Neural Causal Models for Counterfactual Identification and Estimation
Authors: Kevin Muyuan Xia
Yushu Pan
Elias Bareinboim
Venue: ICLR 2023 poster (2023)
Keywords: causal inference
deep learning
neural models
neural causal models
causal identification
causal estimation
counterfactual

Abstract: Evaluating hypothetical statements about how the world would be had a different course of action been taken is arguably one key capability expected from modern AI systems. Counterfactual reasoning underpins discussions in fairness, the determination of blame and responsibility, credit assignment, and regret. In this paper, we study the evaluation of counterfactual statements through neural models. Specifically, we tackle two causal problems required to make such evaluations, i.e., counterfactual identification and estimation from an arbitrary combination of observational and experimental data. First, we show that neural causal models (NCMs) are expressive enough and encode the structural constraints necessary for performing counterfactual reasoning. Second, we develop an algorithm for simultaneously identifying and estimating counterfactual distributions. We show that this algorithm is sound and complete for deciding counterfactual identification in general settings. Third, considering the practical implications of these results, we introduce a new strategy for modeling NCMs using generative adversarial networks. Simulations corroborate with the proposed methodology.

Published as a conference paper at ICLR 2023
NEURAL CAUSAL MODELS FOR COUNTERFACTUAL IDENTIFICATION AND ESTIMATION
Kevin Xia and Yushu Pan and Elias Bareinboim Causal Artificial Intelligence Laboratory Columbia University, USA
(kevi wmmniaaa yushupan, eb) 8cs. wcw.cclluumc..... wamluumaa.e..
ABSTRACT
Evaluating hypothetical statements about how the world would be had a different course of action been taken is arguably one key capability expected from mod- ern Al systems. Counterfactual reasoning underpins discussions in fairness, the determination of blame and responsibility, credit assignment, and regret. In this paper, we study the evaluation of counterfactual statements through neural models. Specifically, we tackle two causal problems required to make such evaluations, i.e., counterfactual identification and estimation from an arbitrary combination observational and experimental data. First, we show that neural causal models (NCMs) are expressive enough and encode the structural constraints necessary for performing counterfactual reasoning. Second, we develop an algorithm for simultaneously identifying and estimating counterfactual distributions. We show that this algorithm is sound and complete for deciding counterfactual identification in general settings. Third, considering the practical implications of these results, we introduce new strategy for modeling NCMs using generative adversarial networks. Simulations corroborate with the proposed methodology.
INTRODUCTION
Counterfactual reasoning is one of human's high-level cognitive capabilities, used across wide range of affairs, including determining how objects interact, assigning responsibility, credit and blame, and articulating explanations. Counterfactual statements underpin prototypical questions of the form "what if_" and "why-". which inquire about hypothetical worlds that have not necessarily been realized (Pearl & Mackenzie 2018). If a patient Alice had taken a drug and died, one may wonder, "why did Alice die?": "was it the drug that killed her?"; "would she be alive had she not taken the drug?". In the context of fairness, why did an applicant, Joe, not get the job offer? Would the outcome have changed had Joe been Ph.D.? Or perhaps of different race? These are examples of fundamental questions about attribution and explanation, which evoke hypothetical scenarios that disagree with the current reality and which not even experimental studies reconstruct.
We build on the semantics of counterfactuals based on a generative process called structural causal model (SCM) (Pearl 2000). A fully instantiated SCM M describes collection of causal mecha- nisms and distribution over exogenous conditions. Each M induces families of qualitatively different distributions related to the activities of sceing (called observational), doing (interventional), and imagining (counterfactual), which together are known as the ladder of causation (Pearl & Mackenzie 2018, Bareinboom al., 2022); also called the Pearl Causal Hierarchy (PCH). The PCH a contain- ment hierarchy in which distributions can be put in increasingly refined layers: observational content goes into layer (Cz); experimental to layer (C2): counterfactual to layer 3 (Cz). It is understood that there are questions about layers 2 and 3 that cannot be answered (i.e. are underdetermined), even given all information the world about layer 1; further, layer questions are still underdetermined given data from layers 1 and 2 |Bareinboim al. 2022 lbeling & Icard] 2020).
Counterfactuals represent the more detailed, finest type of knowledge encoded in the PCH, so naturally, having the ability to evaluate counterfactual distributions is attractive proposition. In practice, a fully specified model M is almost never observable, which leads to the question how can counterfactual statement, from L3, be evaluated using combination of observational and experimental data (from C and E3)? This question embodies the challenge of cross-layer inferences, which entail solving two challenging causal problems in tandem, identification and estimation.

