The Causal-Neural Connection: Expressiveness, Learnability, and Inference
Authors: Kevin Muyuan Xia
Kai-Zhan Lee
Yoshua Bengio
Elias Bareinboim
Venue: NeurIPS 2021 Poster (2021)
Keywords: causal inference
deep learning
neural models
causal identification
causal estimation

Abstract: One of the central elements of any causal inference is an object called structural causal model (SCM), which represents a collection of mechanisms and exogenous sources of random variation of the system under investigation (Pearl, 2000). An important property of many kinds of neural networks is universal approximability: the ability to approximate any function to arbitrary precision. Given this property, one may be tempted to surmise that a collection of neural nets is capable of learning any SCM by training on data generated by that SCM. In this paper, we show this is not the case by disentangling the notions of expressivity and learnability. Specifically, we show that the causal hierarchy theorem (Thm. 1, Bareinboim et al., 2020), which describes the limits of what can be learned from data, still holds for neural models. For instance, an arbitrarily complex and expressive neural net is unable to predict the effects of interventions given observational data alone. Given this result, we introduce a special type of SCM called a neural causal model (NCM), and formalize a new type of inductive bias to encode structural constraints necessary for performing causal inferences. Building on this new class of models, we focus on solving two canonical tasks found in the literature known as causal  identification and estimation. Leveraging the neural toolbox, we develop an algorithm that is both sufficient and necessary to determine whether a causal effect can be learned from data (i.e., causal identifiability); it then estimates the effect whenever identifiability holds (causal estimation). Simulations corroborate the proposed approach.

