Variational Causal Autoencoder for Interventional and Counterfactual Queries

We propose the Variational Causal Autoencoder ( VCAUSE ) , a novel class of1 variational graph autoencoders for causal inference in the absence of hidden con-2 founders , when only observational data and the causal graph are available . Without3 making any structural assumption , VCAUSE mimics the necessary properties of4 a Structural Causal Model ( SCM ) to provide a framework for performing inter-5 ventions ( do-operator ) and abduction-action-prediction steps . As a result , and as6 shown by our empirical results , VCAUSE provides a practical and accurate pipeline7 for estimating the interventional and counterfactual distributions of diverse SCMs.8 Finally , we apply VCAUSE to evaluate counterfactual fairness in classification9 problems and also to learn accurate and fair classifiers.10 1 Introduction11 Predicting causal effects of actions ( interventions ) is a central problem in scientific research in a12 broad variety of fields [ 4 , 5 , 7 , 23 , 51 ] , and machine learning is no exception [ 44 ] . As an example,13 fundamental machine learning questions—such as fairness [ 6 , 9 , 19 , 24 , 25 ] and interpretability14 [ 17 ] — , are increasingly being formulated as causal queries.15 Research on causal reasoning has predominantly focused on causal discovery , a.k.a . structure16 learning , aimed at discovering the underlying causal graph from data ( see , e.g. , [ 15 , 30 , 49 , 60 ] ) .17 An alternative line of work instead aims to answer causal queries under different assumptions , e.g.,18 assuming access to partial causal knowledge [ 17 , 18 ] or to a randomized trial [ 16 ] . Here , we focus19 on the latter line of research , that is , on answering the following two types of causal questions:20 interventional queries , e.g. , “ What is the effect of a universal unconditional basic income of 1k21 EUR on the health of the population ? ” ; and counterfactual queries , e.g. , “ Had Kim received an22 unconditional basic income of 1k EUR , what would have been the effect on Kim ’ s health ? ” .23 Unfortunately , predicting causal effects from observational data alone is in general difficult and24 often requires strong and impractical causal assumptions . In this context , the Structural Causal25 Model ( SCM ) [ 39 ] is a framework that allows to answer causal queries from observational data , but26 requires complete causal knowledge . That is , knowledge not only on the parent-children ( cause-effect ) 27 relationship between every pair of observed variables ( i.e. , on the causal graph ) , but also on how28 these relationships are ( i.e. , on the structural equations ) . As a consequence , randomized controlled29 studies are today still considered to be the gold standard for estimating causal effects . Unfortunately,30 real world experiments are often expensive to conduct , unethical , or directly impossible.31 In this work , we aim at answering the above causal queries , when only observational data and32 the causal graph are available . Note that the causal graph can often be inferred from domain33 knowledge [ 62 ] or via one of the numerous approaches for causal discovery [ 27 , 54 ] . We assume34 causal sufficiency , i.e. , that there are no hidden confounders , which are unobserved variables that35 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . affect more than one observed variable . We propose the novel Variational Causal Autoencoder36 ( VCAUSE ) , a variational graph autoencoder that leverages the causal graph structure and yields37 accurate estimates of the observational , interventional and counterfactual distributions induced by an38 unkonwn causal model.39 Importantly , we provide the necessary conditions for the design of the encoder and decoder graph40 neural networks ( GNNs ) , so that the resulting VCAUSE behaves like an SCM . As a result , and41 without making any assumptions on the true structural equations , VCAUSE provides a practical42 framework to perform interventions ( do-operator ) and abduction-action-prediction steps , which are43 necessary to evaluate interventional and counterfactual queries.44 We evaluate the performance of the proposed VCAUSE model in extensive experiments using45 observational data from different SCMs , with diverse causal graphs and structural equations . Our46 experiments show that VCAUSE outperforms competing methods [ 17 , 18 ] at estimating not only47 the mean of the interventional/counterfactual distribution , but also the overall distribution , as shown48 by the quality of its samples ( in terms of Maximum Mean Discrepancy , MMD ) . We finally show a49 use-case in which VCAUSE is used to assess counterfactual fairness of different classifiers trained on50 the German Credit dataset [ 10 ] as well as to learn accurate and counterfactually fair classifiers.51 Related work . There are numerous works on causal discovery [ 15 , 18 , 27 , 30 , 33 , 40 , 49 , 54 , 56,52 58 , 60 , 63 ] . In addition , extensive work focuses on interventional and/or counterfactual queries53 using