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In the more traditional literature of causal inference, there are different symbolic methods for solving these
Unobserved NatureTrutt
Leamedr
Hypothesized
problems in various settings and under different as- sumptions. In the context of identification, there ex- ists an arsenal of results that includes celebrated meth- ods such as Pearl's do-calculus (Pearl 1995), and go through different algorithmic methods when consid-
SCM MM" (FF,, P
Carsi
NCM V
(.P, Prty)
G
G.Cueazuis
PCH:
4
La
ering inferences for L2- (Tian & Pearl Shpitser
& Pearl 2006 Huang& & Valtorta| 2006] Bareinboim
Traning
& Pearl 2012 Lee et 2019
Lee & Bareinboim
2020: 2021 j and La-distributions (Heckman 1992
Figure I: The Lh.s. contains the true SCM M that induces PCH's three layers. The r...s. contains a neural model M constrained by inductive bias g (entailed by MM") and matching M on L and L2 through training.
Pearl 2001 Avin et al.
2005
[Shpitser & Pearl 20099
Shpitser & Sherman 2018
Zhang & -Bareinboim
Correa et al.J
(2021). On the estimation side,
there are various methods including the celebrated
Propensity Score/IPW for the backdoor case (Rubin)
1978: Horvitz & Thompson 1952 Kennedy| 2019] Kallus & Uchara| 2020). and some more relaxed settings (Fulcher et al. 2019 Jung et al. 2020 (021), )), but the literature is somewhat scarcer and less developed. In fact, there is a lack of estimation methods for L3 quantities in most settings.
On another thread in the literature, deep learning methods have achieved outstanding empirical success in solving wide range of tasks in fields such as computer vision (Krizhevsky et al. 2012), speech recognition (Graves & Jaitly 2014), and game playing (Mnih al. 2013)). One key feature of deep learning is its ability to allow inferences to scale with the data to high dimensional settings. We study here the suitability of the neural approach to tackle the problems of causal identification and estimation while trying to leverage the benefits of these new advances experienced in non-causal settings. [ The idea behind the approach pursued here is illustrated in Fig. [ Specifically, we will search for a neural model M (r.h.s.) that has the same generative capability of the true, unobserved SCM MM (l.h.s.); in other words, M should be able to generate the same observed/inputted data, ie., Li = Cj and C2 = Cz. To tackle this task in practice, we use an inductive bias for the neural model in the form of causal diagram (Pearl 2000; Spirtes et al., 2000 BBariinboi & Pearl 2016). which is a parsimonious description of the mechanisms (F*) ) and exogenous conditions (P(U ")) of the generating SCM. The question then becomes: under what conditions can model trained using TCe
this combination of qualitative inductive bias and the available data be suitable to answer questions about hypothetical counterfactual worlds, as we had access the true M
There exists a growing literature that leverages modern neural methods to solve causal inference tasks.! Our approach based on proxy causal models will answer causal queries by direct evaluation through a parameterized neural model M fitted on the data generated by Mr For instance, some recent work solves the estimation of interventional (C2) or counterfactual (Ca) distributions from observational (L1) data in Markovian settings, implemented through architectures such as GANs, flows, GNNs, and VGAEs (Kocaoglu et al. 2018 Pawlowski et al., 2020; Zecevic et al.] 1021 Sanchez-Martin et al., 2021). In some real-world settings, Markovianity is a too stringent condition (see discussion in App. DD.4) and may be violated, which leads to the separation between layers 1 and 2. and,, in turn, issues of causal identification. P The proxy approach discussed above was pursued in Xia et al. (2021) to solve the identification and estimation of interventional distributions (C2) from observational data (C₂) in non-Markovian settings. 66) This work introduced an object we leverage throughout this paper called Neural Causal Model (NCM, for short), which is a class of SCMs constrained to neural network functions and fixed distributions over the exogenous variables. While
One of our motivations is that these methods showed great promise at estimating effects from observational
data under backdoor/ignorabiity conditions (Shalit el al.) 2017 Louizos el al. 2017 Li & Fu 2017 Johansson
ct al.
2016; Yao et al. 2018) Yoon et al. 2018 Kallus 2020 Shi et al. 2019 Du et al. 2020 Guo et al. (020).
his represents an extreme case where all 2 and Ep-distributions are provided as data. In practice, this
may be unrealistic, and our method takes as input any arbitrary subset of distributions from Li and L2.
'When imposed on neural models, they enforce equality constraints connecting layer and layer quantities,
defined formally through the causal Bayesian network (CBN) data structure (Barcinboim et al. 2022 Def. 16).
*In general, M does not need to, and will not be equal to the true SCM M'
$ "Laeer 3 differs from lower layers even in Markovian models; see Bareinhoim et al. 2002 Ex. 7).
Wiity et al. (202)) shows related approach taking the Bayesian route; further details, see Appendix
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NCMs have been shown to be able solve the identification and estimation tasks for Le queries. their potential for counterfactual inferences is still largely unexplored, and existing implementations have been constrained to wwwdimensional settings.
Despite all the progress achieved so far, no practical methods exist for estimating counterfactual (Ca) distributions in the general setting where an arbitrary combination of observational (C1) and experimental (L2) distributions is available, and unobserved confounders exist (i.e. Markovianity does not hold). Hence, in addition to providing the first neural method of counterfactual identification, this paper establishes the first general counterfactual estimation technique even among non-neural methods, leveraging the neural toolkit for scalable inferences. Specifically, our contributions are: 1. We prove that when fitted with graphical inductive bias, the NCM encodes the L3 consttatassss necessary for performing counterfactual inference (Thm. D. and that they are still expressive enough to model the underlying data -ata-genn model, which is not necessarily neural network (Thm..2) ...