The Causal-Neural Connection: Expressiveness, Learnability, and Inference
Kevin Xia CausalAí Lab Columbia University
Kai-Zhan Lee Bloomberg L.P.
Yoshua Bengio MILA Université Montréal
Elias Bareinboim
CausalAT Lab
Columbia University k127928columbia. edu
Columbia University
kmx20008columiia.com edu
bengioomii la. quebec
ebCcs. cww.mbia.aaaaaaaaamm edu
Abstract
One of the central elements of any causal inference is an object called structural causal model (SCM), which represents a collection of mechanisms and exogenous sources of random variation of the system under investigation (Pearl, 2000). An important property of many kinds of neural networks universal approximability: the ability to approximate any function to arbitrary precision. Given this property, one may be tempted to surmise that a collection of neural nets is capable learning any SCM by training on data generated by that SCM. In this paper, we show this is not the case by disentangling the notions of expressivity and learnability. Specifically, we show that the causal hierarchy theorem (Thm. 1, Barcinboim et al., 2020), which describes the limits of what can be learned from data, still holds for neural models. For instance, an arbitrarily complex and expressive neural net is unable to predict the effects of interventions given observational data alone. Given this result, we introduce special type of SCM called neural causal model (NCM), and formalize a new type of inductive bias to encode structural constraints necessary for performing causal inferences. Building on solving two canonical tasks found in the literature known as causal identification and estimation. Leveraging the neural toolbox, develop algorithm that is both sufficient and necessary to determine whether a causal effect can be learned from data (i.e., causal identifiability); it then estimates the effect whenever identifiability holds (causal estimation). Simulations corroborate the proposed approach.
this new class of models, we focus
Introduction
One of the most celebrated and relied upon results in the science of intelligence the universality of neural models. More formally, universality says that neural models can approximate any function (e.g., boolean, classification boundaries. continuous valued) with arbitrary precision given enough capacity in terms of the depth and breadth of the network [14, 26, 47, 53]. This result, combined with the observation that most tasks can be abstracted away and modeled input/output i.c. as functions - leads to the strongly held belief that under the right conditions, neural networks can solve the most challenging and interesting tasks in AI. This belief not without merits, and corroborated by ample evidence of practical successes, including in compelling tasks in computer vision [43], speech recognition [22], and game playing [54]. Given that the universality of neural nets is such compelling proposition, we investigate this belief in the context of causal reasoning.
To start understanding the causal-neural connection - i.e., the non-trivial and somewhat intricate relationship between these modes of reasoning two standard objects in causal analysis will be instrumental. First, evoke class of generative models known the Structural Causal Model (SCM, for short) [58, Ch. 7]. In words, an SCM M is representation of a system that includes collection of mechanisms and probability distribution over the exogenous conditions (to be formally defined later on). Second, any fully specified SCM M induces a collection of distributions known as the Pearl Causal Hierarchy (PCH) [5, Def. 9]. The importance of the PCH that it formally delimits
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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distinct cognitive capabilities (also known as layers; not to be confused with neural nets layers) that can be associated with the human activities of "seeing" (layer 1). "doing" (2), and "imagining" (3) [59, Ch. IJ. Each of these layers can be expressed as a distinct formal language and represents queries that can help to classify different types of inferences [5, Def. 8). Together, these layers form a strict containment hierarchy [5, Thm. 1]. We illustrate these notions in Fig. ((a) (left side), where SCM M induces layers Li, Lig, L3 of the PCH.
Even though each possible statement within these capabilities has well-defined semantics given the true SCM MM' [58, Ch. 71, a chal- lenging inferential task arises when one wishes to recover part of the PCH when MM is only par- tially observed. This situation is typical in the real world aside from some special settings in physics and chemistry where the laws of nature are understood with high precision.
(a) Unobwerved Nature/Truth
(b) Leamed Hypothesized
Structural Causal Model _M-
Neural Model W
PCH
2
4
4
Inaring (C) =n ALD)
For concreteness, consider the setting where one needs to make a statement about the effect of new intervention (i.e., about layer 2), but only has observational data from layer 1, which is passively collected. Going back to the causal- neural connection, one could try to learn a neural model A using the observational dataset (layer 1) generated by the true SCM MM,, as illustrated in Fig. 1(b). Naturally, a basic consistency require- ment is that / should be capable of generating the same distributions as MM: in this case, their layer I predictions should match (i.e., L1 = Lÿ). Given the universality of neural models, it is not hard to believe that these constraints can be satisfied in the large sample limit. The question arises of whether the learned model N' can act as proxy, having the capability of predicting the effect of interventions that matches the L2 distribution generated by the true (unobserved) SCM M 3 The answer to this question cannot be ascertained in general, as will become evident later on (Corol. 1). The intuitive reason behind this result is that there are multiple neural models that are equally consistent w.r.t. the La distribution of M but generate different dz-ddistribution. Even though /N may be expressive enough to fully represent M (as discussed later on), generating one particular parametrization of N consistent with L, is insufficient provide any guarantee regarding higher-layer inferences, i.e., about predicting the effects of interventions (L2) or counterfactuals (La).
The I.h.s. contains the unobserved true SCM .M" that induces the three layers of the PCH. The r.h.s. contains an NCM that is trained to match in layer 1. The matching shading indicates that the two models agree w.r.t. L1 while not necessarily agreeing w.r.t. layers 2 and 3.
Figure 1:
The discussion above entails two tasks that have been acknowledged in the literature, namely, causal effect identification and estimation. The first - causal identification has been extensively studied, and general solutions have been developed, such as Pearl's celebrated do-calculus [57]. Given the impossibility described above, the ingredient shared across current non-neural solutions is to represent assumptions about the unknown M in the form of causal diagrams [58, 65, 7] their equivalence classes [28, 60, 29, 71]. The task is then to decide whether there is a unique solution for the causal query based such assumptions. There are no neural methods today focused on solving this task. The second task- causal estimation - is triggered when effects are determined be identifiable by the first task. Whenever identifiability is obtained through the backdoor criterion/conditionll ignorability [58. Sec. 3.3.1J. deep learning techniques can be leveraged to estimate such effects with impressive practical performance [63, 52, 48, 31, 69, 70, 35, 64, 15, 25, 37, 30J. For effects that are identifiable through causal functionals that are not necessarily of the backdoor- (e.g., frontdoor,
'This structure is named after Judea Pearl and is central topic in his Book of Why (BoW), where it is also called the "Ladder of Causation" [59]. For more technical discussion the PCH, we refer readers [5].
"The full inferential challenge is, in practice, more general since an agent may be able to perform interventions and obtain samples from a subset of the PCH's layers, while its goal is to make inferences about some other parts of the layers [7, 46, 5]. This situation not uncommon in RL settings [66, 17, 44, 45]. Still, for the sake of space and concreteness, we will focus on two canonical and more basic tasks found in the literature.
We defer more formal discussion on how neural models could be used to assess the effect of interventions Sec. 2. Still, this is neither attainable in all universal neural architectures nor trivially implementable.
*Pearl shared a similar observation in the BoW (59, p. 32]: "Without the causal model, we could not go from rung (layer) one to rung (layer) two. This is why learning systems (as long as they use only rung- rung-one data and do not have causal model) will never be able to answer questions about interventions (...)".