non-parametric methods [ 1 , 32 , 46 , 47 ] , and more recently , tractable probabilistic models [ 59 ] .54 Moreover , deep generative models are enjoying increasing attention for causal queries in complex55 data [ 31 , 35 ] . Existing approaches focus on estimating the Average Treatment Effect ( ATE ) by56 assuming a fixed causal graph that includes the treatment variable [ 19 , 29 , 42 , 45 , 53 ] ; on discovering57 and intervening on the causal latent structure of the ( e.g. , image ) data [ 19 , 35 , 37 , 48 , 56 ] ; or58 on addressing interventional and/or counterfactual queries by fitting a conditional model for each59 observed variable given its causal parents [ 11 , 17 , 22 , 36 , 38 ] . In the most recent work related60 to ours [ 18 ] , the authors propose CAREFL , an autoregressive normalizing flow ( ANF ) for causal61 discovery and queries , which focuses on bi-variable scenarios with affine relationships between62 observed and unobserved variables . In our experiments , we compare VCAUSE with CAREFL ( as63 well as [ 17 ] ) in more general settings . Finally , up to the best of our knowledge , GNNs have previously64 been used for causal discovery [ 58 , 61 ] , but have not yet been exploited to address counterfactual and65 interventional queries , like VCAUSE does.66 2 Background67 In this section , we first provide a brief overview on structural causal models ( SCMs ) and then68 introduce the main building block of VCAUSE , i.e. , variational graph autoencoders ( VGAEs ) .69 2.1 Structural causal models70 An SCM M = ( p ( U ) , F̃ ) determines how a set of d endogenous ( observed ) random variables X : =71 { X1 , . . . Xd } is generated from a set of exogenous ( unobserved ) random variables U : = { U1 , . . . Ud } 72 ( with prior distribution p ( U ) ) via the set of structural equations F̃ = { Xi : = f̃i Xpa ( i ) , Ui } di=1.73 Here Xpa ( i ) refers to the set of variables directly causing Xi , i.e. , parents of i . Every SCM M is74 associated with a directed acyclic graph ( DAG ) : a causal graph G : = ( X , E ) , for which the nodes75 ( vertices ) correspond to endogenous variables X and the directed edges E account for the causal76 parent-child relationship between variables [ 39 ] . Given an SCM , there are two types of causal queries77 of general interest : interventional queries , e.g. , “ What would happen to the population X , if variable78 Xi would be set to a fixed value ↵ ? ” ; and counterfactual queries , e.g. , “ What would have happened to79 a specific factual sample xF , had Xi been set to a value ↵ ? ” .80 More in detail , interventional queries aim to evaluate changes in the causal world , or equivalently,81 manipulations of a subset of the endogenous variables I ✓ [ d ] : = { 1 , . . . , d } at the population82 level . Interventions on an SCM M are often represented with the do-operator do ( Xi = ↵i ) and83 lead to a new distribution over the set of endogenous variables p ( X | do ( Xi = ↵i ) ) , which is84 referred to as the interventional distribution . In G an intervention removes incoming edges to node85 i and sets Xi = ↵ ( see Figure 1c ) . A counterfactual query for a given factual instance xF aims to86 estimate what would have happened had XI instead taken value ↵ . This effect is captured by the87 counterfactual distribution p ( xCF | xF , do ( XI = ↵ ) ) , which can be computed using the abduction-88 action-prediction approach by Pearl [ 39 ] . Refer to Section 3 for further details on the computation of89 the interventional and counterfactual distributions.90 2.2 Variational Graph Autoencoder and Graph Neural Networks91 Variational Autoencoders ( VAEs ) [ 20 ] are powerful latent variable models based on neural networks92 ( NNs ) for jointly i ) learning complex and expressive density estimators p ( X ) ⇡ R p✓ ( X | Z ) p ( Z ) dZ,93 where the likelihood function ( a.k.a . decoder ) is parameterized using a NN with parameters ✓ ; and ii ) 94 performing approximate posterior inference over the latent variables Z using a variational distribution95 ( a.k.a . encoder ) q ( Z | X ) parameterized using a NN with parameters . The parameters ✓ and 96 are usually learned by maximizing a lower bound on the log-evidence [ 3 , 34 , 41 , 52 ] .97 Variational Graph Autoencoders ( VGAEs ) [ 21 ] extend VAEs to account for graph-structure in-98 formation on the data [ 58 ] . VGAEs define a ( potentially multidimensional ) latent variable Zi per99 observed variable Xi , i.e. , Z : = { Z1 , . . . , Zd } . Additionally , VGAEs rely on an adjacency ma-100 trix A , which is used by two Graph Neural Networks ( GNNs ) , one for the encoder and one for101 the decoder , to enforce structure on the posterior approximation q ( Z | X , A ) and the likelihood102 p✓ ( X | Z , A ) . More in detail , A 2 { 0 , 1 } d⇥d encodes the graph structure among the observed103 variables X : = { X1 , . . . Xd } , so that Aij = 1 if there is a directed edge from Xj to Xi , and Aij = 0,104 otherwise . Hence , A determines which variables Xi influence Zj ( i , j 2 [ d ] ) , and vice versa.105 Graph Neural Networks ( GNNs ) have generated a lot of attention during the last years , as they106 achieved significant improvements in graph representation learning [ 2 , 12 , 14 , 43 , 57 ] , While the107 taxonomy of GNNs is immense [ 55 ] , in this work we focus on message passing GNNs which allow108 us to work with directed graphs . In its most general form , a message-passing GNN calculates the109 output hli for node i in layer l in three steps : i ) compute the set of incoming messages arriving to node110 i from its neighbors Ni = { Xj | Aij = 1 } using a message function fm ( a NN with parameters ✓lm ) ,111 that is { mlij } j2Ni = { fmi ( h l 1 i , h l 1 j ; ✓ l m ) | j 2 Ni } ; ii ) combine the set of messages into a single112 message M li : = fa ( { mlij } j ) using an aggregation function fa ( e.g . adding up the messages ) ; and iii ) 113 update the node state hli : = fu ( h l 1 i , M l i ; ✓ l u ) , using an update function fu ( a NN with parameters114 ✓ l u ) . As a result , the output hli can be written as115 h l i = f u h l 1 i , f a { fm ( hl 1i , h l 1 j ; ✓ l m ) | j 2 Ni } ; ✓lu . ( 1 ) Note that the above expression assures that the output for each node i is computed using information116 from its neighbors Ni according to A . Moreover , if the GNN has Nh hidden layers , then the117 output for node i not only depends on its direct neighbors Ni , but also on its neighbors up to order118 Nh + 1 ( hops ) . As an example , if Nh = 0 then the output for each node only depend on its direct119 neighbors ( parents ) . If instead Nh = 1 , then the output for each node depends on 2-hop neighbors120 ( grand-parents ) . For a detailed description of GNNs , please refer to Appendix A.121 3 Observational , interventional and counterfactual distributions122 In this section , we introduce the observational , interventional and counterfactual distributions ( trig-123 gered by any intervention do ( XI = ↵ ) ) that are induced from an SCM M : = { p ( U ) , F̃ } . Specifi-124 cally , we summarize the main properties of an SCM that will allow us to propose a novel class of125 VGAEs , the variational causal autoencoders ( VCAUSE ) , to compute accurate estimates of these126 distributions using observational data and a known causal graph . To this end , we assume the absence127 of hidden confounders , i.e. , we assume that p ( U ) = Qd i=1 p ( Ui ) .128 Observational distribution . The SCM M determines the observational distribution p ( X ) over the129 set of endogenous variables X = { X1 , . . . Xd } , which satisfies causal factorization [ 44 ] , i.e. , p ( X ) =130 Qd i=1 p ( Xi | Xpa ( i ) ) . That is , after marginalizing out the exogenous variables U , the distribution of131 each endogenous variable Xi depends only on its parents , i.e. , Xpa ( i ) . The observational distribution132 can alternatively be written only in terms of the exogenous variables U as133 p ( X ) = Z F ( U ) p ( U ) dU , ( 2 ) where F : U ! X corresponds to the set of structural equations , equivalent to F̃ , that directly134 transform the exogenous variables U into the endogenous variables X . Let us denote by an ( i ) the set135 of indexes of the ancestors of i , and an⇤ ( i ) : = an ( i ) [ { i } . Then , the causal factorization induced136 by the SCM M leads to the following property of F ( U ) :137 Property 1 . Each endogenous variable Xi can be expressed as a function of its exogenous variable138 Ui and the ones of all its causal ancestors , i.e. , F ( U ) : = { Xi = fi ( { Uj | j 2 an⇤ ( i ) } ) } . This,139 together with the causal sufficiency assumption , implies that Xi is statistically independent of140 Uj , 8j /2 an⇤ ( i ) .141 Interventional distribution . As stated in Section 2.1 , interventions on a set of variables I can be142 performed using the do-operator , which can be seen as a mapping do ( XI = ↵ ) : M 7 ! MI =143 ( p ( U ) , F̃I ) where F̃I = { f̃j | j 62 I } [ { ↵i | i 2 I } . As above , we can represent the resulting set144 of intervened structural equations FI = { fj | j 62 I } [ { ↵i | i 2 I } in terms of only the exogenous145 variables U , so that we can write the interventional distribution as:146 p ( X | do ( XI = ↵ ) ) = Z FI ( U ) p ( U ) dU . ( 3 ) Assuming an intervention do ( XI = ↵ ) on M , then the resulting structural equations FI ( U ) satisfy:147 Property 2 . After an intervention do ( XI = ↵ ) on M , all the causal paths from Uj 8j 2 an⇤ ( i ) to148 Xi that include an intervened variable in XI ( i.e. , the causal paths where XI is a mediator ) are149 severed in FI , while the rest of causal paths remain untouched.150 The above property is illustrated in Figure 1 , where we can easily observe that after an intervention151 do ( X2 = ↵ ) , the indirect causal path ( in red ) from X1 , and thus from U1 , to X3 via X2 is severed,152 while the direct path ( in green ) remains.153 Counterfactual distribution . Assuming the SCM M = { p ( U ) , F̃ } to be known , the following154 three steps defined by Pearl [ 39 ] allow us to compute counterfactuals xCF as : i ) Abduction : infer the155 values of the exogenous variables U for a factual