2. We show that counterfactual identification within neural proxy model setting equivalent to established symbolic approaches (Thm. 3. We leverage this duality to develop an optimization procedure (Alg. M for counterfactual identification and estimation that is both sound and complete (Corol 2. The approach is general in that it accepts any combination of inputs from L1 and C2, it works in any causal diagram setting, and it does not require the Markovianity assumption to hold. 3. We develop new approach modeling the NCM using generative adversarial networks (GANs) (Goodfellow et al., 22014), capable of robustly scaling inferences to high dimensions (Alg. (3). We then show how GAN-NCMs can solve the challenging optimization problems in identifying and estimating counterfactuals in practice. Experiments are provided in Sec. 5 and proofs in Appendix A
All supplementary material can be found in the full technical report (Xia et al. 2022).
Preliminaries. We now introduce the notation and definitions used throughout the paper. We use uppercase letters (X) to denote random variables and lowercase letters (x) denote corresponding values. Similarly, bold uppercase (X) and lower case (x) letters are used to denote sets of random variables and values respectively. We use Dx to denote the domain of X and Dx = Dx, x... x Dx, for the domain of X = (X1,.... J.). We denote P(X = x) (which we will often shorten to P(x)) as the probability of X taking the values x under the probability distribution P(X).
We utilize the basic semantic framework of structural causal models (SCMs), as defined in (Pearl 2000, Ch. 7). An SCM M consists of endogenous variables V, exogenous variables U with distribution P(U), and mechanisms F. F contains a function fv, for each variable V, that maps endogenous parents Pay, and exogenous parents Uv. to V4. Each M induces a causal diagram G. where every Vi V is vertex, there is directed arrow (V3 + V.) for every V; E V and V E Pav. and there is a dashed--biirretted arrow (V, + a----a VA2) for every pair V, V E V such that Uy, and Uv, are not independent. For further details, see (Bareinboim al. 2022 Def. 13/16, Thm. 4). The exogenous Uv, 's are not assumed independent (i.e. Markovianity IS not required). Our treatment is constrained recursive SCMs (implying acyclic causal diagrams) with finite domains over V (see Apps. A for details). Each SCM M assigns values to each counterfactual distribution as follows: Definition (Layer Valuation). An SCM M induces layer C3(M), set of distributions over V. each with the form P(Y.) = P(Y ((x.), Y xxxal....) such that
p. (y Y2[xa] / 1 |X (u) = yI, (u)x y2. Id..... dP(u). Du
(1)
where (ixx (u) is evaluated under Fx = Uv; VV, EVXXUUUXX + +x:
Each Y, corresponds to set of variables in a world where the original mechanisms fx are replaced with constants x; for each X E X,; this is also known as the mutilation procedure. This procedure corresponds to interventions, and we use subscripts denote the intervening variables (c.g. Yx) or subscripts with brackets when the variables are indexed (e.g. Y (((x)) D. For instance, P(ya, V'y)) ) is the probability of the joint counterfactual event Y = had X been x and Y = s/ had X been x'.
SCM M2 is said to be p(..))((.oo))) -consiittttt (for short, L.-consistent) with SCM Mj if (MM)) = (MM2). We will use Z to denote set of quantities from Layer 2 (i.e. Z = (P(V,, 2, }) )===)) and use Z(M) to denote those same quantities induced by SCM M (i.e. Z(M) = (PM (V 2. 1))
We use neural causal models (NCMs) as a substitute (proxy) model for the true SCM, as follows: Definition 2 (G-Constrained Neural Causal Model (G-NCM) (Xia et al. 2021] Def. 7)). Given causal diagram G, G-constrained Neural Causal Model (for short, G-NCM) M()) over variables V
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with parameters 0 = {0y, V2 E V) is an SCM (Ü, V. such that Ü= (Üc C C(
where C(g) is the set of all maximal cliques over bidirected edges of g, and Dj = [0, 1] for all
Ú E Ü: F = {fv. V2 V), where each fv, is feedforward neural network parameterized by By, E mapping values of Uv, U Pay, to values of V; for Uv; = {Üc Üc Ü s.t. E CJ and
Pay; Puc(Vi); P(U) is defined s.t. Ö ~ Unif(O, 1) for each Ü E Ü.
2 NEURAL CAUSAL MODELS FOR COUNTERFACTUAL INFERENCE
We first recall that inferences about higher layers of the PCH generated by the true SCM M cannot be made in general through an NCM M trained only from lower layer data (Bareinboim et al. 2022 Xia et al. 2021). 1). This impossibility motivated the use of the inductive bias in the form of causal diagram G in the construction of the NCM in Def. 2 which ascertains that the G-consistency property holds. (See App. for further discussion.) We next define consistency w.r.t. to each layer, which will be key for more fine- tne-grained discussion later on.
