The Causal-Neural Connection: Expressiveness, Learnability, and Inference

1 Introduction . One of the most celebrated and relied upon results in the science of intelligence is 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 as input/output – i.e. , 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 is not without merits , and is 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 a 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 , we evoke a class of generative models known as the Structural Causal Model ( SCM , for short ) [ 58 , Ch . 7 ] . In words , an SCMM∗ is a representation of a system that includes a collection of mechanisms and a probability distribution over the exogenous conditions ( to be formally defined later on ) . Second , any fully specified SCMM∗ induces a collection of distributions known as the Pearl Causal Hierarchy ( PCH ) [ 5 , Def . 9 ] . The importance of the PCH is that it formally delimits 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 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 . 1 ] . 1 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 . 1 ( a ) ( left side ) , where SCMM∗ induces layers L∗1 , L∗2 , L∗3 of the PCH . Even though each possible statement within these capabilities has well-defined semantics given the true SCM M∗ [ 58 , Ch . 7 ] , a challenging inferential task arises when one wishes to recover part of the PCH whenM∗ is only partially 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 . For concreteness , consider the setting where one needs to make a statement about the effect of a new intervention ( i.e. , about layer 2 ) , but only has observational data from layer 1 , which is passively collected.2 Going back to the causalneural connection , one could try to learn a neural model N using the observational dataset ( layer 1 ) generated by the true SCMM∗ , as illustrated in Fig . 1 ( b ) . Naturally , a basic consistency requirement is that N should be capable of generating the same distributions asM∗ ; in this case , their layer 1 predictions should match ( i.e. , L1 = L∗1 ) . 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 a proxy , having the capability of predicting the effect of interventions that matches the L2 distribution generated by the true ( unobserved ) SCMM∗ . 3 The answer to this question can not 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 L1 distribution ofM∗ but generate different L2-distributions . 4 Even though N may be expressive enough to fully representM∗ ( as discussed later on ) , generating one particular parametrization of N consistent with L1 is insufficient to provide any guarantee regarding higher-layer inferences , i.e. , about predicting the effects of interventions ( L2 ) or counterfactuals ( L3 ) . 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 unknownM∗ in the form of causal diagrams [ 58 , 65 , 7 ] or their equivalence classes [ 28 , 60 , 29 , 71 ] . The task is then to decide whether there is a unique solution for the causal query based on such assumptions . There are no neural methods today focused on solving this task . The second task – causal estimation – is triggered when effects are determined to be identifiable by the first task . Whenever identifiability is obtained through the backdoor criterion/conditional ignorability [ 58 , Sec . 3.3.1 ] , 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 , 30 ] . For effects that are identifiable through causal functionals that are not necessarily of the backdoor-form ( e.g. , frontdoor , 1This structure is named after Judea Pearl and is a central topic in his Book of Why ( BoW ) , where it is also called the “ Ladder of Causation ” [ 59 ] . For a more technical discussion on the PCH , we refer readers to [ 5 ] . 2The 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 is 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 . 3We defer a more formal discussion on how neural models could be used to assess the effect of interventions to Sec . 2 . Still , this is neither attainable in all universal neural architectures nor trivially implementable . 4Pearl 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 deep-learning systems ( as long as they use only rung-one data and do not have a causal model ) will never be able to answer questions about interventions ( ... ) ” . napkin ) , other optimization/statistical 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 to 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 SCMM∗ . It is our goal here to 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 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 to gradient descent called a neural causal model ( NCM ) . We prove basic properties of this class of models , including its universal expressiveness and ability to encode an 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 a sound and complete algorithm to jointly train and decide effect identifiability for an NCM ( Alg . 1 , 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 presentation 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 x its corresponding values . We use DX to denote the domain of X and DX = DX1 × · · · × DXk for X = { X1 , . . . , Xk } . We denote P ( X ) as a probability distribution over a 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 ( X ) . For simplicity , we will mostly abbreviate P ( X = x ) as simply P ( x ) . The basic semantic framework of our analysis rests on structural causal models ( SCMs ) [ 58 , Ch . 7 ] , which are defined below . Definition 1 ( Structural Causal Model ( SCM ) ) . An SCMM is a 4-tuple 〈U , V , F , 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 , V2 , . . . , Vn } of ( endogenous ) variables of interest that are determined by other variables in the model – that is , in U ∪V ; F is a set of functions { fV1 , fV2 , . . . , fVn } such that each fi is a mapping from ( the respective domains of ) UVi ∪PaVi to Vi , where UVi ⊆ U , PaVi ⊆ V \ Vi , and the entire set F forms a mapping from U to V. That is , for i = 1 , . . . , n , each fi ∈ F is such that vi ← fVi ( paVi , uVi ) ; and P ( u ) is a probability function defined over the domain of U . Each SCMM induces a causal diagram G where every Vi ∈ V is a vertex , there is a directed arrow ( Vj → Vi ) for every Vi ∈ V and Vj ∈ Pa ( Vi ) , and there is a dashed-bidirected arrow ( Vj L9999K Vi ) for every pair Vi , Vj ∈ V such that UVi and UVj are not independent . For further details on this construction , see [ 5 , Def . 13/16 , Thm . 4 ] . The exogenous UVi ’ s 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 SCMM 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 SCMM induces layer L2 ( M ) , a set of distributions over V , one for each intervention x . For each Y ⊆ V , PM ( yx ) = ∑ { u|Yx ( u ) =y } P ( u ) , ( 1 ) where Yx ( u ) is the solution for Y after evaluatingFx : = { fVi : Vi ∈ V\X } ∪ { fX ← x : X ∈ X } . The specific distribution P ( V ) , where X is empty , is defined as layer L1 ( M ) . In words , an external intervention forcing a set of variables X to take values x is modeled by replacing the original mechanism fX for each X ∈ 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 , L3 ( M ) , see Def . 9 in Appendix A or [ 5 , Def . 7 ] .

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.