sample xF , i.e. , compute p ( U | xF ) ; ii ) Action:156 intervene with do ( XI = ↵ ) : M 7 ! MI = ( p ( U ) , F̃I ) ; and iii ) Prediction : use the posterior157 distribution p ( U | xF ) and the new structural equations F̃I to compute p ( xCF | xF ) . The prediction158 step can be alternatively computed using the new set of structural equations FI defined in terms of159 the exogenous variables U , so that we can write the counterfactual distribution as:160 p ( xCF | xF , do ( XI = ↵ ) ) = Z FI ( U ) p ( U | xF ) dU . ( 4 ) Importantly , the resulting posterior distribution p ( U | xF ) satisfies:161 Property 3 . In the abduction step , statistical independence implies that conditioned on the endoge-162 nous variables of the factual sample xF , each exogenous variable Ui is independent of the factual163 value x F j if j 6= i and the variable Xj is not a parent of Xi , i.e. , j 62pa⇤ ( i ) : = pa ( i ) [ { i } .164 4 Variational Causal Autoencoder ( VCAUSE ) 165 In this section , we present a novel variational causal graph autoencoder ( VCAUSE ) to approximate166 the observational , interventional and counterfactual distributions given in ( 2 ) , ( 3 ) and ( 4 ) , respectively.167 While the underlying SCM M is unknown , we assume access to : the causal graph G and observational168 data { xn } Nn=1 , i.e. , i.i.d . samples of the observational distribution induced by M.169 Definition 4.1 . ( VCAUSE ) . Given a causal graph G over a set of endogenous variables X =170 { X1 , . . . , Xd } , which establishes the set of parents pa⇤ ( i ) for each variable Xi ( including the i-th171 node ) . A variational causal graph autoencoder ( VCAUSE ) is defined by:172 • A causal adjacency matrix A , which is a d ⇥ d binary matrix with elements Aij = 1 if173 j 2 pa⇤ ( i ) , i.e. , when i = j or j is a parent of i . Otherwise , Aij = 0.174 • A prior distribution p ( Z ) = Q i p ( Zi ) over the set of latent variables Z = { Z1 , . . . , Zd } .175 • A decoder p✓ ( X | Z , A ) , which is a GNN ( parameterized by ✓ ) that takes as input the set176 of latent variables Z and the causal adjacency matrix A , and outputs the parameters of the177 likelihood p✓ ( X | Z , A ) .178 • An encoder q ( Z | X , A ) , which is a GNN ( parameterized by ) that takes as input the179 endogenous variables X and the causal adjacency matrix A , and outputs the parameters of180 the posterior approximation q ( Z | X , A ) .181 Given observational data { xn } Nn=1 , one may learn the parameters ✓ and that best estimate the182 density p ( X ) . We here rely on the partially importance weighted auto-encoder ( PIWAE ) [ 41 ] .183 Next , we discuss how to design VCAUSE such that it is able to capture the observational , inter-184 ventional , and counterfactual distribution induced by an unknown SCM . Importantly , we derive the185 necessary conditions on the design of both the encoder and decoder GNNs such that VCAUSE fulfills186 the SCM properties introduced in Section 3.187 4.1 Observational distribution188 VCAUSE approximates the observational distribution in ( 2 ) using the generative model as189 p ( X ) ⇡ Z p✓ ( X | Z , A ) p ( Z ) dZ = Z dY i=1 p✓ ( Xi | Z , A ) p ( Z ) dZ . ( 5 ) Figure 2a depicts this generative process . If we compare ( 5 ) with the true observational distribution190 in ( 2 ) , we observe that the latent variables Z play a similar role to the exogenous variables U,191 and the decoder p✓ ( X | Z , A ) plays a similar role to the structural equations F. Yet , we remark192 that Z does not need to correspond to the exogenous variables , i.e. , p ( U ) 6= p ( Z ) , in order for ( 5 ) 193 to provide a good approximation of the observational distribution in ( 2 ) . In fact , standard VAEs194 perform accurate density estimation using observational data , without the need for capturing causal195 information . However , in this paper , we seek to ensure that our observational distribution induced196 by VCAUSE complies causal factorization ( Property 1 ) . To that end , we need to make sure that197 p✓ ( Xi | Z , A ) = p✓ ( Xi | Zan⇤ ( i ) ) . That is , Xi depends only on Zj if j = i or Xj is an ancestor of198 Xi in the causal graph . To fulfill this property , the GNN of the decoder should satisfy the following:199 Proposition 1 . ( Causal factorization ) . VCAUSE satisfies causal factorization , p✓ ( X | Z , A ) =200 Q i p✓i ( Xi | Zan⇤ ( i ) ) , if and only if the number of hidden layers in the decoder is greater or equal201 than 1 , with being the longest shortest directed path between any two endogenous nodes.202 The above proposition ( proved in Appendix B ) is based on the fact that , in a GNN with Nh hidden203 layers ( and Nh + 1 layers in total ) , the output for the i-th node depends on its neighbors of up204 to Nh + 1 hops . As an example , consider the following chain causal graph : X1 ! X2 ! X3,205 such that = 2 . In the decoder , the first layer yields a hidden representation for the 3-rd node206 h 1 3 : = f ( f ( Z2 ) , Z3 ) that only depends on Z2 and Z3 . Thus , we need a second layer for its output207 h 2 3 : = f ( h2 , Z3 ) = f ( f ( f ( Z1 ) , Z2 ) , Z3 ) to depend on Z1 ( note that X1 is an ancestor of X3 ) .208 4.2 Interventional distribution209 VCAUSE approximates the interventional distribution in ( 3 ) as ( illustrated Figure 3 ) :210 p ( X | do ( XI = ↵ ) ) ⇡ Z p✓ ( X | { Zi } i 62I , { ZIi } i2I , AI ) p ( Z ) q ( ZI | AI , XI ) dZ , ( 6 ) where the do-operator is performed on the causal adjacency matrix as do ( XI = ↵ ) : A 7 ! AI =211 { Aij } 8i 62I , j [ { Aij = 0 } 8i2I , j . This ensures that Xi for i 2 I is independent of Zj for all212 j 6= i . Note that in order for ( 6 ) to be able to approximate the interventional distribution in ( 3 ) , an213 intervention on a variational causal autoencoder should satisfy Property 2 , i.e . :214 Proposition 2 . ( Causal interventions ) . VCAUSE can capture causal interventions if and only if the215 number of hidden layers in its decoder is greater than or equal to 1 , with being the longest216 directed path between any two endogenous nodes in G.217 To illustrate this , Figure 3 depicts how messages218 are exchanged in a one-hidden-layer decoder219 GNN corresponding to the causal graph G in220 Figure 1 ( triangle with = 2 ) , both ( a ) without221 and ( b ) with an intervention on X2 . We high-222 light in green the direct messages ( sent via direct223 causal path in G ) , and in red the indirect mes-224 sages ( sent via indirect causal path in G ) from225 Z1 to X3 . Observe that , similarly to Figure 1 , in226 ( a ) there is an indirect path ( via h2 ) from Z1 to227 X3 ; while in ( b ) this path is severed . Hence , the228 hidden layer ( h1 , h2 , h3 ) allows to differentiate229 between direct and indirect paths and thus to230 capture interventional effects.231 As the condition in Proposition 2 is more restrictive than the one in Proposition 1 , in order for232 VCAUSE to be able to capture observational and interventional distributions , it should satisfy that:233 Design condition 1 : The decoder GNN of VCAUSE has at least as many hidden layers as 1 , with234 being the longest directed path in the causal graph G.235 4.3 Counterfactual distribution236 VCAUSE approximates the counterfactual distribution in ( 4 ) as ( illustrated in Figure 2c ) :237 p ( xCF | do ( XI = ↵ ) , xF ) ⇡ Z p✓ ( X | { ZFi } i 62I , { ZIi } i2I , AI ) q ( ZI | xI , AI ) | { z } action q ( Z F | xF , A ) | { z } abduction dZ | { z } prediction , where xF represents a sample from X for which we seek to compute the distribution over counterfac-238 tual xCF . Note here that two different passes of the encoder are necessary : one for the abduction239 step of the factual instance q ( ZF | xF , A ) ; and another one for the action step ( intervention ) 240 q ( ZI | xI , AI ) with xIi = ↵i 8i 2 I ( we remark that the rest of the values in xI do not affect the241 overall counterfactual computation ) . We then evaluate the likelihood , making sure that the resulting242 counterfactual sample xCF only depends on the { ZFi } i 62I ✓ ZF and { ZIi } i2I ✓ ZI . Importantly,243 in order for VCAUSE to be able to approximate the counterfactual distribution , we need its abduction244 ( and action ) step ( s ) to comply with Property 3 , i.e . :245 Proposition 3 . ( Abduction ) . The abduction step of an observed sample x = { x1 , . . . , xd } in a246 variational causal autoencoder satisfies that for all i the posterior of Zi is independent on the subset247 { xj } j 62pa⇤ ( i ) ✓ x , if and only if the encoder GNN has no hidden layers.248 The above result ( proved in Appendix B ) can be shown by the message passing algorithm computed249 by the encoder GNN , and leads to the second condition that VCAUSE should satisfy by design:250 Design condition 2 : The encoder GNN of VCAUSE has no hidden layers.251 Note that while the above condition may look restrictive and limiting the capacity of our encoder , we252 may choose arbitrarily complex NNs to model the message fm and update fu functions , as well as253 one or more aggregation functions fa , e.g. , sum , mean or max , to model the encoder [ 8 ] .254 4.4 Practical considerations255 Next , we briefly discuss practical implementation considerations to handle complex causal models,256 which often appear in real world applications–see the causal graph of the German Credit dataset [ 10 ] 257 in Section 6 for an example . For further details on VCAUSE implementation , refer to Appendix C.258 Heterogeneous endogenous variables : In general GNNs are parametrized such that the parameters259 of the message fm and update fu functions are shared for all the nodes and edges in the graph.260 However , similarly as in the structural causal equations F , we can define a different message function261 f m ij for every edge in the causal graph by assuming a different set of parameters ✓mij per edge in ( 1 ) .262 Similarly , we can also assume different update functions fui for each node i , by considering different263 update parameters ✓ui for each node . This allows us to use different functions for each node , and264 thus model heterogeneous endogenous variables , in terms of their continuous/discrete distribution,265 