Definition 3 (g(C.).Consistene
Let G be the causal diagram induced by the SCM M For any SCM M, .M is said to be gonssscctttt (w.r.t MM) if (MM satisfies all layer equality constraints implied by G.
This generalization is subtle since regardless of which C; is used with the definition, the causal diagram 9 generated by .M" is the same. The difference lies in the implied constraints. For instance, if an SCM M is giEs)-consistent, that means that G is a Bayesian network for the observational distribution of M, implying independences readable through d-separation Pearl (1988). If M is g(Ca)-consistent, that means that G is a Causal Bayesian network (CBN) (Bareinboim et al. 2022 Def. 16) for the interventional distributions of M. While several SCMs could share the same d-separation constraints as MM", there are fewer that share all L2 constraints encoded by the CBN. g- consistency at higher layers imposes stricter set of constraints, narrowing down the set of compatible SCMs. There also exist constraints of layer 3 that are important for counterfactual inferences.
To motivate the use of such constraints, consider an example inspired by the multi- aumdi-armed bandit problem. A casino has 3 slot machines, labeled "0", "1", and "2". Every day, the casino assigns one machine a good payout, one a bad payout, and one an average payout, with chances of winning represented by exogenous variables U+, U_, and U= respectively. A customer comes every day and plays a slot machine. X represents their choice of machine, and Y is a binary variable representing whether they win. Suppose a data scientist creates model of the situation, and she hypothesizes that the casino predicts the customer's s choice based on their mood (U)) and will always assign the predicted machine the average payout to maintain profits. Her model is described by the SCM M':
U
=(UM, UUM U.U E {0, 1,2), U+, U_ E 10, =1X,1, X (0,,1, E {0,
V
(fx(uM)
uM
M'
U==
x = UM
(2)
F
fr uM, U4, u-)
u_
I = (uM - 1)%
ut = (uM + 1)%3
P(U): P(UM =) = 3. P(U+= = =0.6, P(U_ =1 1) = 0.4, P(U_= =1) =1 0.2
It turns out that in this model P(yz) ply 22. For example, P(Y
X 0) P(U=
1) = 0.4, and P = 1) = P(UM P)P =1 + P(UM = P(U- + P(UM
2), =) 1) (0.4) (0.2) + (0.6) = 0.4.
Suppose the true model M employed by the casino (and unknown by the customers and data scientist) induces graph g = -Y Interestingly enough, M'' would be ((Ca)) consistent with M' since M' is compatible with all L=-constraints, including P(yz) P(y 2)) and P(z,) = P(z). However, and perhaps surprisingly, it would fail to be g(La)-consisten A further constraint implied by G on the third layer is that P(y. x') = P(yx). which is not true of M'. To witness, note that P(Y==0 1 X = 2) P(U+ = 1) 0.6 in M', which means that if the customer chose machine 2, they would have had higher payout had they chosen machine O. This does not match P(Yx=u = 1) = 0.4, computed earlier, so M' fails to encode the Lo-cosstraints implied by G.
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In general, the causal diagram encodes family of L3 Co-ossttras which we leverage to make cross-layer inferences. A more detailed discussion can be found in Appendix D We show next that NCMs encodes all of the equality constraints related to L3. in addition to the known L2-constraints.
S2
Theorem (NCM g(Ca)) Consisseet), Any G-NCM M() is g(as) consistent.
This will be key result for performing inferences at the counterfactual level. Similar to how constraints about layer 2 distributions help bridge the gap between layers and 2, layer 3 constraints allow us to extend our inference capabiliitess into layer 3. (In fact, most of L is distributions are not obtainable through experimentation.) While this graphical inductive bias is powerful, the set of NCMs constrained by G is no less expressive than the set of SCMs constrained by G, as shown next.
Figure 2: Model-
theoretic visualization of Thms. [ 2
Theorem 2 (C3-g Expressiveness). For any SCM M that induces causal diagram G, there exists G-NCM M() (Ū, V F, P(Ū)) s.r. M is C3-consistent w.r.f. .M.
This result ascertains that the NCM class is as expressive, and therefore, contains the same generative capabilitiees as the original generating model. More interestingly, even if the original SCM M does not belong to the NCM class, but from the higher space, there exists a NCM M0) that will be capable of expressing the collection of distributions from all layers of the PCH induced by it.
A visual representation of these two results is shown in Fig. - The space of all SCMs is called D2,, and the subspace that contains all SCMs (((C.).consisten w.r.t. the true SCM M (black dot) is called Ω (g(A)) Note that the ((C.) space shrinks with higher layers, indicating a more constrained space with fewer SCMs. Thm. states that all G-NCMs (S)))) are within 12' (G(C))), and Thm. 2 states that all SCMs in 52" (C( can be represented by corresponding G-NCM on all three layers.
may seem intuitive that the G-NCM has these two properties by construction, but these properties are nontrivial and, in fact, not enjoyed by many model classes. Examples can be found in Appendix D Together, these two theorems ensure that the NCM has both the constraints and the expressiveness necessary for counterfactual inference, elaborated further in the next section.
