization of L2.25/L2.5 as refinements of PCH’s L3 is novel, especially the introduction of AMWN for inferential machinery and the CRS criterion (Lemma 1).\n2. The paper clearly distinguishes its work from prior models and situates them within an expanded PCH framework (Thm 3/4). This contextualizes the contribution without unnecessary redundancy.", "was_revised": true, "final_justification": "See the response to the authors.", "rebuttal": "We appreciate your positive assessment and valuable feedback. Please find our detailed responses to your questions and comments on weaknesses below.\n\n >*Quality*: The paper does not provide empirical validation or experiments to demonstrate the practical utility of these models. While theoretical completeness is ensured, real-world performance remains unexplored.\n\nWe acknowledge that the primary focus of this paper is on establishing the proper foundations for a hierarchy of graphical models in causal reasoning. We believe the foundations are important so that the applications can flourish. Additionally, no paper can do everything (i.e., solve all problems), and we hope the reviewer can appreciate this point. Although the paper is theoretically profound and very mathematical, we still attempted to include as many illustrative examples as possible to highlight the basic concepts and how the various moving parts connect, as well as to help the community start visualizing connections with real-world applications. For instance, Examples 5 and 7 demonstrate how the effect of treatment on the treated (ETT) and the natural direct effect (NDE) can be identified using the constraints encoded in CBN2.25 and CBN2.5, respectively. We agree that a broader empirical evaluation is a crucial future step, and we look forward to continuing the empirical work. \n\n>*Clarity*: Dense mathematical notation in Def 1/2 and equations for distributions. For example, the distinction between P^L_{2.25} and P^L_{2.5} could benefit from clearer intuitive explanations alongside formal definitions.\n\nWe acknowledge that the notation in Definitions 1 and 2 is somewhat dense, and we aimed to provide as much intuitive explanation as possible to support the reader's understanding. For example, immediately following Definition 2, we highlight that “the key difference between $\\mathcal{L}\\_{2.25}$ and $\\mathcal{L}\\_{2.5}$ lies in how the distributions are indexed: $\\mathcal{L}\\_{2.25}$ are indexed by specific interventional value sets $\\mathbf{x} \\in Val(\\mathbf{X})$, while $\\mathcal{L}\\_{2.5}$ are indexed by interventional variable sets $\\mathbf{X}$. The more refined index for $\\mathcal{L}\\_{2.25}$ creates more restrictions on the expressiveness for its distributions, which is also reflected in the differences between conditions of the two definitions.” This distinction is further illustrated in Example 1, where distributions from the two layers are compared based on a specific SCM. \n\nHowever, due to space constraints, additional explanations and examples were included in Appendix B.2. In particular, we believe that Figure 10 offers an intuitive visual comparison of Layers 2, 2.25, and 2.5. Given the additional one page allowed and if the reviewers deem it suitable, we would be happy to relocate this figure to the main text in the revised version to enhance clarity.\n\n>*Significance*: The paper focuses almost exclusively on theoretical contributions without illustrating their application to real-world problems. For instance, it lacks examples of how these models improve over existing methods, like CTFBN, in scenarios like estimating ETT or NDE.\n\nAs explained earlier, we agree that the paper is primarily theoretically positioned. That said, we did include concrete examples, Example 5 (ETT) and Example 7 (NDE), to illustrate that the constraints encoded in CBN2.25 and CBN2.5 are sufficient to identify ETT and NDE, respectively. This demonstrates that some of the constraints encoded by CTFBN are, in fact, not necessary for identifying queries that lie in $\\mathcal{L}\\_{2.25}$ and $\\mathcal{L}\\_{2.5}$. \n\nThe same idea is also illustrated in Table 1 (reproduced below; see also Table 4, for an even more detailed view), where we show that to identify an $\\mathcal{L}\\_{2.5}$ query, constraints encoded in CBN are not sufficient, while those encoded in CTFBN are sufficient but not necessary. In contrast, CBN2.5 provides precisely the right set of constraints, which are both necessary and sufficient. These examples clarify how CBN2.25 and CBN2.5 refine the space of graphical models by precisely aligning the level of constraints with the layer of the query.\n\n| Q Layer | GM           | Suff.                        |Nec.      |\n|----------|---------------|----------|--------------|\n| $\\mathcal{L}\\_{2.5}$      | CBN | x   | $\\checkmark$ |\n| $\\mathcal{L}\\_{2.5}$    | CBN2.5 | $\\checkmark$   | $\\checkmark$ |\n| $\\mathcal{L}\\_{2.5}$   | CTFBN | $\\checkmark$  |  x  |\n\n>*Question 1*: How do the computational costs of ctf-calculus in CBN2.25/CBN2.5 compare to Pearl’s do-calculus or ctf-calculus in CTFBN? Could the additional constraints impact scalability?