and also of their structural equations , e.g. , linear/non-linear.266 Heterogenous causal nodes : So far , we have modeled each endogenous variable Xi as a node in267 the causal graph G , and thus in the VCAUSE GNNs . However , in some application domains the268 relationships between a subset of variables may be unknown , or they may be affected by hidden269 confounders , leading to an undirected path between them . In such cases , the subset of ( ki ) variables270 is modeled as a multidimensional and potentially heterogeneous node Xi = { Xi1 , . . . , Xiki } . Note271 that all the variables in the multidimensional node Xi share the same latent random variable Zi.272 5 Evaluation273 In this section , we conduct extensive experiments to evaluate the performance of VCAUSE at274 estimating the outcomes of causal queries . Please refer to Appendix D for a complete description275 of the experimental set-up . Moreover , to ease the reproducibility of our experiments , our code is276 publicly available at https : //github.com/XXXX/XXXXX.277 Datasets . We consider different synthetic causal graphs that differ in the number of nodes d , diameter278 , and longest path : synthetic collider ( d = 3 , = 1 , = 1 ) , M-graph ( d = 3 , = 1 , = 1 ) ,279 triangle ( d = 3 , = 1 , = 2 ) , chain ( d = 3 , = 2 , = 2 ) , and a semi-synthetic loan ( d = 7 , = 2,280 = 3 ) from [ 17 ] . For all of the synthetic datasets ( i.e. , except loan ) , we consider three different types281 of structural equations with increasing complexity : linear additive noise ( LIN ) , non-linear additive282 noise ( NLIN ) and non-additive noise ( NADD ) .283 Metrics . We evaluate the observational distribution using the Maximum Mean Discrepancy ( MMD ) 284 [ 13 ] as distance-measure between the true and estimated distributions as a whole , i.e. , the lower the285 MMD the better the distributions match . For the interventional distribution , we additionally report the286 estimation squared error for the mean and for the standard deviation ( MeanE and StdE respectively ) 287 for the children of the intervened variables . For the counterfactual distribution we report the mean288 square error ( MSE ) as well as the standard deviation of the squared error ( SSE ) between the true and289 the estimated counterfactual value . We compute all results over 10 independent runs.290 Validating VCAUSE design conditions . In a first step we empirically validate our design choices291 for the VCAUSE encoder and decoder . We show how the number of hidden layers Nh in the decoder292 affect the quality of the estimation of the observational and interventional distributions for three293 SCMs , with different values of longest shortest directed path and longest directed path . In Table 1,294 we observe that as expected : i ) the collider ( = = 1 ) does not need any hidden layer to provide295 accurate estimate of both the observational and interventional distributions . In contrast , the triangle296 ( = 1 , = 2 ) , which according to Proposition 2 needs at least one hidden layer to get a more297 accurate estimate of the interventional distribution ( while an improvement in the observational is not298 as evident ) . Finally , as stated by Propositions 1 and 2 , the chain ( = = 2 ) requires at least one299 hidden layer to accurately approximate both the observational and interventional distributions.300 5.1 Estimating interventional and counterfactual distributions301 In the following we evaluate the potential of VCAUSE to model interventional and counterfactual302 queries . We consider interventions of the form do ( xi = ↵i ) for several values of ↵i on both root and303 non-root nodes . Here we report the results for the triangle and loan graphs . Refer to Appendix E for304 the remaining results.305 Baselines . We compare our VCAUSE with two competing methods : i ) MultiCVAE , which trains a306 conditional VAE for each endogenous variable that is not a root node in the causal graph [ 17 ] ; and ii ) 307 CAREFL [ 18 ] , which relies on autoregressive causal flows to estimate counterfactual queries.308 Results for interventional distributions . Table 2 ( middle columns ) reports the MMD , MeanE , and309 StdE for the interventional distribution . Here we can observe that VCAUSE consistently outperforms310 other methods in terms of MMD . Note that the three methods provide comparable results in capturing311 the mean of the interventional distribution ( MeanE ) ( except for the more complex loan graph , where312 VCAUSE outperforms the others ) . However , it can also be seen that CAREFL and MultiCVAE313 often fail to capture the standard deviation of the interventional distribution ( StdE ) , while VCAUSE314 provides a more accurate estimate of the overall interventional distribution.315 Results for the counterfactuals . Table 2 also316 reports the results for the counterfactual dis-317 tribution . Here , we first observe that MultiC-318 VAE slightly underperforms the other two mod-319 els . Second , we observe that CAREFL provides320 more accurate estimates than VCAUSE in terms321 of