3 NEURAL COUNTERFACTUAN IDENTIFICATION
The problem of identification is concerned with determining whether a certain quantity is computable from a combination of assumptions, usually encoded in the form of causal diagram, and collection of distributions (Pearl 2000 p. 77). This challenge stems from the fact that even though the space of SCMs (or NCMs) is constrained upon assuming certain causal diagram, the quantity of interest may still be underdetermined. In words, there are many SCMs compatible with the same diagram G but generate different answers for the target distribution. In this section, we investigate the problem of identification and decide whether counterfactual quantities (from L3) can be inferred from a combination of subset of L2 and L1 datasets together with G. as formally defined next.
Definition 4 (Neural Counterfactual Identification). Consider an SCM M and the corresponding causal diagram G. Let 2 {P )=== be collection of available interventional (or observational if 2 = 8) distributions from M The counterfactual query P(Y. y. X. x.) is said to be
neural identifiable (identifiable, for short) from the set of G-constrained NCMs (G) and Z if and
only if M (y. x.) = p (y= xx) for every pair of models M,, M2 e ((G) s.t. they match M on all distributions in Z (i.e. Z (MM)) = Z(M1) = 2(M2) > 0).
From a symbolic standpoint, a counterfactual quantity Ply. xx)) is identifiable from G and Z if all SCMs that induce the distributions of Ż and abide by the constraints of G also agree on P(y. x.). This is illustrated in Fig. 3 In the definition above, the search is constrained to the NCM subspace (shown in light gray) within the space of SCMs (dark gray). It may be concerning that the true SCM M might not be an NCM, as we alluded to carlier. The next result ascertains that identification within the constrained space of NCMs is actually equivalent to identification in the original SCM-space.
Theorem 3 (Counterfactual Graphical-Nert
Equivalence (Dual ID)). Let Ω Ω be the spaces including all SCMs and NCMs, respectively. Consider the true SCM M and the corresponding
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causal diagram G. Let Q P(y. x.) be the target query and Z the set of observational and interventional distributions available. Then, Q is neural identifiable from ((G) and Z if and only if is identifiable from G and Z.
Interestingly, this result connects the new concept of neural counterfactual identification (Def. 4 with estab- lished non-neural results. If a counterfactual quantity is determined to neural identifiable, then it also iden- tifiable from G and ž through other non-neural means, and vice versa Practically speaking, counterfactual in- ference can be performed while constrained in the NCM space, and the obtained results will be faithful to existing symbolic approaches. This broadens the previous results connecting NCMs to classical identification.
f
Stractural Assumptions
M
G
M
phiyy bx.) (y.lx))
X(M")= (a)
(C:. Distristionn
Dura
(Ex)
Conmerfactual Query
Corollary (Neural Counterfactual Mutilation (Opera- tional ID), Consider the true SCM M E EL" causal diagram G, set of available distributions Z. and tar- get query Q equal to PMd (y- x.). Let M E ((G) be G-constrained NCM such that (MM) ((M)). IfQ is identifiable from 9 and Z, then Q is computable via Eq. from M.
Figure 3: P(y.) is identifiable from Z and (2()) if for any SCM M Ω2" and NCMs M2 E Q (top left), Mr, M, M" match in E (bottom left) and (top right), then the NCMs M1, M2 also match P(y-) (bottom right).
Corol. states that once identification is established, the counterfactual query can inferred through the NCM M, as if it were the true SCM M by directly applying layer 3's definition to M (Def. 11 Remarkably, this result holds even if .M" does not match M in either the mechanisms F or the
exogenous dist. P(U), and it only requires some specific properties: ((Ca) -oonssccoss,,,ncc, matching Z,
and identifiability. Without these properties, inferences performed on M would bear meaningful information about the ground truth. To understand this subtlety, refer to examples in App. D
Building on these results, demonstrate through the procedure NeuralID (Alg. [ how to decide the identifiability of counter- factual quantities. The specific optimization procedure searches explicitly in the space of NCMs for two models that respectively min- imize and maximize the target query while maintaining consistency with the provided data distributions in Z. the two models match in the target query Q, then the effect is identifiable, and the value is returned; oth- erwise, the effect is non- non-identifiable.
Algorithm I: NeuralID Identifying/essiiaainng counterfactual queries with NCMs.
query P(y. [x.), C2 datasets Z(M") and cousal dingram g Output: P.M (y xx. if identifiable, FA1l otherwise.
Input
M + NCN.V. 9)
1/ from Def. 2
0 -rarg ming p2 (y.|xx)) st. 2(M
0 + arg maxe p5r (y-/x-)) (0)
p (y (y.llx.) ) o (0 ( ty-1e (x..) then
5
return PACL.