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napkin). other opúmizztioon'satisiicii techniques can be employed that enjoy properties such as double robustness and debiasedness [32, 33, 34]. Each of these approaches optimizes a particular estimand corresponding to one specific target interventional distribution.
Despite all the great progress achieved so far, it is still largely unknown how perform the tasks of causal identification and estimation in arbitrary settings using neural networks as a generative model, acting as a proxy for the true SCM .MM'. It is our goal here develop a general causal-neural framework that has the potential to scale to real-world, high-dimensional domains while preserving the validity of its inferences, as in traditional symbolic approaches. In the same way that the causal diagram encodes the assumptions necessary for the do-calculus to decide whether a certain query is identifiable, our method encodes the same invariances as an inductive bias while being amenable to gradient-based optimization, allowing us to perform both tasks in an integrated fashion (in a way. addressing Pearl's 's concerns alluded to in Footnote 4). Specifically, our contributions are as follows:
1. [Sec. 2] We introduce a special yet simple type of SCM that is amenable gradient descent called a neural causal model (NCM). We prove basic properties of this class of models, including its universal expressiveness and ability encode inductive bias representing certain structural invariances (Thm. 1-3). Notably, we show that despite the NCM's expressivity, it still abides by the Causal Hierarchy Theorem (Corol. 1).
2. [Sec. 3) We formalize the problem of neural identification (Def. 8) and prove a duality between identification in causal diagrams and in neural causal models (Thm. 4). We introduce an operational way to perform inferences in NCMs (Corol. 2-3) and sound and complete algorithm to jointly train and decide effect identifiability for an NCM (Alg. I, Corol. 4).
3. [Sec. 4] Building on these results, we develop a gradient descent algorithm to jointly identify and estimate causal effects (Alg. 2).
There are multiple ways of grounding these theoretical results. In Sec. 5, we perform experiments with one possible implementation which support the feasibility of the proposed approach. All appendices including proofs, experimental details, and examples can be found in the full technical report [68].
1.1 Preliminaries
In this section, we provide the necessary background to understand this work, following the presenta- tion in [58]. An uppercase letter X indicates a random variable, and a lowercase letter x indicates its corresponding value; bold uppercase X denotes a set of random variables, and lowercase letter its corresponding values. We use Dx to denote the domain of X and Dx = Dx, X .... x Dxx for set of random variables X and P(X = x) as the probability of X being equal to the value of x under the distribution P P(X). For simplicity,
X = [X ) We denote P(X) as a probability distribution
will mostly abbreviate P(X = x) simply P The basic semantic framework of our analysis rests on structural causal models (SCMs) [58, Ch. 7], which are defined below.
Definition (Structural Causal Model (SCM)). An SCM M is a 4-tuple (U, V, P(U)), where U is a set of exogenous variables (or "latents") that are determined by factors outside the model; V is a set (V1, Vi,..., Vn) of (endogenous) variables of interest that are determined by other variables in the model - that is, in U VU F set of functions ffv,,f2. mapping from (the respective domains of) Uv. U Pay, to V,, where Uv, U, Pav, CV) \ V,, and the entire set F forms a mapping from U to V. That is, for i = l,.... each fi E F is such that v < (pay,, uy;); and P(u)) is a probability function defined over the domain of U.
fv. such that each fi is a
Each SCM M induces a causal diagram G where every V, E V is vertex, there is directed arrow (Vj V.) for every V42 V and V3 E Pa(V)), and there is a dashed-bidirected arrow (V, +----+ Vi) for every pair Vi, V3 E V such that Uv, and Uy, are not independent. For further details on this construction, see [5, Def. 13/16, Thm. 4]. The exogenous Uv, are not assumed independent (i.e. Markovianity does not hold). We will consider here recursive SCMs, which implies acyclic diagrams, and that the endogenous variables (V) are discrete and have finite domains.
We show next how an SCM M gives values to the PCH's layers; for details on the semantics, see [5, Sec. 1.2]. Superscripts are omitted when unambiguous.
Definition 2 (Layers 1, 2 Valuations), An SCM M induces layer L2(M), a set of distributions over V, one for each intervention x. For each Y C V.
PMM(yx)
Σ
P(u),
(u)Y)
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where Yx (u) is the solution for Y after evaluating Fx (fv : V. The specific distribution P(V). where X is empty, defined as layer (MM.
vixju[fx + -2:XeX
In words, an external intervention forcing a set of variables X to take values x modeled by replacing the original mechanism fx for each X E X with its corresponding value in x. This operation is represented formally by the do-operator, do(X = x), and graphically as the mutilation procedure. For the definition of the third layer, L2(M), see Def. 9 in Appendix A or [5, Def. 7).
2 Neural Causal Models and the Causal Hierarchy Theorem
In this section, aim to resolve the tension between expressiveness and learnability (Fig. 1). To that end, we define a special class of SCMs based on neural nets that is amenable to optimization and has the potential as proxy for the true, unobserved SCM M
Definition 3 (NCM). A Neural Causal Model (for short, NCM) (()) over variables V with parameters {0v; Vi E V) is an SCM (Ū, V, P(Ū)) such that
Ú g (Ūc C C V), where each Ü is associated with some subset of variables C S V, and Dj = [0, 1J for all Ö E Ü. (Unobserved confounding is present whenever C| 1.) f 1fv. V; V). where each fv. is feedforward neural network parameterized
by Ov. E 0 mapping values of Uv. U Pay, to values of VÚ for some Pay, S V and
Uv = (Ūc Uc E U.V. E C};
P(Ū) is defined s.t. Ú ~ (0, 1) for each 0 Ü.
There number of remarks worth making at this point.
1. [Relationship NCM - SCMJ By definition, all NCMs are SCMs, which means NCMs have the capability of generating any distribution associated with the PCH's layers.
2. [Relationship SCM ~ NCM] On the other hand, not all SCMs are NCMs, since Def. 3 dictates that follows uniform distributions in the unit interval and F are feedforward neural networks.
[NonMMakoviianity] For any two endogenous variables V; and Vj, it is the case that Uv; and Uv, might share an input from U. which will play a critical role in causality, not ruling out a priori the possibility of unobserved confounding and violations of Markovianity.
4. [Universality of Feedforward Nets] Feedforward networks are universal approximators [14, 26] (see also [19)), and any probability distribution
be generated by the uniform one (e-g., see probability integral transform [1]). This suggests that the pair (F, P(Ū)) may be expressive enough for modeling ' mechanisms F and distribution P(U) without loss of generality.
[Generalizations Other Model Classes] The particular modeling choices within the definition above were made for the sake of explanation, and the results discussed here still hold for other, arbitrary classes of functions and probability distributions, as shown in Appendix D.
To compare the expressiveness of NCMs and SCMs, we introduce the following definition.
Definition 4 ConsistCons)) Consider two SCMs, Mi and M2. M2 is said be p(L:). consistent (for short, I4-consistent) w.r.t. Mi if L (Mi)) = Tx((22).
This definition applies to NCMs since they are also SCMs. As shown below, NCMs can not only approximate the collection of functions of the true SCM MM but they can perfectly represent all the observational, interventional, and counterfactual distributions. This property is, in fact, special and not enjoyed by many neural models. (For examples and discussion, see Appendix C and D.1.)
Theorem (NCM Expressiveness). For any SCM M = (U, V.F,F,()), there exists an NCM M(()) = (Ū, v, P(Ū)) s.t. M is L3-consistent w.r.f. M
Thm. ascertains that there is no loss of expressive power using NCMs despite the constraints imposed over its form, i.e., NCMs are expressive as SCMs. One might be tempted surmise, therefore, that an NCM can be trained on the observed data and act as a proxy for the true SCM M and inferences about other quantities of M can be done through computation directly in M. Unfortunately, this almost never the case:
Multiple examples of this phenomenon discussed in Appendix C.I and [5, Sec. 1.2]

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Corollary (Neural Causal Hierarchy Theorem (N-CHT)). Let Qt" and QI be the sets of all SCMs and NCMs, respectively. We say that Layer of the causal hierarchy for NCMs collapses to Layer i
(i < j) relative to M E Di if (MM") (M) implies that [,(M*) = (M) for all M E Q Then, with respect to the Lebesgue measure over (a suitable encoding of L L-gequivalence classes of)
SCMs, the subset in which Layer j of NCMs collapses to Layer i has measure zero.
This corollary highlights the fundamental challenge of performing inferences across the PCH layers
even when the target object (NCM M) is a suitable surrogate for the underlying SCM M in terms of expressiveness and capability of generating the same observed distribution. That is, expressiveness does not mean that the learned object has the same empirical content as the generating model. For concrete examples of the expressiveness of NCMs and why it is insufficient for causal inference, see Examples I and 2 in Appendix C.I. Thus, structural assumptions are necessary to perform causal inferences when using NCMs, despite their expressiveness. We discuss next how to incorporate the necessary assumptions into an NCM to circumvent the limitation highlighted by Corol.
2.1 A Family of Neural-Inneeventional Constraints (Inductive Bias)
In this section, we investigate constraints about MM' that will narrow down the hypothesis space and possibly allow for valid cross-layer inferences. One well-studied family of struc- tural constraints comes in the form of a pair comprised of a collection of interventional dis- tributions P and causal diagram G, known as causal bayesian network (CBN) (Def. 15; see also [5, Thm. 4J)). The diagram G encodes constraints over the space of interventional distributions P which are useful perform cross-layer inferences (for details, see Appendix C.2). For simplicity, we focus on interventional inferences from observational data. To com- pare the constraints entailed by distinct SCMs, we define the following notion of consistency:
Definition 5 (G-Consistency), Let G be the causal diagram induced by SCM MM.. For any SCM M. we say that M is G-consistent (w.r.. M*) if G is a CBN for (2M).
Unobserved Nature/Truth
Leamedd Hypothesized
SCM M (JF', P(U")
Cund
NOM V R.P. P(Ú)
In the context of NCMs, this means that M would impose the same constraints over P as the true SCM M (since G is also a CBN for L2(M") by [5. Thm. 4)). Whenever the corresponding diagram g is known, one should only consider NCMs that are G-consistent. We provide below a systematic way of constructing G-consistent NCMs.
G
G.Coeasiis
PCH:
4
C
The Lh.., contains the true SCM M that induces PCH's three layers. The r.h.s. contains an NCM that is trained with layer I data. The matching shading indicates that the two models agree with respect to L while not necessarily agreeing in layers 2 and 3. The causal diagram G entailed by M is used as inductive bias for M.
Figure 2:
Definition 6 (C**-Component For a causal diagram G. a subset C C V is complete confounded component (for short, c²=component) if any pair V., Vy E C is connected with a bidirected arrow in and is maximal (i.e. there is C2-component C' for which C C C.))
Definition 7 (g C--Consrrinee NCM (constructive)).
Let 9 be the causal diagram induced by SCM M Construct NCM M as follows. (1) Choose Ô s.t. Üc E Ú if and only if C is a C2-component in G. (2) For each variable V/2 V, choose Pay, C V s.t. for every V, E V, V, E Pay, if and only if there is a directed edge from V, to V; in G. Any NCM in this family is said to be G-constrained.
Note that this represents a family of NCMs, not unique one, since (the parameters of the neural networks) are not yet specified by the construction, only the scope of the function and independence relations among the sources of randomness (Ü). In contrast to SCMs where both (F, P(u)) can freely vary, the degrees of freedom within NCMs come from 0."
'Otherwise, the causal diagram be learned through structural learning algorithms from observational data [65, 61] or experimental data [41, 40. 27]. See the next footnote for neural take on this task.
'There is growing literature that models SCMs using neural nets as functions, but which differ in nature and scope work. Broadly, these works assume Markovianity, which entails strong constraints over P(U)
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We show next that an NCM constructed following the procedure dictated by Def. 7 encodes all the constraints of the original causal diagram.
Theorem 2 (NCM G- G-Consistency). Any G-constrained NCM M() is G-consistent.
We show next the implications of imposing the structural constraints embedded in the causal diagram. Theorem 3 (L2-ÿ Representation) For any SCM M that induces causal diagram G, there exists G-constrained NCM (0) (U,V V F, P(Ū)) that is Logconsistent w.r.l. M
The importance of this result stems from the fact that despite constraining the space of NCMs to those compatible with G, the resultant family is still expressive enough represent the entire Layer of the original, unobserved SCM .M..
Fig. 2 provides mental picture useful understand the results discussed far. The true SCM M generates the three layers of the causal hierarchy (left side), but in many settings only observational data (layer 1) is visible. An NCM M trained with this data is capable of perfectly representing L (right side). For almost any generating MM sampled from the space S2,, there exists an NCM M that exhibits the same behavior with respect to observational data (M is -i-cosiistent) but exhibits a different behavior with respect to interventional data. In other words, I, underdetermines T2. (Similarly, L, and 122 underdetermine L3 [5, Sec. 1.3].) Still, the true SCM M also induces a causal diagram & that encodes constraints over the interventional distributions. If we use this collection of constraints as inductive bias, imposing G-consistency in the construction of the NCM, M may agree with those of the true M under some conditions, which we will investigate in the next section.
3
The Neural Identification Problem
We now investigate the feasibility of causal inferences in the class of G-constrained NCMs. The first step is to refine the notion of identification [58, pp. 67] to inferences within this class of models.
Definition (Neural Effect Identification). Consider any arbitrary SCM MM and the corresponding causal diagram G and observational distribution P(V). The causal effect P(y | do(x)) is said to be
ccural-identifiable from the set of G-constrained NCMs DGG) and observational distribution P(V)
if and only if pM (y do(x)) ( do(x)) for every pair of models M1, M2 E !(G) s.t.
P.M" (V) = Mi (V) = Ma (V).
In the context of graphical identifiability [58, Def. 3.2.4] and do-calculus, an effect is identifiable if any SCM in Ω² compatible with the observed causal diagram and capable of generating the observational distribution matches the interventional query. If we constrain our attention to NCMs, identification in the general class would imply identification in NCMs, naturally, since needs to hold for all SCMs. On the other hand, it may be insufficient to constrain identification within the NCM class, like in Def. 8, since it is conceivable that the ef- fect could match within the class (perhaps in a not very expressive neural architecture) while there still exists an SCM that generates the same observational distribution and induces the same diagram, but does not agree in the interventional query; see Example 7 in Appendix C. The next result shows that this never the case with NCMs, and there is no loss of generality when deciding identification through the NCM class.
Strectural Assumption
G
M
PVV) (VV)-A())- A(V
Pitylda(x))
Paly/do(x)),
Inserveniinal
(21) Distpuutoo
(E2) Disglaaions
Figure 3: P(y do(x)) is identifiable from P(V) and ((G) if for any SCM M E Ω and NCMs M,, My E Q (top left), Mr, M2, match in P(V) (bot- tom left) and ç (top right), then the NCMs M, M2 also match in P(y do(x)) (bot- tom right).
and, in the context of identification, implies that all effects are always identifiable; see Corol. 3. For instance, [21] attempts to learn the entire SCM from observational (L,) data, while [8, 10] also leverages experimental (L2) data. On the inference side, [42] focuses estimating causal effects of labels images.
*This is akin to what happens with the non nen-neural CHT (5, Thm. 1J and the subsequent use of causal diagrams to encode the necessary inductive bias, and in which the do-calcula allows for cross-layet inferences directly from the graphical representation [5, Sec. 1.4].

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Theorem& 4 (Graphical-Neunl Equivalence (Dual ID),, St" be the set of all SCMs and St the set of NCMs. Consider the true SCM M and the corresponding causal diagram G. Let = P(y do(x)) be the query of interest and P(V) the observational distribution. Then, Q is neural identifiable from ((g) and P(V) if and only if is identifiable from 9 and P(V).
In words, Theorem 4 relates the solution space of these two classes of models, which means that the identification status of a query is preserved across settings. For instance, if an effect is identifiable from the combination of causal graph G and P(v), it will also be identifiable from G-constrained NCMs (and the other way around). This is encouraging since our goal is to perform inferences directly through neural causal models, within (G)), avoiding the symbolic nature of do-calculus computation; the theorem guarantees that this is achievable in principle.
Corollary 2 (Neural Mutilation (Operational ID)). Consider the true SCM M E S2,, causal diagram G. the observational distribution P(V). and target query Q equal to PM' (y do(x)). Let M E (i(g) be G-constrained NCM that is L concossist with M IfQ is identifialee from g and P(V), then Q is computable through a mutilation process on proxy NCM M, i.e., for each X X, replacing the equation fz with constant x (Q PROC-MUTILATION M: X x, Y)).
Following the duality stated by Thm. 4, this result provides a practical, operational way of evaluating queries in NCMs: inferences may be carried out through the process of mutilation, which gives semantics to queries in the generating SCM M (via Def. 2). What is interesting here is that the proposition provides conditions under which this process leads to valid inferences, even when M is unknown, or when the mechanisms F and exogenous distribution P(U) of MM and the corresponding functions and distribution of the proxy NCM M do not match. (For concreteness, refer to example 5 in Appendix. C.) In words, inferences using mutilation on M would work as if they were on M itself, and they would be correct so long as certain stringent properties were satisfied - -onosiiittn G-constraint, and identifiability. As shown earlier. if these properties are not satisfied, inferences within a proxy model will almost never be valid, likely bearing no relationship with the ground truth. (For fully worked out instances of this situation, refer to examples 2, 3, or 4 in Appendix C).
Still, one special class of SCMs in which any interventional distribution is identifiable is called Markovian, where all U, are assumed independent and affect only one endogenous variable V,.
Corollary 3 (Markovian Identification). Whenever the G-constrained NCM M is Markovian, P(y do(x)) is always identifiable through the process of mutilation in the proxy NCM (via Corol. 2).
This is obviously not the case for general non- Markovian models, which leads to the very problem of identification. In these cases, we need to decide whether the mutilation procedure (Corol. 2) can, in principle, produce the correct answer. We show in Alg. a learning procedure that decides whether a certain effect is identi- fiable from observational data. Intuitively, the procedure searches for two models that respec- tively minimize and maximize the target query while maintaining L Li-consistency with the data distribution. If the L.2 query values induced by the two models are equal, then the effect is identifiable. and the value is returned; otherwise, the effect con-identifiable. Remarkably, the procedure both necessary and sufficient, which means that all, and only, identifiable effects are classified as such by our procedure. This implies that, theoretically, deep learning could be as powerful as the do-caleulus in deciding identifiability. (For a more nuanced discussion of symbolic versus optimization- based approaches for identification, see Appendix C.4. For non- con-identifiaiilityy examples and further discussion, see C.3.)
Algorithm 1: Identifying/essimatinng queries with NCMs.
Input causal query Q = Ply do(x)), Ln data P(V), and cansal diagriam 9
Output: P.M (y do(x)) if iideetiifable,, FAIL otherwise.
// from Def. 7 6 arg ming prr(() (y [do(x)) st. L. (M(()
M KCH(V. ;)
0imnn earg maxe pMi0) do(x)) s... L, (M (0)) P(V)
if phrteminl do(x)) zf p (y| do(x)) then
return FAIL
else
return P prrto do(x)) // choose min Bax
arbitrarily
Corollary 4 (Soundness and Completeness). Let Ωn be the set of all SCMs, M E Q2 be the true SCM inducing causal diagram G. Q P(y do(x)) be a query of interest, and Q be the result from running Alg. I with inputs p* P*(V) = Li(M)) > 0, G. and Q Then Q is identifiable from & and Pt (V) if and only if Q is not FAIL. Moreover, if Q is not FAIL, then Q = pM' (y do(x)).
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4 The Neural Estimation Problem
While identifiability is fully solved by the asymptotic theory discussed fär (i.e., it is both necessary and sufficient), we now consider the problem of estimating causal effects in practice under imperfect optimization and finite samples and computation. For concreteness, case with binary variables, but our construction extends naturally to categorical and continuous variables (see Appendix B). We propose next construction of G-constrained NCM (G; 0) = (Ü, V. F, P(U)), which is possible instantiation of Def. 7:
discuss next the discrete
V
V, U == (Ve : C E (9()) U (Gv, : Vi E V).
log (dv, (pay, u 0v;))
i == 1
F
= fv. == arg m0x 93,, +
llog(1 - (Øv, (pay, u dv,)))
j = 0f (2)
P(Ù)
{Uc (n, 1): Uc E U
[G ~ Gumbel 1): e. (0,1)),
where are the nodes G; a: R + (0, 1) is the sigmoid activation function; (²()) is the set of Ce-components of 9; each Gj,,, is standard Gumbel random variable [24]; each Φv (:: fv, is neural net parameterized by By, E 0: pay, are the values of the parents of Vi; and uy; are the values of V == (Vc : Uc E U s.t. V E C). The parameters are not yet specified and must be learned through training enforce L coonsisstyy (Def. 4).
Let U and G denote the latent C componeetttt variables and Gumbel random variables, respectively. To estimate PM(v) (v) and pM (y do(x)) given Eq. 2. we may compute the probability mass of datapoint with intervention do(X x) (X is empty when observational) as:
pāt (v do(x))
E r(ue)
II VEVIX
1
m
II
T
(3)
VEVIX
To(o:(pay ug,
U/ and {ug} are samples from P(U9). Here, Us =
where T. 1 (6. u Bv))) assume consistent with (the values of X E X in match the corresponding ones of x). Otherwise, PM(G:0) (v do(x)) = 0. For numerical stability of each (x(+), we work in log-space and use the log-sum-exp trick.
Alg. (lines 2-3) requires non-trivial evaluations of expressions like arg maxę pM (y do(x)) while en- forcing Li Li-consistency. Whenever only finite sam- ples are available (vx === ~ (V).), the parameters of an conssisttt NCM may be estimated by min-
Algorithm 2: Training Model
Input
Data 77 variables V. X V.
x Dx. Y Y. Dy. causal diagram G.
number of Monie Carlo sumples regulatization
constant A. learning rate
MJ ECN(V, G)
11 from Def. 7
imizing data negative log-likelihood:
Initialize parameters Omiu Omnx 3 for k +- to n. do
// Estinate from Eq.
arg min E (v) [ log o pAo(0:0))))
Pmii +- Estinate ( (M (0miin) ) V. v4. 0,0, m))
Pmax Eetinste (M (max)), V, m)
4min +-1
n
Imax
arg min log
for Dv do
0
72
Consistent(v. y) then
10
4min 8min
Eatinata (0min)) V, v, X, x, m))
To simultaneously maximize pM (y do(x)),
11
dmax + Umax
subtract a weighted second term log PM
m (y do(x)),
Estinate (0 m.
m))
resulting in the objective C([v=) equal to
/f from Eq. 5
log Punin
Alog(1 Auin))
13
Lmax +- - log Pmax A log Auaa
fM log PM (vx) - Alog m (y do(x)), (5)
Omin 0mi + 1/ Lmin
15
Omxx +- Omax + 7V Lmax
where À is initially set to a high value and decreases during training. To minimize, instead subtract A Alg((1 PM (y do(x))) from the log-likelihood.
8

---

Identifiable
Non-Identifiable
(c)
2000 30003 2000
2000 3300
3500
1308
3000
3500
35000
1000
300
Figure 4: Experimental results on deciding identifiability with NCMs. Top: Graphs from left to right: (ID cases) back-door, front-door, M. napkin; (not ID cases) bow, extended bow, IV, bad M. Middle: Classification accuracy over 3,000 training epochs from running hypothesis test on Eq. with 0.01 (blue). 0.03 (green), 0.05 (red). Bottom: (1, 5, 10, 25, 50, 75, 90, 95, 99)-percenssles for max max-min gaps over 3000 training epochs.
Alg. 2 is one possible way of optimizing the parameters A required in lines 2,3 of Alg. 1. Eq. is amenable to optimization through standard gradient descent tools, e.g., [38, 51, 50].
9
One way of understanding Alg. 1 is search within the S2(G) space for two NCM parameterizations, the interventional distribution, respectively. Whenever the
Onia and iaax. that minimizes/maximinnn optimization ends, we can compare the corresponding P(y do(x)) and determine whether an effect identifiable. With perfect optimization and unbounded resources, identifiability entails the equality between these two quantities. In practice, rely hypothesis testing step such as
(e - f (M
<
(6)
for quantity of interest f and certain threshold 7. This threshold is somewhat similar a significance level in statistics and can be used to control certain types of errors. In our case, the threshold can be determined empirically. For further discussion, see Appendix B.
5 Experiments
We start by evaluating NCMs (following Eq. 2) in their ability to decide whether an effect is identifiable through Alg. 2. Observational data is generated from different SCMs, and their corresponding causal diagrams are shown in Fig. 4 (top part), and Appendix B provides further details of the parametrizations. Since the NCM does not have access to the true SCM, the causal diagram and generated datasets are passed to the algorithm to decide whether an effect is identifiable. The target effect P(Y | do) and the quantity we optimize is the average treatment effect (ATE) of X on Y. T.ATEM(X. Y)= EM do(X =]] E do(X 0)}. Note that if the outcome Y is binary, as in our examples, E[Y do(X = x)] = P(Y = 1 (X = x)). The effect is identifiable through do-calculus in the settings represented by Fig. 4 in the left part, and not identifiable in right.
The bottom row of Fig. shows the max-min gaps, the hh.s of Eq. with f(M) A Y), over 3000 training epochs. The parameter À is set to 1 at the beginning, and decreases logarithmically over each epoch until it reaches 0.001 at the end of training. The max-min gaps can be used to classify the quantity as "ID" or "no--ID" using the hypothesis testing procedure described in Appendix B. The classification accuracies per training epoch are shown in Fig. 4 (middle row). Note that in identifiable settings, the gaps slowly reduce to 0 while the gaps rapidly grow and stay high throughout training in the unidentifiable ones. The classification accuracy for ID cases then gradually increases as training
Our approach is flexible and may take advantage of these different methods depending on the context. There are number of alternatives for minimizing the discrepancy between / and ph including minimizing diver- gences, such as maximum mean discrepancy [23] or kernelized Stein discrepancy [49], performing variational inference [9]. or generative adversarial optimization [20].
10hh NCM can be extended to the continuous case by replacing the Gumbel-max trick on (()(-)) with a model that directly computes probability density given data point, e.g., normalizing flow [62] or VAE [39].

---

progresses, while accuracy for non-ID cases remain high the entire time (perfect in the bow and IV cases).
In the identifiable settings, we also evaluate the performance of the NCM estimating the correct
(N)
(b)
(9
(4)
Eis
causal effect, as shown in Fig. 5. As a generative model, the NCM is capable of generating samples from both P(V) and identifiable L2 distributions like P(Y do(X)). We compare the NCM naïve generative model trained via likelihood maximization fitted on P(V) without using the inductive bias of the NCM. Since the naive model is not defined to sample from Pyy do(x)), this shows the implications of arbitrarily choosing P(y do(x)) = P(y æ²). Both models improve fitting P(V) with more samples, but the naive model fails to learn the correct ATE except in case (c), where P(y do(x)) = P(y x). Further, the NCM is competitive with WERM [33], a state- of the-art estimation method that directly tar- gets estimating the causal effect without generating samples.
Figure 5: NCM estimation results for ID cases. Columns a, b, c, d correspond to the same graphs as a, b, c, d in Fig. 4. Top: KL divergence of P(V) induced by naïve model (blue) and NCM (orange) compared to PM (V). Bottom: MAE of ATE of naive model (blue), NCM (orange), and WERM (green). Plots in log-log scale.
6 Conclusions
In this paper, we introduced neural causal models (NCMs) (Def. 3, 18), special class of SCMs trainable through gradient- based optimization techniques. We showed that despite being as expres- sive as SCMs (Thm. 1), NCMs are unable to perform cross-layer inferences in general (Corol. 1). Disentangling expressivity and learnability, we formalized a new type of inductive bias based on non- parametric, structural properties of the generating SCM, accompanied with a constructive procedure that allows NCMs to represent constraints over the space of interventional distributions akin to causal diagrams (Thm. 2). We showed that NCMs with this bias retain their full expressivity (Thm. 3) but are now empowered to solve canonical tasks in causal inference, including the problems of identification and estimation (Thm. 4). We grounded these results by providing training procedure that is both sound and complete (Alg. 1, 2, Cor. 4). Practically speaking, different neural implementations combination of architectures, training algorithms. loss functions can leverage the framework results introduced in this work (Appendix D.I). We implemented one of such alternatives as a proof of concept, and experimental results support the feasibility of the proposed approach. After all, we hope the causal-neura framework established in this paper can help develop more principled and robust architectures empower the next generation of AI systems. We expect these systems to combine the best of both worlds by (I) leveraging causal inference capabilities of processing the structural invariances found in nature construct more explainable and generalizable decision-making proce- dures, and (2) leveraging deep learning capabilities to scale inferences to handle challenging, high dimensional settings found in practice.
Acknowledgements
We thank Judea Pearl, Richard Zemel, Yotam Alexander, Juan Correa, Sanghack Lee, and Junzhe Zhang for their valuable feedback. Kevin Xia and Elias Bareinboim were supported in part by funding from the NSF, Amazon, JP Morgan, and The Alfred P. Sloan Foundation. Yoshua Bengio was supported in part by funding from CIFAR, NSERC, Samsung, and Microsoft.
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14


Reviews:
[{"review_id": "zxuuZK1d4z-", "reviewer": "Reviewer_ue4r", "summary": "The authors investigate the connection between structural causal models and neural networks from an learning perspective. Their theoretical results emphasize, notably, that the impossibility to learn neural networks that are interventionally equivalent to the true SCM based on observations only. The paper provides an interesting perspective on the connection between causal inference and learning with neural network, although clarity could be improved, notably regarding the limitations and the contribution with respect to previous work.", "questions": "", "limitations": "", "rating": 6, "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": "While the overall effort of this paper is clearly valuable, I struggled with the presentation of the theoretical results. It might be partly due to my partial understanding of the framework, although I think the main text lacks some key information.\n1/ First, only by going through the appendix one can notice that the theory applies to variables in countable numerical domains (lemma 4). Besides, I tend to think this is not fully accurate, as the proof of lemma 4 seems to assume *finite* domains.\nWhile one can argue that binary/finite domains can approximate continuous ones with arbitrary precision, it is important to specify it in the main text, as results would appear very confusing to readers familiar with technical issues, while unfamiliar readers who see the use of neural networks would naturally assume every functions defined can act on, say, Euclidean domains.\n2/ It is confusing to have theorem 1 on L3 consistency while this property is not defined in main text. An informal presentation of Li would help.\n3/ Corollary 1 conveys an important point, but only to those not familiar with causal inference (pointing to structural identifiability issues), for whom the level of abstraction will be difficult to parse. On the other hand, I tend to think that the setting of this result is too “coarse” to capture when causal inference is useful (and I assume the authors do agree with me that sometimes it is). If I did not miss anything, if we restrict ourselves to an SCM with two nodes, and we add additional information on the causal graph (e.g. uncounfounded, and the direction of causation in known), then we are in good shape to be able to collapse the interventional level to the observational one. This kind of positive result is not captured by the corollary one because a) we consider “all SCMs” at once, thus ignoring the informativeness of certain graph structures, b) we do not include the possibility to incorporate assumptions on the graph structure at inference, which is key in practice. I do get that the next sections address this issue, but I question the benefit this side result. Overall, I tend to think the target readers would benefit more from a high-level discussion, possibly in relation to the examples in supplemental, that from such abstract result (besides largely not self-contained in the paper). \n4/ Overall, it is difficult to disentangle the contribution of the paper with respect to what is already known regarding learnability of SCMs. Once we have established the ability of neural networks to perfectly fit functions with finite domains (which can be found in the appendix, and are typically not too challenging to show), classical result from causality seem to apply as usual. I certainly may have missed key novel results that are fundamentally different from previous work, but they are not really emphasized in main text. It looks like the lemmas in appendix reflect better the gist of the contribution than the main text results. \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Assumptions of finite domain seems to be central for the proofs, but does not seem to be mentioned in main text, while only a countability assumption is mentioned in only one lemma in supplemental, although finite domain seems to be exploited in the proof. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4 hours"}, "extra_text": {}}, {"review_id": "u6k3JH5CUor", "reviewer": "Reviewer_Amv3", "summary": "The paper considers a restricted class of SCMs (structural causal models) where each variable in a graph is generated by applying a neural-network-parameterized function of the variable's parents, called Neural Causal Models (NCMs). The paper shows a wealth of results mapping the relationship between the full class of SCMs and the smaller class of NCMs in terms of when and how identification holds and how estimation of the interventional distribution can be achieved. The method proposed to identify and estimate effects takes a graph as an input, computes the maximum and minimum of the effect of a particular intervention within the class of NCMs that are consistent with the observed data distribution (at the lowest level in the causal hierarchy, observational); if the maximum and minimum match, the effect is identified. The method is evaluated on a few experiments.", "questions": "", "limitations": "", "rating": 7, "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": "The paper is well-written and puts forth an interesting perspective of estimating interventional distributions. However, it is unclear why the notion of neural networks is central here in theory. I understand that practically there are appealing properties in terms of optimization and techniques used to quickly build good predictors using neural networks. However, all of the theory about NCMs being an interesting class of causal models, i.e. with identification and estimation, holds for any sufficiently flexible class of functions (lemmas 4,5 are constructive and nice but any large enough function class contains the constructed functions). \n\nI think the distinction above is important because under this perspective, the main claim of the paper is that neural networks help learn interventional distributions better than other existing causal effect estimation methods that do not rely on the machinery of the NCMs; for example, one could check causal identifiability based on the given graph and then apply causal effect estimation methods like the following: like https://arxiv.org/abs/2102.12034 and https://arxiv.org/abs/2001.07426. \n\nAs the theory holds for any flexible class of functions, it seems to me that the experiments should show that the proposed estimation method is better than other baseline methods as suggested above.  As these baselines are not included, the experiments, while encouraging, are insufficient to establish the improvement that the proposed method (and NCM-based methodology) brings to causal effect estimation.\n\n\n--- UPDATES after rebuttal\n\nscore updated after clarifying the position of the paper.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I am concerned about the applicability of the proposed method to estimation of interventional distributions in general seeing that the experiments are limited. It is unclear why WERM is the only (non-naive) method compared against and how it is state-of-art. If the authors could explain why no other baselines were considered, that would help clarify the standing of the paper; for example, compare against interventional distributions using knowledge of the graph and effect estimation methods like like https://arxiv.org/abs/2102.12034 and https://arxiv.org/abs/2001.07426. ", "review_text": "", "extra_scores": {"ethical_concerns": "None", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "gp4vfU2S6gh", "reviewer": "Reviewer_kLWB", "summary": "This paper studies the expressivity, identification, estimation of causality with neural models. For expressiveness, it shows that Neural Causal Model (NCM) is expressive enough to represent any structural causal model (SCM), but in general, given observation data alone, without further constraints, the NCM is not able to identify higher layers of Pearl Causal Hierarchy (PCH). The authors then develop a necessary and sufficient condition to determine whether a causal effect can be learned from data (i.e., causal identifiability); if identifiability holds, it also estimates the effect from data (causal estimation). Empirical experiments on a simple synthetic dataset support the theoretical results.", "questions": "", "limitations": "", "rating": 7, "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": "In terms of novelty, to my knowledge, the theory and the algorithms introduced are novel. \n\nIn terms of quality, I did not check through the theoretical proof in the Appendix, but I believe they are correct. For the experiments, they are evaluate on simple scenarios which demonstrate the effectiveness of the algorithm's capability of causal identification and estimation. I wonder how well the algorithm is able to scale to more complex scenarios where the observed number of variables and their edges are larger, and its run-time compared to training a standard network of similar structure. Furthermore, for the neural architecture, it would offer more insight if the authors can perform experiments that test if and how the expressivity of the neural models (number of layers, number of neurons per hidden layer, etc.) affects the results, and how many examples are needed for confident identification and estimation.\n\nIn terms of clarity, the paper is very theory-heavy. To improve intuition, if possible, it may be useful to use a running example, starting from the beginning, to accompany the proved theorems, so as to make it more friendly for readers not in the causal community. For example, the examples in Appendix C may be good examples.\n\nOverall, I think this paper is well-written and of high-quality, and makes useful contribution to causal identification and estimation with neural methods. If the more thorough experiments are performed, it will further strengthen the paper.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors did not address the limitation of the paper.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5 hours"}, "extra_text": {}}, {"review_id": "Tp3bh0lVYnT", "reviewer": "Reviewer_fTRd", "summary": "In this paper a subclass of SCMs whose structural equations are based on feed-forward NNs are considered.\nThe authors show the expressiveness of this class and address the problem of training those model and evaluated identifiability of a causal task. Experiments on models (with few variables) are promising.", "questions": "", "limitations": "", "rating": 6, "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": "Causal machine learning is definitely an important field of research for the NeurIPS community. This paper offers an interesting class of models to practically achieve that. Considering that NNs are universal approximators, the results in the technical part are not surprising, but I think the paper might really help researchers in this area. The fact that the experiments are only focused on very small models leave some doubts about the practical scalability of these techniques.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Nothing.", "review_text": "", "extra_scores": {"ethical_concerns": "Nothing.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1"}, "extra_text": {}}, {"review_id": "53PKBzX3U7z", "reviewer": "Reviewer_Z2BF", "summary": "A new class of models called Neural Causal Models is proposed which retains causal reasoning properties of Structural Causal Models but can be learned through neural net based methods. contributions include formally proving connections between SCMs and NCMs, disentangling expressiveness with learnability of NCMs and deriving constrained classes of NCMs, a practical learning algorithm for NCMs.", "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": "The paper proposes a new model called Neural Causal model that has the properties similar to a structural causal model but can be learned through gradient-based methods. NCMs are a subset of SCMs but have causal reasoning power like SCMs. Formal properties are proved about the expressive power of NCMs, i.e., they can represent distributions as an SCM. However, this expressiveness does not mean that NCMs are learnable and require additional constraints imposed on the causal diagram. The constraints give rise to a learnable class of NCMs. It is shown that in this class it is possible to perform causal reasoning using observational data. Soundness and completeness results are presented for neural inference in NCMs and it is shown that it is equivalent to symbolic inference in SCMs. A gradient-based training method is presented for approximately learning NCMs.\n\nExperiments are performed that show if an effect can be identified from a cause accurately through NCMs for different causal settings. For non-identifiable effects it is shown that the effects cannot be accurately identified through NCM learning. A generative model through NCM is compared with a generative model that does not use the causal graph (only data-driven) and also WERM, a state-of-the-art causal learning model.\n\nThe paper is well written and covers a lot given the space limits. The clarity of writing makes it accessible even to non-experts in this area I think. The significance of this work seems quite remarkable since it proves several fundamental connections between SCMs and neural nets. I can see that this work can have impact in many other domains such as explainability in AI. A case study of a real-world application would have been beneficial for understanding the impact in a practical setting. Overall, though this seems to be a highly significant paper connecting Neural and Symbolic AI.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Societal impact not directly addressed, but since the paper addresses fundamental connections between symbolic and neural models, I cannot think of any negative societal impact or limitation to be concerned about.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper considers the Pearl causal hierarchy layers, with $L_1$ representing the observations, $L_2$ the interventions and $L_3$ the counter-factuals. 

The contribution establishes that the fact that a powerful model (neural nets in the following (1)) fits the true causal model 
${\cal M}^*$ regarding $L_1$, does not imply that it fits ${\cal M}^*$ regarding $L_2$. The proof, following from [6], is based on that fact that the set of models fitting ${\cal M}^*$ on (at least two) layers has measure 0.


In order to tackle the second layer $L_2$ of the Pearl causal hierarchy, one must consider the causal identification task (predicting the causal diagram (2)) and the causal estimation (predicting the effects of intervention). 

The second contribution is to establish identifiability conditions (including the knowledge of the causal diagram) for an intervention query. 

The algorithmic condition for this identifiability is stated as the fact that the neural models fitting the interventional data and maximizing / minimizing the outcome of the intervention, converge toward each other. 

The significance of these contributions has been challenged and very convincingly argued by the authors. 

Some limitations regarding the domain of the observed variables (discrete and finite) have been discussed and will be stated in the revised version. 

(1) The neural net class is considered in the paper, though the discussion suggests that any class with sufficient expressivity could be considered instead.

(2) The paper claims that there exist no neural methods today focused on solving the causal identification task, which is not exact (though restricted to Markovian), consider https://arxiv.org/abs/1803.04929. 

Decision: Accept (Poster)