MSE , which may be explained by the fact that322 CAREFL performs exact inference . However,323 CAREFL presents high variance in its results324 ( see SSE ) . Note that to perform interventions,325 CAREFL sets the parents of the intervened vari-326 ables to zero , which may not completely severe the causal paths to the intervened nodes . In contrast,327 as further illustrated in Figure 4 , VCAUSE leads to consistent counterfactual estimations across fac-328 tual samples and interventions . Figure 4 also shows that CAREFL fails severely for some intervention329 values , despite of intervening on a root node.330 6 Use case : counterfactual fairness331 Finally , we showcase the practical use of VCAUSE for assessing counterfactual fairness and also for332 training a counterfactually fair classifier . To this end , we use the German Credit dataset publicly avail-333 able at the UCI repository [ 50 ] . We rely on the causal model with the following random variables X334 as proposed in [ 6 ] ( see Figure 5 ) : sensitive feature S = { sex } , and non-sensitive features C = { age } ,335 R = { credit amount , repayment history } and H = { checking account , savings , housing } . Then , we336 aim to predict the binary feature Y = { credit risk } from X . See Appendix F for further details.337 Counterfactual fairness . Let S ⇢ X be a sensitive attribute ( e.g. , gender ) , then a classifier h : X ! 338 Y is considered ✏-counterfactually fair [ 24 ] if:339 P ( h ( xCF ) = y | do ( S = ↵ ) , xF ) P ( h ( xCF ) = y | do ( S = ↵0 ) , xF ) ✏ , 8xCF , ↵0 6= ↵ , y . A classifier is counterfactually fair ( ✏ = 0 ) , if , given a factual xF with sensitive attribute S = ↵,340 had its sensitive attribute been different S = ↵0 , the classifier prediction would remain the same . As341 VCAUSE allows us to generate counterfactual samples , we can thus use it to audit the fairness level of342 a classifier . Moreover , we can use the VCAUSE encoder to learn a fair classifier hVCAUSE : Z\ZS ! 343 Y , which takes as input the latent variables generated by VCAUSE without the one of the sensitive344 attribute ZS . Following [ 24 ] , we compare our VCAUSE fair classifier hVCAUSE with : i ) a full model345 hfull : X ! Y that takes as input the complete variable set ; ii ) an unaware model hunaw : X\S ! Y346 that takes as input all variables but the sensitive one ; iii ) and a fair model hfair : { Xi|S 62an⇤ ( i ) } ! Y347 that takes as input all non-descendant variables of the sensitive attribute.348 Results . The results for logistic regression ( LR ) and support vector machine ( SVM ) classifiers are349 summarized in Table 3 . Note that VCAUSE correctly ranks the different methods based on their350 unfairness level , showing that the full classifier is consistently less fair than the unaware and the fair351 classifiers , respectively . Moreover , the VCAUSE classifier leads to a fair classifier , while keeping352 the f1-score comparable to the unfair classifier . Therefore , VCAUSE does not only allow us to audit353 counterfactual fairness but also provides a practical approach to train accurate and fair classifiers.354 7 Conclusion356 In this work , we have proposed VCAUSE a variational causal autoencoder based on GNNs that : i ) is357 specially designed to capture the properties of SCMs ; ii ) inherently handles heterogeneous causal358 graphs and data ; and iii ) provides accurate estimates of interventional and counterfactual distributions359 for SCMs of different complexities . As demonstrated by extensive experiments , VCAUSE provides360 accurate results for a wide variety of interventions in diverse SCMs leading to significantly more361 robust results than competing methods [ 17 , 18 ] . Finally , we have shown a practical use-case of362 VCAUSE in a problem of increasing interest for the machine learning community , namely , fairness363 in classification . In particular , we have shown how to use VCAUSE to both assess counterfactual364 fairness and to train counterfactually fair classifiers.365 Moreover , our work opens up many interesting venues for future work . First , as we have assumed a366 known causal graph and the absence of hidden confounders , it would be important to evaluate the367 sensitivity of VCAUSE to the violation of these assumptions in order to avoid its misuse . 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For all authors ... 547 ( a ) Do the main claims made in the abstract and introduction accurately reflect the paper ’ s548 contributions and scope ? 549 [ Yes ] Our contributions are : i ) identifying the properties of an SCM that allow us550 to propose our model ( Section 3 ) , ii ) proposing the Variational Causal Autoencoder551 ( VCAUSE ) for answering causal queries given observational data and the causal552 graph ( Section 4 ) , iii ) evaluating the estimation of interventional and counterfactual553 distributions as a whole on different SCMs ( Section 5 ) , iv ) showing how VCAUSE can554 be used for counterfactual fairness evaluation and classification ( Section 6 ) .555 ( b ) Did you describe the limitations of your work ? 556 [ Yes ] As stated throughout the paper , our approach relies on two causal assumptions,557 i.e. , causal sufficiency and access to the true causal graph . As these assumptions may558 be seen as a practical limitations , in Section 7 , we discuss our plans to address them559 in future work . Moreover , Section 7 states that our approach inherits the limitations560 of GNNs , and thus propose future ablation studies to investigate the scalability of the561 proposed approach to more complex causal graphs.562 ( c ) Did you discuss any potential negative societal impacts of your work ? 563 [ Yes ] The proposed approach allows for causal reasoning under certain assumptions,564 which we make explicit to avoid its misuse . Moreover , as discussed in Section 7 , we565 consider the relaxation of such assumptions as future work.566 ( d ) Have you read the ethics review guidelines and ensured that your paper conforms to567 them ? 568 [ Yes ] We have read the ethics review guidelines and ensured that our paper conforms569 to them.570 2 . If you are including theoretical results ... 571 ( a ) Did you state the full set of assumptions of all theoretical results ? 572 [ Yes ] In section 4 , we explicitly state the assumptions that we make throughout the573 paper as well as the necessary conditions for the proposed approach to yield reliable574 results.575 ( b ) Did you include complete proofs of all theoretical results ? 576 [ Yes ] Appendix B provides complete proofs for all the theoretical results in the paper.577 3 . If you ran experiments ... 578 ( a ) Did you include the code , data , and instructions needed to reproduce the main experi-579 mental results ( either in the supplemental material or as a URL ) ? 580 [ Yes ] We upload the code with the supplementary material and will be releasing the581 code on GitHub.582 ( b ) Did you specify all the training details ( e.g. , data splits , hyperparameters , how they583 were chosen ) ? 584 [ Yes ] Appendix D provide a detailed description of the experimental set-up , the im-585 plementation and validation of the methods and the computation of the performance586 metrics.587 ( c ) Did you report error bars ( e.g. , with respect to the random seed after running experi-588 ments multiple times ) ? 589 [ Yes ] In Tables 1 and 2 we report the average and standard deviation for 10 different590 runs.591 ( d ) Did you include the total amount of compute and the type of resources used ( e.g. , type592 of GPUs , internal cluster , or cloud provider ) ? 593 [ Yes ] See Appendix D. However , note that experiments are light-weight and we do not594 require any GPU.595 4 . If you are using existing assets ( e.g. , code , data , models ) or curating/releasing new assets ... 596 ( a ) If your work uses existing assets , did you cite the creators ? 597 [ Yes ] Our implementation of the baselines in Section 5 is built on the publicly available598 code provided by [ 17 ] and [ 18 ] . The semi-synthetic dataset loan is taken from [ 17 ] . In599 Section 6 we use the publicly available German Credit dataset from the UCI repository600 [ 10 ] .601 ( b ) Did you mention the license of the assets ? 602 [ Yes ] See Appendix D. We rely on research code from [ 17 ] and [ 18 ] provided under603 MIT License.604 ( c ) Did you include any new assets either in the supplemental material or as a URL ? 605 [ Yes ] Our code can be found in the supplemental material and we will make it publicly606 available on GitHub.607 ( d ) Did you discuss whether and how consent was obtained from people whose data you ’ re608 using/curating ? 609 [ No ] Does not apply . The only real world data we use is the publicly available German610 Credit dataset from the UCI repository [ 10 ] , which is published in anonymized form.611 ( e ) Did you discuss whether the data you are using/curating contains personally identifiable612 information or offensive content ? 613 [ N/A ] Does not apply.614 5 . If you used crowdsourcing or conducted research with human subjects ... 615 ( a ) Did you include the full text of instructions given to participants and screenshots , if616 applicable ? 617 [ N/A ] Does not apply.618 ( b ) Did you describe any potential participant risks , with links to Institutional Review619 Board ( IRB ) approvals , if applicable ? 620 [ N/A ] Does not apply.621 ( c ) Did you include the estimated hourly wage paid to participants and the total amount622 spent on participant compensation ? 623 [ N/A ] Does not apply.624

Summary: The authors propose a framework for the use of variational graph autoencoders for causal inference in settings without hidden confounders when observational data and the causal graph are available. The framework aims to mimic the properties of the matching Structural Causal Model (SCM), including the ability to evaluate observational, interventional and counterfactual queries. To evaluate their proposed framework, the authors perform an experimental evaluation in estimating the outcomes of causal queries on multiple small synthetic causal graphs and in assessing counterfactual fairness.