7
return phrto (y. (x.) // choose min oz max (y. arbicrarily
The implementation of how to enforce these consistency constraints in practice is somewhat chal- lenging. We note two nontrivial details that are abstracted away in the description of Alg. [ First, although training to fit a single, observational dataset is straightforward, it is not as clear how to simultancously maintain consistency with the multiple datasets in Σ. Second, unlike with simpler interventional queries, it is not clear how to search the parameter space in way that maximizes or minimizes counterfactual query, which may be more involved due to nesting (c.g. P(Y2x-o) or evaluating the same variable in multiple worlds (e.g. P(Yx=o, Yx=1)). The details of how to solve these issues are discussed in Sec.
Interestingly, this approach is qualitatively different than the case of classical, symbolic methods that avoid operating in the space of SCMs directly. Still, in principle, this alternative approach does not imply any loss in functionality, as evident from the next result.
Corollary (Soundness and Completeness). Let O be the set of all SCMs, M E S2 be the true SCM inducing causal diagram G, Q Ply. x.) be a query of interest, and Q be the result from
7 We say identification from G and Z instead of S2 (g) and Z, because existing symbolic approaches (c.g. do-calculus) directly solve the identification prohlem on top of the graph instead of the space of SCMs.

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running Alg. with inputs Z(M)) O, G, and Q. Then Q is identifiable from 9 and Z if and only if Q is not FAIL. Moreover, if Q is not FAIL, then pM' (y= x.).
In words, the procedure NeuralID is both necessary and sufficient in this very general setting, implying that for any instances involving any arbitrary diagram, datasets, or queries, the identification status of the query is always classified correctly by this algorithm.
4 NEURAL COUNTERFACTUAL ESTIMATION
We developed a procedure for identifying counterfactual quantities that is both sound and complete under ideal con- ditions- e.g., unlimited data, perfect optimization, which is encouraging. In this section, we build these results and establish a more practical approach to solving these tasks under imperfect optimization and finite samples.
Algurillum NCM Counterfactual Sampling
Input :NCM
M(0) (Ú. V (()),
eounterfactual conditional
x., number of samples m
Output: samples from p (Y, (x.)
Function M. Mamppll (Y m):
Consider a G-NCM M constructed as specified by Def. Any counterfactual statement Y. = (Yi(xi) X2(xa].-..
2
0
while (S] m do P(m))sample11
can be evaluated from an NCM M for a specific setting of
5 if i x x. then
Sadd ( (())))
Ü = u by computing the corresponding values of Y, in the mutilated submodel My for each i. That is,
return S:
Y. Ld (u) = -Y [(x] (u), Y 2(xx) (u)....)
(3)
Then, sampling can be done following the natural approach delineated by Alg. In words, the
distribution P(Y. x.) can be sampled from NCM M by (1) sampling instances of P(Ū), (2) computing their corresponding value for X. (via Eq. 3) while rejecting cases that do not match x.,
and (3) returning the corresponding value for Y. (via Eq. 3 for the remaining instances.
Following this procedure, counterfactual P(Y. y. X. = =x.) can be estimated from the NCM through a Monte-Carlo approach, instantiating Eq. as follows:
m
Y (ū,) = y., (ū,) = X.
xM (y. x.) ~
(4)
ΣØ 1
X (ū,) =x,
where {ūr} are set of m samples from P(Ù).
Alg. 3 demonstrates how to solve the challenging optimization task in lines 2 and 3 of Alg. The first step is to learn parameters such that the distributions induced by the NCM M match the true distributions in Z. While Alg. [
I
describes the inputs in the form of L2-distributions, Z(M*) (PM ) in most settings, one has the empirical versions of such distributions in the form of finite datasets, {p. (Vz.)= = [V2cs) L8
One way train M match M in the distribution P(V,,) x.))) is compare the distribution of data points in P.d (Vzz) with the distribution of samples from M, PM (Vz.). The two empirical distributions can be compared using a divergence function Dp. which returns a smaller value when the two distributions are "similar". The goal is then to minimize WD pD (PM (Vz,). ), PM' (V )) for each & {I....,{). L). In this work, generative adversarial approach (Goodfellow al. 2014 is taken to train the NCM, and Dp is computed using discriminator network.
In addition to fitting the datasets, the second challenge of Alg. [ to simultaneously maxi- mize or minimize the query of interest Q = P(y. x.). This can be done by first comput- ing samples of P(Y. x.) from M via Alg. 2 denoted Q. and then minimizing (or maxi- mizing) the "distance" " between Q and Q. Essentially, samples from Y, are penalized based on how similar they
to the correct values y.. For example, if the query to maximize is P(Y 1). and a value of 0.6 is sampled for Y from M, then the goal could be minimize squared error, (1 - 2. In general, a distance metric Do is used to compute the distance between @ and Q, and we use log loss for Bg our experiments involve binary variables.