\n\nBased on the number of encoded constraints in the models, the computational cost of ctf-calculus in CBN2.25 and CBN2.5 is expected to lie between that of do-calculus in CBN and ctf-calculus in CTFBN. However, as with do-calculus, exhaustively searching for all possible derivations based on the rules is known to be computationally hard in general. \n\nFor both CBN and CTFBN, prior work has developed algorithmic identification approaches based on tree structures that significantly improve scalability and efficiency by solving the problem in a more systematic way (for details, please see Section 4.4 and 5.4 in [Bareinboim, 2025]. Although such methods are not yet fully explored for CBN2.25 and CBN2.5, we believe it may be possible to apply similar approaches to these two models as well. This is a natural extension of the foundational work presented in this paper, and we plan to explore these computational aspects further in future work.\n\n>*Question 2*: What concrete applications have been tested with these models, and how do their results compare to prior methods like CTFBN? For instance, in estimating ETT (Example 5), are there cases where CBN2.25/CBN2.5 fail due to insufficient assumptions?\n\nAs this paper is focused on establishing the theoretical foundations of the proposed hierarchy of graphical models, we have not yet conducted systematic empirical evaluations based on real-world datasets. The examples included are primarily designed to provide a clear picture of the relative positions of the models in the hierarchy, so they do offer information about comparisons between the two models and the existing models, CBN and CTFBN. \n\nOn the more substantive aspect regarding Example 5, we confirm that there are cases where the ETT is not identifiable. For instance, when there is an unobserved confounder between $X$ and $Y$, the counterfactual independence constraint $Y_x\\perp\\\\!\\\\!\\\\!\\perp X$ no longer holds. As a result, $P(Y_x,X)$ will not be identifiable from the observational distribution $P(\\mathbf{V})$. However, the key point is that ETT lies in $\\mathcal{L}\\_{2.25}$, and its identifiability can be determined by a CBN2.25. In other words, the constraints encoded in CBN2.25 are sufficient for deciding if ETT is identifiable or not, and there is no need to assume additional constraints from higher layers, $\\mathcal{L}\\_{2.5}$ and $\\mathcal{L}\\_{3}$, as encoded in CBN2.5 and CTFBN.\n\n>*Question 3*: The authors discuss falsifiability but do not address potential societal risks (like biases in counterfactual reasoning). Are there scenarios where CBN2.25/CBN2.5 could amplify ethical concerns compared to prior models?\n\nThank you for raising this important issue. We agree that the ethical implications of causal modelling, particularly those involving counterfactuals, deserve careful evaluation. Specifically, in situations where the assumptions are not well-justified, there could be potential risks of encoding or amplifying biases. \n\nThat said, we believe that the newly proposed framework can actually aid in mitigating such risks by making assumptions more transparent and explicit. The models we introduce offer a systematic approach to classifying assumptions based on their empirical falsifiability. This transparency allows practitioners to more rigorously assess which assumptions are warranted in a given application, whether via data, expert knowledge, or domain-specific validation.\n\nStill, we recognize that ethical and societal implications must be addressed more fully in follow-up work, especially when applying these models in sensitive domains, such as,  healthcare and criminal justice. We will include a brief discussion of these concerns in the revised version to highlight the potential risks and the responsibility that comes with using these models involving counterfactual assumptions. Thank you again for sharing this thought. \n\n*References*:\n- [Bareinboim, 2025] Elias Bareinboim. Causal Artificial Intelligence: A Roadmap for Building Causally Intelligent Systems, 2025."}, {"review_id": "bfbWQXCpR6", "reviewer": "Reviewer_pvuV", "summary": "PCH consists of three layers, observational, interventional, and counterfactual. When the underlying SCM is fully known, then we can compute any query to the model. However, in reality, the model might not be fully known. At the interventional level, it is possible, at least in principle, to obtain full knowledge by doing experiments. This it not true for the counterfactual level. in [19], the concept of counterfactual realizability was introduced, which formalizes the concept of being able to sample from a counterfactual distribution by doing experiments. Not all counterfactual distributions are realizable. In the present paper, the authors formalize two subclasses of the counterfactual layer of PCH by restricting the kind of counterfactual worlds that can be present. These subclasses are realizable in the sense of [19]. \n\nThe main results are the following:\n1) The proof that two corresponding classes of causal Bayesian networks capture the distributions defined by the new layers (Thm 1).\n2) The adaption of the counterfactual calculus to the subclasses and the proof that a quantity is identifiable if it can be expressed in terms of the given distribution using the rules (Thm 2). \n3) The proof that the distributions defined in terms of SCMs of the different layers form a hierarchy (Thm 3) as well as the constraints encoded in the corresponding graphical models (Thm 4).", "questions": "No particular question, but I am thankful for any comments regarding the mentioned weaknesses.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper makes interesting new theoretical contributions by defining sublayers of the counterfactual layer of PCH which might be easier to falsify by experiments. These layers might be easier to understand, since the way counterfactual worlds can be used is restricted.  \n\nWeaknesses:\n- On the technical side, I think the contributions are rather incremental.\n  - The proof of Theorem 1 is just a check of the properties, which is similar to proofs for the standard layers. Which is to some extend to be expected, as the construction procedures for the CBNs are the same.\n  - In the proof of Theorem 2, soundness is obvious and completeness follows by the same argument as for layer 3, even using the same algorithm.\n  - For the proof of Theorem 3, one only has to check that layer 2.25 is subsumed by layer 2.5, which is quite obvious since Definition 2 generalizes Definition 1, as is the fact that it transfers to the corresponding CBN.\n- One of the selling points of the paper is that the new layers allow empirical falsifiability. The example in the paper are of mathematical/explanatory nature, which is fine. However, the examples one finds in other papers (like changing the color of cars in images) are a little contrived.\n\nComments:\n\n- I think the naming of the new layers is not good. The numbers are arbitrary. Why not 2.33 and 2.66? Furthermore, if someone next introduces a layer between interventional and counterfactual that is not comparable to your new layers, you created a mess, since real numbers have a total order. That being said, already numbering the layers of PCH might have been a mistake, since there is no natural numerical parameter.\n- The paper is not easy to read (which to some part cannot be avoided, since counterfactual reasoning is complicated). Section 2 is pretty dense, while I find section 3 rather lengthy. While Appendix A contains lots of information, I find it to lengthy to be a good reference manual but contains to few explanations to be a good introduction. I do not think that the [..]-notation is explained anywhere, but I might have overlooked it.\n\nMinor comments:\n\n- line 110: V -> U\n- line 113: Val -> \\mathit{Val} (or whatever you prefer)\n- line 118: \\in -> \\subseteq\n- line 158: maybe use x and x'. The indices do not correspond to the indices used in Definition 1\n- page 5, footnote: An \"Appendix\" is missing (most likely a c in \\cref)\n- line 222: Definition 11 only appears in the appendix\n- Eq (35): fi -> f_i\n- Definition 12: The second sentence is not a proper mathematical definition\n- It might be helpfull in Def. 18 to discuss the relation to Def. 13\n- Definition 24 is also not a proper mathematical definition. What is a randomization device is this context?\n- line 693: a blank is missing\n- Lemma F.1 seems to be folklore to me.\n- line 1018 2.25 -> 2.5\n- line 1247: looks like some latex issue", "was_revised": true, "final_justification": "I have raised my score by 1 taking the discussions with the authors into account.\n\nThe paper introduces two new layers in PCH between interventional and counterfactual reasoning. Per se, this is a valuable contribution. However, the paper still has a few shortcomings, but I think the new conceptual contributions outweigh these shortcomings. \n\nMy main points of critic are:\n- shortcomings in the presentation, which will be hopefully improved\n- some of the proofs are of incremental nature (which to some extend cannot be helped)\n- relevance of the new introduced layer (which time has to tell)", "rebuttal": "We thank the reviewer for the time and effort in reviewing our paper. While we appreciate the thoughtful feedback, we believe a few misunderstandings may have contributed to the evaluation being a bit harsh. We respectfully ask the reviewer to reconsider our paper based on the clarifications below. \n\n>*Weakness 1*: On the technical side, I think the contributions are rather incremental.\nThe proof of Theorem 1 is just a check of the properties, which is similar to proofs for the standard layers. Which is to some extent to be expected, as the construction procedures for the CBNs are the same.\nIn the proof of Theorem 2, soundness is obvious and completeness follows by the same argument as for layer 3, even using the same algorithm.\nFor the proof of Theorem 3, one only has to check that layer 2.25 is subsumed by layer 2.5, which is quite obvious since Definition 2 generalizes Definition 1, as is the fact that it transfers to the corresponding CBN.\n\nThank you for your careful reading of the proofs and detailed comments. While we agree the proof techniques share structural similarities with CTFBNs, we respectfully disagree that the contributions are merely incremental. The resemblance arises naturally because layers 2.25 and 2.5 also concern counterfactuals. Yet, the novelty of the work does not lie solely in the complexity of the proofs, but more in the implications of highlighting the heterogeneity of the counterfactual space, a space that has remained largely underexplored at least for the past three decades, since Pearl’s groundbreaking 1995 Biometrika paper that started the field in the context of AI.\n\nAs discussed in the introduction, the formal definitions of the intermediate two layers provide a foundation for “a finer-grained understanding of the region between $\\mathcal{L}_2$ and $\\mathcal{L}_3$ of the PCH”, and “a precise formalization of the empirical heterogeneity of distributions between” these two layers. The corresponding graphical models inherit the empirical properties in their assumptions, which equip data scientists with a systematic way to evaluate and justify assumptions based on empirical evidence, expert knowledge, or domain-specific reasoning.\n\nTo illustrate the practical value of these layers, we provide examples throughout the main text. In particular, we illustrate in Examples 5 (ETT) and 7 (NDE) that important causal quantities across disciplines lie within the two new layers and can be identified by the corresponding graphical models. These examples show that our framework is not only theoretically motivated but also practically meaningful in real-world applications. A significant amount of discussion in the literature follows precisely from the lack of these clearly delineated spaces that we circumscribe in our work.\n\nWe believe that recognizing the two intermediate layers refines our understanding of the causal hierarchy and opens up new directions for working with counterfactual assumptions in a more granular and transparent way. We will clarify this perspective more explicitly in the revised version and appreciate your feedback for prompting it.\n\n>*Weakness 2*: One of the selling points of the paper is that the new layers allow empirical falsifiability. The examples in the paper are of mathematical/explanatory nature, which is fine. However, the examples one finds in other papers (like changing the color of cars in images) are a little contrived.\n\nThanks for the positive recognition of the examples in our paper. \n\nRegarding the “changing the color of cars in images” example, we believe you are referring to Example 1 in [Raghavan & Bareinboim, 2025], since it is not in our paper. Based on our understanding, that example is designed for illustrative purposes, showing how counterfactual randomization can be implemented in real-world contexts via a counterfactual mediator. \n\nWe agree that having natural and intuitive examples is important for conveying the practical value of the new layers. The empirical accessibility of Layers 2.25 and 2.5 can also be illustrated through many widely recognized real-world studies. One example is the field experiment on labor market discrimination by [Bertrand & Mullainathan,2004]. In this study, the authors experimentally manipulated perception of race by changing the names of fictitious job applicants to study racial discrimination in hiring. With access to both the original names and the hiring outcomes based on altered names, the resulting data corresponds to a counterfactual distribution in Layer 2.25. As a result, assumptions made over this distribution are also empirically testable. Another example showing empirical access to NDE is the gender bias study in language models by [Vig et al., 2020].\n\nExamples like these highlight how counterfactual randomization is feasible in real-world studies and support the practical relevance of the two new layers. They also demonstrate that the empirical falsifiability of counterfactual assumptions, like those in CBN2.25/2.5, is not only a theoretical construct but a property that can be exploited in practice.\n\nWe recognize that examples in our paper like Example 5(ETT) and Example 7(NDE) are primarily explanatory, without details on specific application domains. However, the causal quantities in these examples are common across scientific fields, and our results directly apply whenever they are of interest. We will add discussion in the revised version to clarify the connection between our theoretical results and real-world applications, and citations to empirical studies where such quantities are evaluated. We also plan to explore broader empirical evaluations in future work. Thanks for highlighting this important point.\n\n>*Comment 1*: I think the naming of the new layers is not good. The numbers are arbitrary. Why not 2.33 and 2.66? Furthermore, if someone next introduces a layer between interventional and counterfactual that is not comparable to your new layers, you created a mess, since real numbers have a total order. That being said, already numbering the layers of PCH might have been a mistake, since there is no natural numerical parameter.\n\nRight, the numbers may be a bit arbitrary, there is indeed no natural numerical parameter. The current naming arose organically: we introduced Layer 2.5 first as a natural midpoint between Layers 2 and 3, and then named Layer 2.25 to reflect its position between Layers 2 and 2.5. While the absolute values are not meaningful, the relative ordering reflects the intended hierarchy among the layers. That said, we are happy to consider alternatives and suggestions from the reviewers, maybe more interpretable names for the two new layers (perhaps following a similar approach as the other three layers of the PCH: observational, interventional, and counterfactual.) \n\n>*Comment 2*: The paper is not easy to read (which to some part cannot be avoided, since counterfactual reasoning is complicated). Section 2 is pretty dense, while I find section 3 rather lengthy. While Appendix A contains lots of information, I find it too lengthy to be a good reference manual but contains too few explanations to be a good introduction. I do not think that the [..]-notation is explained anywhere, but I might have overlooked it.\n\nWe acknowledge that some sections are a bit dense, largely due to the need to compress content within the page limit while preserving the technical contributions. Section 2 is a bit dense as it introduces core definitions and theorems (otherwise, based on our experience, the paper would be deemed as not self-contained). Section 3 was given slightly more space to highlight the significance of the two new layers in their role of refining the hierarchy of causal models. With the additional page allowance for the revised version, in case the paper is accepted, we plan to rebalance the sections and incorporate comments from reviewers to improve clarity and flow. \n\nFor Appendix A, we agree that it currently sits between a reference and an introduction. It is primarily meant to serve as a concise reference for readers who want a quick overview of the SCM framework and the existing graphical models within it. For a more in-depth understanding, we encourage readers to consult the original sources cited in the appendix (e.g., [Bareinboim et al., 2022]). That said, we recognize that it would benefit from improved structure and narrative flow, and we will revise it accordingly to make it more reader-friendly.\n\nWe also welcome any suggestions to improve the paper’s structure and flow that the reviewer may see fit.\n\nAs for the [..]-notation, we apologize for the potential oversight. The square brackets indicate the variables being intervened upon to generate the potential outcome variables. For example, in $V\\_{i\\_{[\\mathbf{x}\\_i]}}$, $i$ is the index for the variable $V_i$, while $\\mathbf{x}\\_i$ inside the square bracket denotes the interventional value set. This was introduced to distinguish intervention targets from subscripts used for indexing variables. We will add a clarifying note in the revised version to ensure the notation is clear.\n\n> *Minor Comments*:\nline 110: V -> U\n(...)\n\nMany thanks, we will correct these typos in the revised version.\n\n*References*:\n- [Raghavan & Bareinboim, 2025] Arvind Raghavan and Elias Bareinboim. Counterfactual Realizability. ICLR, 2025.\n- [Bertrand & Mullainathan, 2004] Marianne Bertrand and Sendhil Mullainathan. Are Emily and Greg more employable than Lakisha and Jamal? A field experiment on labor market discrimination. AER, 2004.\n- [Vig et al., 2020] Jesse Vig, Sebastian Gehrmann, Yonatan Belinkov, Sharon Qian, Daniel Nevo, Yaron Singer, Stuart Shieber. Investigating Gender Bias in Language Models Using Causal Mediation Analysis. NeurIPS, 2020.\n- [Bareinboim et al., 2022] Elias Bareinboim, Juan D Correa, Duligur Ibeling, and Thomas Icard. On Pearl’s Hierarchy and the Foundations of Causal Inference. ACM, Special Turing Series, 2022."}]

Author rebuttal:

Meta-review: (a) Scientific claims and findings
The paper addresses falsifiability in Pearl’s counterfactual layer by introducing two novel classes of models for experimentally testable counterfactuals and a sound and complete calculus based on Bareinboim’s CTF framework.

(b) Strengths
Novel theoretical contribution; addresses a fundamental question in causal inference; introduces new model classes and a rigorous calculus; reviewers viewed the work as original and timely.

(c) Weaknesses
Clarity and quality of writing are limited, making the paper hard to follow. The exposition would benefit from clearer explanations, illustrative examples, and stronger contextualization. The work is purely theoretical with no empirical evaluation, and the examples provided are mathematical rather than illustrative. Reviewer pvuV noted that while this is acceptable, the lack of more concrete or less contrived examples (as sometimes seen in related literature) limits accessibility.

(d) Reasons for decision
I recommend acceptance. The contribution is theoretically significant and the reviewers were positive. However, the paper does not merit spotlight/oral due to issues of clarity, absence of evaluation, and lack of accessible examples.

(e) Rebuttal and discussion
During the rebuttal stage, reviewers raised concerns about the clarity of writing, the heavy reliance on a long appendix, and the lack of illustrative examples or empirical/theoretical evaluation. Reviewer pvuV additionally questioned the incremental nature of some of the theorems. The authors responded to these points, clarifying the role of theorems and committing to improving exposition in the main text. While the rebuttal did not introduce new results, the discussion addressed reviewers’ doubts, and by the end all reviewers were satisfied and maintained their positive recommendations.
Decision: Accept (poster)