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ATE Elgall El
ETT E
NDE
CtIE
z
E 1x= -=
Z=
2= (PIVAAV
Z= [P())
Z= (AVAM(V))
(P(V)
(P(V)
(PVA)P13))
(P(V)
(AVLANA))
10
65
1.0
0.2
Accuracy
6.5
300
10000
500
1000
900
10300
500
1000
0
502
104000
300
1000
3
500
10000
340
1000
Training Iteration
Figure 4: Experimental results on deciding identifiability on counterfactual queries with NCMs. GAN-NCM (green) is compared with MLE-NCM (orange) for settings with d = 1. GAN-NCM performance is also shown for d = 16 (dashed green). Blue (resp. yellow) backgrounds on plots correspond to ground truth of ID (resp. non-ID). ID cases are numbered for reference in later plots.
For this reason, NCM trained with this approach will be referred as a GAN-NCM. R More details about architecture and hyperpa- rameters used throughout this work can be found in Appendix B
Algorithm Training Model
Input :
i-1 = P(y.,xx, ). diagram G. number of Monte Carlo samples m, regularization constant A, learning
Data (BM' (Va,) = {Vag. 1 T query
rate n, training epochs T
+ VV
11 from Def.
Putting Dr and TDo together, we can write that
Initialize parameters Pmin and Amnx 3 for ( +- T do
the objective L, M,, PP. (Vz))) is
Limi for ls < to do
Lmax +
11 Semple via Alg.
é Σ
Dr (MM Pa)))
Posin (Vz,) + Mi sempll Vx
+ ATQ (ē.0)
Panx M (0mmmm) (Vz, mp)
Liin
(5)
Limn Dr (amnn(Vn (Vz)). BM9 (Vzz))
where À is initially set to high value, and decreases during training. Optimization may be done using gradient descent. After training, the two values of Q induced by (0 and (0max) are compared with hypothesis test- ing procedure to decide identifiability. Eq. 4 is used as Q's 'ss estimate, whenever identifiable.
Lmax +
Liax + Dp P (Vec) ). BM" (Vaz))
Quin M((min)) (Y.,m)
11
Qmac +- (0 damppl (Y m)
I. from Eq.
12
Liiin +- Lunin xD2 (Qmin Q)
Liuxx + Limax ADQ (Quax, Q)
Own to Omn 7 Limin 14
Omax Omax V Lemax
5 EXPERIMENTAL EVALUATION
We first evaluate the NCM's ability to identify counterfactual distributions through Alg. 3 9 Each setting consists of target query (Q), a causal diagram (9), and a set of input distributions (2). In total, we test 32 variations. Specifically, we evaluate the identifiability of four queries Q: (1) Average Treatment Effect (ATE), (2) Effect of Treatment on the Treated (ETT) (Paal 2000, Eq. 8.1),, (3)
Natural Direct Effect (NDE) (Peall 2001 Eq. 6), and (4) Counterfactual Direct Effect (CtfEE (Zhang & Bareinboim 2018 Eq. 3); each expression is shown on the top of Fig. 4 The four graphs used are shown on the figure's 's left side, and represent general structures found throughout the mediation and fairness literature (Pearl 200T Zhang & Bareinboim] 2018). The variable X encodes the treatment/decisionn, Y the outcome, Z observed features, and W mediating variables. Lastly, consider a setting in which only the observational data is available (Z = (P(())) and another in which additional experimental data on X is available (Z (P(V), P(V+))). In the experiments shown, all variables are d-dimensional binary variables except Z, whose dimensionality d is adjusted
*Other choices of DD include KI. divergence or Maximum Mean Discrepancy (MMD) (Grettonn et al. 2012 'hhe code publicly available at: https::////....... //gi thub. con wtt.//////aaaaa A l.ab/ hwt.conneeraacttaaaaa
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in experiments. The background color of each setting indicates that the query Q is identifiable (blue) or is not identifiable (yellow) from the inputted 9 and Z. Given the sheer volume of variations, we summarize the experiments below and provide further discussion and details in Appendix B
We implement two approaches to ground the discussion around NCMs, one based on GANs (GAN-NCM), and another based on maximum likelihood (MLE-NCM). The former was discussed in the previous section and the latter is quite natural in statistical settings, The experiments (Fig. 4) show that GAN-NCM has on average higher accuracy. The MLE-NCM performs slightly better in ID cases (blue), but the performance drops significantly for non-ID cases (yellow), suggesting it may be biased returning ID for all cases. The GAN-NCM is also shown to achieve decent performance in 16-d, where the MLE--CM fails to work. We plot the run time of these two approaches in Fig. 5 which shows that the MLE-NCM scales poorly compared to the GAN-NCM; this pattern is observed in all settings. Intuitively, this is not surprising since the MLE-NCM explicitly computes a likelihood for every value in every variable domain, the size of which grows exponentially w.r.t. the dimensionality (d), while the GAN-NCM avoids this by implicitly fitting distributions through P(Ú) and F and directly outputting samples.
800
Time On poor dal
600
220
Dimmention (6)
Figure 5: Results comparing run times of 100 epochs of training between GAN-NCM (green) and MLE-NCM (or- ange) in the first graph of Fig. J the dimensionality d of Z scales higher.
For the identifiable cases (blue background), the tar- get Q is estimated through Eq. 4 after training. Re- sults are shown in Fig. 6 The MLE-NCM serves as a benchmark for 1-dimensional cases since, intuitively, the data distributions can be learned more accurately when modeled explicitly. Still, even when = 1, the GAN-NCM achieves competitive results in most settings and consistently achieves an error under 0.05
MAE Query Estimation
20"
10
10'
with more samples. The GAN-NCM is able to main- tain this consistency even at d = 16, demonstrating its robustness scaling to higher dimensions.
13
15
30
10
30
1030
After all, the GAN-NCM is shown to be effective at identifying and estimating counterfactual distri- butions even in high dimensions. As expected, the MLE-NCM may achieve lower error in some 1-d set- tings, but the GAN-NCM may be preferred for scala- bility. Moreover, an incorrect ID conclusion in a non- ID case may be dangerous for downstream decision- making as the resulting estimation will likely be in- correct or misleading. The GAN-NCM is evidently more robust in such non-ID cases while still performing competitively in ID cases. Further experi- ments and discussions are provided in App. B
Samples
Figure 6: Results on estimating identifiable cases from Fig. 4 (corresponding numbers shown on the right). Mcan Absolute Error (MAE) is plotted (with 95% confidence) for each setting for varying sample sizes. Results are shown for GAN-NCM (solid green) and MLE-NCM (orange) with d = 1 and also GAN-NCM with 16 (dashed green).
CONCLUSIONS
We developed in this work a neural approach to the problems of counterfactual identification and estimation using neural causal models (NCMs). Specifically, we first showed that with the graphical inductive bias, NCMs are capable of encoding counterfactual (Lz) constraints and are still expressive so as to represent any generating SCM (Thms. 2 We then showed that NCMs have the ability of solving any counterfactual identification instance (Thm. 3 Corol T. Given these theoretical properties, we introduced a sound and complete algorithm (Alg- [ C for identifying and estimating counterfactuals in general non-Markovian settings given arbitrary datasets from L and
L2. We developed an approach based on GANs to implement this algorithm in practice (Alg. 3) and empirically demonstrated its ability to scale inferences. From a neural perspective, counteriactual reasoning under causal inductive bias allows for deep models to be trained with an improved understanding of interpretability and generalizability. From a causal perspective, neural nets can now provide tools solve counterfactual inference problems previously only understood in theory.

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ACKNOWLEDGEMENTS
This research was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation.
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14


Reviews:
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The authors also provide new identification results and Algorithm 1 that uses supervised learning techniques to tell if the causal prior constraints are sufficient for identification. Finally, they propose GAN-NCM to scale causal models to high dimensional data.", "summary_of_the_review": "This paper advances the theory of neural causal models and prove that they can recover the counterfactuals (L3 in Pearl's ladder of causality). The authors also provide new identification results and Algorithm 1 that uses supervised learning techniques to tell if the causal prior constraints are sufficient for identification. Finally, they propose GAN-NCM to scale causal models to high dimensional data.", "strength_and_weaknesses": "The overall arguments of the paper seem intuitive and correct. Although, the authors do not provide short overview of the proofs and the paper looks a bit jammed (better fits a journal paper). The paper heavily relies on the familiarity of the audience with the recent prior works such as (Bareinboim et al., 2022). Thus, some of the statements of the paper are unclear and need more explanations. E.g., \"Our treatment is constrained to recursive SCMs, which implies acyclic causal diagrams, with finite domains over V.\"\nThe use of the term neural seems to be mostly for marketing purposes and the results of the paper may hold for general function estimators. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "hHptb9_9DR0", "reviewer": "Reviewer_452t", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper builds the basic theoretical result for neural counterfactual inference. Based on the theoretical results, the authors present the algorithm to determine the identification of counterfacual and give the algorithm to estimate it if it is identifiable.", "summary_of_the_review": "After the rebuttal, I increase my score.\n\n#-----------\nCurrently, I can only give a baseline score because I am not sure about the studies in this paper. I look forward to the response of the authors.", "strength_and_weaknesses": "Strength:\n1. Counterfactual inference is an interesting topic. This paper takes deep learning techniques into counterfactual inference, which could inspire more related studies.\n\n2. The paper presents solid theoretical results.\n\nWeaknesses:\nThe paper is hard to read. I understand that there are many new results, but it will be better if the authors could clearly illustrate the value of their theoretical results and the challenge. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "QCw5oq_xZQY", "reviewer": "Reviewer_AWWR", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Basically, this paper aims to study the problem of evaluating the counterfactual statements based on neural models. To achieve this goal, the authors firstly detail the previous work on the causal neural models, and then propose to identify counterfactual quantities based on the first two causal layers. The authors have also proved the soundness and completeness, which makes the paper very solid. \n", "summary_of_the_review": "see the above comments", "strength_and_weaknesses": "In general, the paper is well-written, and it aims to solve an important and interesting problem. The experiment can support the claims in the paper, and the authors have provided a very detailed appendix, which can help to understand their ideas.\nI do not find significant weaknesses in this paper. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster