Partial Identification of Counterfactual Distributions

This paper investigates the problem of bounding counterfactual queries from a1 combination of observational data and qualitative assumptions about the underlying2 data-generating model . These assumptions are usually represented in the form3 of a causal diagram ( Pearl , 1995 ) . We show that all counterfactual distributions4 ( over finite observed variables ) in an arbitrary causal diagram could be generated5 by a special family of structural causal models ( SCMs ) , compatible with the6 same causal diagram , where unobserved ( exogenous ) variables are discrete , taking7 values in a finite domain . This entails a reduction in which the space where the8 original , arbitrary SCM lives can be mapped to a dual , more well-behaved space9 where the exogenous variables are discrete , and more easily parametrizable . Using10 this reduction , we translate the bounding problem in the original space into an11 equivalent optimization program in the new space . Solving such programs leads to12 optimal bounds over unknown counterfactuals . Finally , we develop effective Monte13 Carlo algorithms to approximate these optimal bounds from a finite number of14 observational data . Our algorithms are validated extensively on synthetic datasets.15 1 Introduction16 This paper studies the problem of inferring counterfactual queries from the combination of non-17 experimental data ( e.g. , observational studies ) and qualitative assumptions about the data-generating18 process . These assumptions are represented in the form of a causal diagram [ 32 ] , which is a19 directed acyclic graph where arrows indicate the potential existence of functional relationships among20 corresponding variables ; some variables are unobserved . This problem arises in diverse fields such21 as artificial intelligence , statistics , cognitive science , economics , and the health and social sciences.22 For example , when investigating the gender discrimination in college admission , one may ask “ what23 would the admission outcome be for a female applicant had she been a male ? ” Such a counterfactual24 query contains conflicting information : in the real world the applicant is female , in the hypothetical25 world she was not . Therefore , it is not immediately clear how to design effective experimental26 procedures for evaluating counterfactuals , let alone how to compute them from observations alone.27 The problem of identifying counterfactual distributions from the combination of data and a causal28 diagram has been studied in the causal inference literature . First , there exist a complete proof system29 for reasoning about counterfactual queries [ 19 ] . While such a system , in principle , is sufficient in30 evaluating any identifiable counterfactual expression , it lacks a proof guideline which determines the31 feasibility of such evaluation efficiently . There are algorithms to determine whether a counterfactual32 distribution is inferrable from all possible controlled experiments [ 41 ] . There exist also algorithms33 for identifying path-specific effects from experimental data [ 1 ] and observational data [ 42 ] .34 In practice , however , the combination of quantitative knowledge and observed data does not always35 permit one to point-identify the target counterfactual queries . Partial identification methods concern36 with deriving informative bounds over the target counterfactual probability , even when the target37 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . itself is non-identifiable . Several algorithms have been developed to bound counterfactuals from the38 combination of observational and experimental data [ 30 , 36 , 3 , 4 , 14 , 35 , 23 , 24 , 16 , 25 , 49 ] .39 In this work , we build on the approach introduced by Balke & Pearl in [ 3 ] , which involves direct40 discretization of the exogenous domains , also referred to as the principal stratification [ 17 , 34 ] . Con-41 sider the causal diagram of Fig . 1a , where X , Y , Z are binary variables in { 0 , 1 } ; U is an unobserved42 variable taking values in an arbitrary continuous domain . [ 3 ] showed that domains of U could be43 discretized into 16 equivalent classes without changing the original counterfactual distributions and44 the graphical structure in Fig . 1a . For instance , despite it being induced by an arbitrary distribution45 P ⇤ ( u ) over a continuous domain of the exogenous variable U , the observational distribution P ( x , y|z ) 46 must be reproduced by a generative model of the form P ( x , y|z ) = P u P ( x|u , z ) P ( y|x , u ) P ( u ) ,47 where P ( u ) is a discrete distribution over a finite exogenous domain { 1 , . . . , 16 } .48 Using the finite-state representation of unobserved variables , [ 4 ] derived tight bounds on treatment49 effects under the condition of noncompliance in Fig . 1a . [ 11 , 21 ] applied the parsimony of finite-state50 representation in a Bayesian framework , to obtain credible intervals for the posterior distribution of51 causal effects in noncompliance settings . Despite their optimal guarantees , these bounds are only52 applicable to the specific noncompliance setting in Fig . 1a . For the most general cases , a systematic53 procedure for bounding counterfactual queries in arbitrary causal diagrams is still missing.54 Our goal in this paper is to overcome these challenges . We investigate the expressive power of discrete55 structural causal models ( SCMs ) [ 33 ] where each unobserved variable is drawn from a discrete56 distribution , takes values in a finite set of states . We show that when inferring about counterfactual57 distributions ( over finite observed variables ) in an arbitrary causal diagram , one could restrict domains58 of unobserved variables to a finite space without loss of generality . This observation allows us to59 develop novel partial identification algorithms to bound unknown counterfactual probabilities from60 the observational data . More specifically , our contributions are as follows . ( 1 ) We introduce a61 special family of discrete SCMs , with finite unobserved domains , and show that it could represent62 all categorical counterfactual distributions in an arbitrary causal diagram . ( 2 ) Using this result , we63 translate the original partial identification task into equivalent polynomial programs . Solving such64 programs leads to informative bounds over unknown counterfactual probabilities , which are provably65 optimal . ( 3 ) We develop an effective Monte Carlo algorithm to approximate optimal counterfactual66 bounds from a finite number of observational data . Finally , our algorithms are validated extensively67 on synthetic datasets . Given space constraints , all proofs are provided in Appendices A and B.68 1.1 Preliminaries69 We introduce in this section some basic notations and definitions that will be used throughout the70 paper . We use capital letters to denote variables ( X ) , small letters for their values ( x ) and ⌦X for71 their domains . For an arbitrary set X , let |X| be its cardinality . For convenience , we denote by P ( x ) 72 probabilities P ( X = x ) ; for an arbitrary subdomain X ✓ ⌦X , P ( X ) ⌘ P ( X 2 X ) . Finally , the73 indicator function 1X=x returns 1 if an event X = x holds true ; otherwise 1X=x = 0.74 The basic semantical framework of our analysis rests on structural causal models ( SCMs ) [ 33,75 Ch . 7 ] . An SCM M is a tuple hV , U , F , P i where V is a set of endogenous variables and U is76 a set of exogenous variables . F is a set of functions where each fV 2 F decides values of an77 endogenous variable V 2 V taking as argument a combination of other variables in the system . That78 is , v fV ( paV , uV ) , PaV ✓ V , UV ✓ U . Exogenous variables U 2 U are mutually independent,79 values of which are drawn from the exogenous distribution P ( u ) . Naturally , M induces a joint80 distribution P ( v ) over endogenous variables V , called the observational distribution . Each SCM81 is associated with a causal diagram G ( e.g. , Fig . 1 ) , which is a directed acyclic graph ( DAG ) where82 solid nodes represent endogenous variables V , empty nodes represent exogenous variables U and83 arrows represent the arguments PaV , UV of each function fV .84 An intervention on an arbitrary subset X ✓ V , denoted by do ( x ) , is an operation where values of85 X are set to constants x , regardless of how they are ordinarily determined . For an SCM M , let86 Mx denote a submodel of M induced by intervention do ( x ) . For any subset Y ✓ V , the potential87 response Yx ( u ) is defined as the solution of Y in the submodel Mx given U = u . Drawing values88 of exogenous variables U following the probability measure P induces a counterfactual variable Yx.89 Specifically , the event Yx = y ( for short , yx ) can be read as “ Y would be y had X been x ” . For any90 subsets Y , . . . , Z , X , . . . , W ✓ V , the distribution over counterfactuals Yx , . . . , Zw is defined as:91 P ( yx , . . . , zw ) = Z ⌦U 1Yx ( u ) =y ^ · · · ^ 1Zw ( u ) =zdP ( u ) . ( 1 ) Distributions of the form P ( yx ) is called the interventional distribution ; when the treatment set92 X = ; , P ( y ) coincides with the observational distribution . Throughout this paper , we assume93 that endogenous variables V are discrete and finite ; while exogenous variables U could take any94 ( continuous ) value . The counterfactual distribution P ( yx , . . . , zw ) defined above is thus a categorical95 distribution . For a more detailed survey on SCMs , we refer readers to [ 33 , Ch . 7 ] .96 2 Discretization of Structural Causal Models97 For a DAG G with endogenous V and exogenous variables U , let P ⇤ denote the collection of all98 counterfactual distributions over variables V . Formally,99 P ⇤ = { P ( yx , . . . , zw ) | 8Y , . . . , Z , X , . . . , W ✓ V } . ( 2 ) Let M be the family of all the SCMs compatible with the causal diagram G , i.e. , M =100 { 8M | GM = G } 1 . Counterfactual distributions in G are defined as the collection { P ⇤M : 8M 2M } 101 that contains all counterfactual probabilities induced by SCMs M in the candidate family M . In this102 section , we will show that counterfactual distributions in any causal diagram G could be generated by103 an alternative family of “ generic ” SCMs compatible with G , which we will define later.104 Definition 1 ( Counterfactual-Equivalence ) . For a DAG G , let M , N be two sets of SCMs compatible105 with G. M and N are said to be counterfactually equivalent ( for short , ctf-equivalent ) if for any106 M 2M , there exists an alternative N 2 N such that P ⇤M = P ⇤ N , and vice versa.107 Our analysis rests on a special family of SCMs where values of each exogenous variable are drawn108 from a discrete distribution over a finite set of states.109 Definition 2 . An SCM M = hV , U , F , P i is said to be a discrete SCM if110 1 . Values of every U 2 U are drawn from a discrete distribution P ( u ) over a domain ⌦U ; let111 ✓u denote the probability P ( U = u ) , for any u 2 ⌦U . f112 2 . Values of every V 2 V are decided by function v fV ( paV , uV ) ⌘ ⇠ ( paV , uV ) V , where for113 8paV , uV , ⇠ ( paV , uV ) V is a constant in the finite domain ⌦V .114 Given a causal diagram G , our goal is to construct a family of discrete SCMs N that is counter-115 factually equivalent to the original family of SCMs M . Our construction utilizes a special type of116 clustering of nodes in the diagram , called the confounded component [ 45 ] .117 Definition 3 . For an DAG G , a subset C ✓ V is a c-component if any pair X , Y 2 C is connected118 in G by a bi-directed path of the form V1 $ V2 $ · · · $ Vn , n = 1 , 2 , . . . , where ( 1 ) V1 = X ,119 Vn = Y ; ( 2 ) { V1 , . . . , Vn } ✓ V ; and ( 3 ) each Vi $ Vj is a sequence Vi Uk ! Vj and Uk 2 U .120 A c-component C in G is maximal if there exists no other c-component that contains C. We denote121 by C ( G ) the collection of all maximal c-components in G. Naturally , c-components in C ( G ) form a122 partition over endogenous variables V , which , in turn , defines a partition { [ V 2CUV | 8C 2 C ( G ) } 123 over exogenous variables U . Therefore , for every U 2 U , there must exist a unique c-component124 in C ( G ) , denoted by CU , such that U 2 [ V 2CUUV . For example , exogenous variables U1 , U2 in125 Fig . 1a corresponds to c-components CU1 = { Z } and CU2 = { X , Y } respectively ; while the causal126 diagram of Fig . 1b only has a single c-component { X , Y , Z } .127 1We will use the subscript M to represent the restriction to a specific SCM M . Therefore , GM represents the causal diagram associated with SCM M ; so does the collection of counterfactuals P ⇤M . Theorem 1 . For a DAG G , consider the following conditions2 : ( 1 ) M is the set of all SCMs128 compatible with G ; ( 2 ) N is the set of all discrete SCMs compatible with G where for every U 2 U ,129 its cardinality |⌦U | = Q V 2CU |⌦PaV 7 ! ⌦V | , i.e. , the number of functions mapping from PaV to130 V for every variable V in the c-component CU . Then , M and N are counterfactually equivalent.131 Thm . 1 establishes the expressive power of discrete SCMs in representing counterfactual distributions132 in a causal diagram G. It implies that the counterfactual distribution P ( yx , . . . , zw ) in any SCM M133 could be generated using a generic model as follows , for dU = Q V 2CU |⌦PaV 7 ! ⌦V |,134 P ( yx , . . . , zw ) = X U2U X u=1 , ... , dU 1Yx ( u ) =y ^ · · · ^ 1Zw ( u ) =z Y U2U ✓u . ( 3 ) Among above quantities , ✓u are parameters of the exogenous distribution P ( u ) over a finite domain135 { 1 , . . . , dU } . Counterfactual variables Yx ( u ) are recursively defined as follows:136 Yx ( u ) = { Yx ( u ) | 8Y 2 Y } , where Yx ( u ) = ( xY if Y 2X ⇠ ( { Vx ( u ) |V 2PaY } , uY ) Y otherwise ( 4 ) where xY is the value assigned to variable Y in constants x . As an example , consider the causal137 diagram G described in Fig . 1b where X , Y , Z are binary variables in { 0 , 1 } . Since G has a single c-138 component { X , Y , Z } , exogenous variables U1 , U2 must share the same cardinality d in the proposed139 family of discrete SCMs N . It follows from Thm . 1 the counterfactual distribution P ( z , xz0 , yx0 ) in140 any SCM compatible with G could be written as follows:141 P ( z , xz0 , yx0 ) = dX u1 , u2=1 1 ⇠ ( u1 ) Z =z ^ 1 ⇠ ( z0 , u1 , u2 ) X =x ^ 1 ⇠ ( x0 , u2 ) Y =y ✓u1✓u2 , ( 5 ) where ⇠ ( u1 ) Z , ⇠ ( z , u1 , u2 ) X , ⇠ ( x , u2 ) Y are parameters taking values in { 0 , 1 } ; ✓ui , i = 1 , 2 , are probabilities142 of the discrete distribution P ( ui ) over the finite domain { 1 , . . . , d } . The cardinality d = |⌦Z | ⇥143 |⌦Z 7 ! ⌦X |⇥ |⌦X 7 ! ⌦Y | = 32 . The total cardinalities of domains for U1 , U2 are thus 2d = 64.144 Comparison with related work One could naïvely apply the discretization procedure in [ 3 ] and145 obtain a family of discrete SCMs that are sufficient in representing distributions in an causal diagram.146 However , such parametrization is not necessarily complete . To witness , consider again the causal147 diagram in Fig . 1b with binary X , Y , Z . Applying the discretization in [ 3 ] leads to a family of discrete148 SCMs compatible with a different diagram in Fig . 1c where the cardinality of exogenous variable149 U is equal to d = 32 ( see Appendix D for details ) . However , this parametrization fails to capture150 some critical constraints over counterfactual distributions since it does not maintain the original151 structure of the causal diagram . For instance , counterfactual variables Z and Yx in the original152 diagram of Fig . 1b are independent due to independence restrictions [ 33 , Ch . 7.3.2 ] ; while Z and153 Yx in Fig . 1c are generally correlated due to the presence of unobserved confounder U . Compared154 with [ 3 ] , the discretization method in Thm . 1 captures all constraints over counterfactual distributions155 while requiring only a factor of |U | increase in the cardinality of exogenous domains.156 More recently , [ 15 ] proved a special case of Thm . 1 for interventional distributions in a specific157 class of causal diagrams that satisfy the running intersection property . When there is no direct arrow158 between endogenous variables , [ 38 ] showed that the observational distribution in a diagram could be159 represented using finite-state exogenous variables . Thm . 1 generalizes these results by showing that,160 for the first time , all counterfactual distributions in an arbitrary causal diagram could be generated161 using discrete exogenous variables taking values from a finite domain , without any loss of generality.162 2.1 Partial identification of Counterfactual Distributions163 To demonstrate the expressive power of discrete SCMs , we investigate the problem of partial iden-164 tification of counterfactual distributions . For an SCM M⇤ = hV , U , F , P i , we are interested in165 evaluating an arbitrary counterfactual probability P ( yx , . . . , zw ) . The detailed parametrization of166 M⇤ is unknown . Instead , the learner only has access to the causal diagram G and the observa-167 tional distribution P ( v ) induced by M⇤ . Our goal is to derive an informative bound [ l , r ] from the168 combination of G and P ( v ) that contains the actual counterfactual probability P ( yx , . . . , zw ) .169 2For every V 2 V , ⌦PaV 7 ! ⌦V is the set of all functions mapping from domains ⌦PaV to ⌦V . Let N denote the family of discrete SCMs defined in Thm . 1 which are compatible with the causal170 diagram G. We derive a bound [ l , r ] over P ( yx , . . . , zw ) from the observational data P ( v ) by solving171 the following optimization problem:172 [ l , r ] = min /max n PN ( yx , . . . , zw ) | 8N 2 N , PN ( v ) = P ( v ) o ( 6 ) For instance , consider again the double-bow diagram G in Fig . 1b . The observational distribution173 P ( x , y , z ) in any discrete SCM in N could be written as:174 P ( x , y , z ) = dX u1 , u2=1 1 ⇠ ( u1 ) Z =z ^ 1 ⇠ ( z , u1 , u2 ) X =x ^ 1 ⇠ ( x , u2 ) Y =y ✓u1✓u2 . ( 7 ) One could derive a bound over the counterfactual distribution P ( z , xz0 , yx0 ) from the observational175 data P ( x , y , z ) by solving polynomial programs which optimize the objective Eq . ( 5 ) over parameters176 ✓u1 , ✓u2 , ⇠ ( u1 ) Z , ⇠ ( z , u1 , u2 ) X , ⇠ ( x , u2 ) Y , subject to the observational constraints Eq . ( 7 ) .177 As a corollary , it follows immediately from Thm . 1 that the solution [ l , r ] of the optimization problem178 Eq . ( 6 ) is guaranteed to be a valid bound over the unknown counterfactual P ( yx , . . . , zw ) .179 Corollary 1 ( Soundness ) . Given a DAG G and an observational distribution P ( v ) , let M be the set180 of all SCMs compatible with G and let Mo = { 8M 2M | PM ( v ) = P ( v ) } . For the solution [ l , r ] 181 of Eq . ( 6 ) , PM ( yx , . . . , zw ) 2 [ l , r ] for any SCM M 2Mo.182 Since the underlying SCM M⇤ 2Mo , Corol . 1 implies that the derived bound [ l , r ] must contain the183 actual counterfactual probability P ( yx , . . . , zw ) . Our next result shows that such a bound [ l , r ] is184 provably tight , i.e. , it can not be improved without additional assumptions.185 Corollary 2 ( Tightness ) . Given a DAG G and an observational distribution P ( v ) , let M be the set186 of all SCMs compatible with G and let Mo = { 8M 2M | PM ( v ) = P ( v ) } . For the solution [ l , r ] 187 of Eq . ( 6 ) , there exist SCMs M1 , M2 2Mo such that PM1 ( yx , . . . , zw ) = l , PM2 ( yx , . . . , zw ) = r.188 Corol . 2 confirms the tightness of the bound [ l , r ] obtained from Eq . ( 6 ) . Suppose there exists a valid189 bound [ l0 , r0 ] strictly contained in [ l , r ] . One could construct from Corol . 2 an SCM M compatible190 with the causal diagram G and the observational distribution P ( v ) , but its counterfactual probability191 P ( yx , . . . , zw ) lies outside [ l0 , r0 ] , which is a contradiction.192 The optimization problem of Eq . ( 6 ) is reducible to equivalent polynomial programs ( see Appendix E ) .193 Despite the soundness and tightness of derived bounds , solving such programs may take exponentially194 long in the most general case [ 29 ] . Our focus here is upon the causal inference aspect of the problem195 and like earlier discussions we do not specify which solvers are used [ 3 , 4 ] . In some cases of196 interest , effective approximate planning methods for polynomial programs do exist . Investigating197 these methods is an ongoing subject of research [ 26 , 31 , 48 , 28 , 27 ] .198 3 Bayesian Approach for Partial Identification199 This section describes an effective algorithm to approximate the optimal counterfactual bound in200 Eq . ( 6 ) , provided with finite samples v̄ = v ( n ) N n=1 drawn from the observational distribution201 P ( v ) , and prior distributions over parameters ✓u and ⇠ ( paV , uV ) V ( possibly uninformative ) .202 We first introduce Markov Chain Monte Carlo ( MCMC ) algorithms that sample the posterior distribu-203 tion P ( ✓ctf | v̄ ) over a counterfactual probability ✓ctf = P ( yx , . . . , zw ) . More specifically , for every204 V 2 V , 8paV , uV , parameters ⇠ ( paV , uV ) V are drawn uniformly over the finite domain ⌦V . For every205 U 2 U , exogenous probabilities ✓u are drawn from a generalized Dirichlet distribution [ 12 ] . We will206 take the view of a stick-breaking construction [ 40 ] which successively breaks pieces off a unit-length207 stick with size proportional to random draws from a Beta distribution . Parameters ✓u are proportions208 of each of the pieces relative to its original size . Formally,209 8u = 1 , 2 , . . . , dU , ✓u = µu u 1Y i=1 ( 1 µi ) , µu ⇠ Beta ⇣ ↵ ( u ) U , ( u ) U ⌘ , ( 8 ) For every endogenous variable V 2 V , ⇠V = n ⇠ ( paV , uV ) V | 8paV , uV o . where dU = Q V 2CU |⌦PaV 7 ! ⌦V | and ↵ ( u ) U , ( u ) U > 0 are hyperparameters . Finally , we truncate210 this construction by setting µdU = 1 . Note from Eq . ( 8 ) that all parameters ✓u for u > dU are equal211 to zero . As an example , Fig . 2 shows a graphical representation of the data-generating process over212 parameters ✓u and ⇠ ( paV , uV ) V associated with SCMs in Fig . 1b , spanning over N observations.213 Gibbs sampling is a well-known MCMC algorithm that allows one to sample posterior distributions.214 For convenience , we introduce the following notations . Let parameters ✓ = { ✓u | 8U 2 U , 8u } 215 and ⇠ = n ⇠ ( paV , uV ) V | 8V 2 V , 8paV , uV o . The set Ū = U ( n ) N n=1 are exogenous variables216 affecting N observations V̄ = V ( n ) N n=1 ; we use ū to represent their realizations . Our blocked217 Gibbs sampler works by iteratively drawing values from the conditional distributions of variables as218 follows [ 22 ] . Detailed derivations of complete conditional distributions are shown in Appendix F.219 Sampling P ( ū | v̄ , ✓ , ⇠ ) . Exogenous variables U ( n ) , n = 1 , . . . , N , are mutually independent220 given parameters ✓ , ⇠ . We could draw each U ( n ) | ✓ , ⇠ , V̄ corresponding to the nth observation221 independently . The complete conditional for U ( n ) is given by222 P ⇣ u ( n ) | v ( n ) , ✓ , ⇠ ⌘ / Y V 2V 1 ⇠ ( pa ( n ) V , u ( n ) V ) V =v ( n ) Y U2U ✓u . ( 9 ) Sampling P ( ⇠ , ✓ | v̄ , ū ) . Parameters ⇠ , ✓ are independent given V̄ , Ū . Therefore , we will derive223 complete conditional ⇠ , ✓ separately . Note that in discrete SCMs , the nth observation of variable224 V 2 V is decided by v ( n ) ⇠ ( paV , uV ) V given pa ( n ) V = paV , u ( n ) V = uV . Thus , draw values of each225 ⇠ ( paV , uV ) V 2 ⇠ from the complete conditional defined as:226 P ⇣ ⇠ ( paV , uV ) V | v̄ , ū ⌘ = ( 1 ⇠ ( paV , uV ) V =v ( i ) if 9i , s.t . pa ( i ) V = paV , u ( i ) V = uV , 1/|⌦V | otherwise. . ( 10 ) Let nu = PN n=1 1u ( n ) =u records the number of values in u ( n ) that are equal to u . By the conjugacy227 of the generalized Dirichlet distribution , the complete conditional of ✓u is given by , for every U 2 U ,228 8u = 1 , 2 , . . . dU , ✓u = µu u 1Y i=1 ( 1 µi ) , µu ⇠ Beta ↵ ( u ) U + nu , ( u ) U + dUX k=u+1 nk ! . ( 11 ) Doing so eventually produces values drawn from the posterior distribution over ✓ , ⇠ , Ū | V̄ . Given229 parameters ✓ , ⇠ , we compute the counterfactual probability ✓ctf = P ( yx , . . . , zw ) following the230 three-step algorithm in [ 33 ] which consists of abduction , action , and prediction . Thus computing ✓ctf231 from each draw ✓ , ⇠ , Ū eventually gives us the draw from the posterior distribution P ( ✓ctf | v̄ ) .232 3.1 Collapsed Gibbs Sampling233 We also describe an alternative sampler that applies to stick-breaking priors with a known Pólya234 urn characterization . Formally , consider stick-breaking priors in Eq . ( 8 ) with hyperparameters235 ↵ ( u ) U = ↵U/dU and ( u ) U = ( dU u ) ↵U/dU for some real ↵U > 0 . Let Ū n denote the set236 difference Ū \U ( n ) ; so does V̄ n = V̄ \ V ( n ) . Our collapsed Gibbs sampler first iteratively draws237 values from the conditional distribution of U ( n ) | Ū n , V̄ , n = 1 , . . . , N , as follows.238

Summary: The paper presents a methodology to obtain bounds on counterfactual probabilities from unknown SCMs using observational data. The authors consider a setting where the causal structure (DAG) of the data-generating model is known and all the endogenous variables are discrete and finite.    In this setting, the authors show that the set of all SCMs compatible with the known DAG, G, and observational data can be mapped to a special class of SCMs, which they call discrete SCMs. These discrete SCMs have the nice property that they are compatible with DAG G and the observational distribution and all exogenous variables are discrete and finite (with known maximum value). The authors show that there must exist a discrete SCM within this family which agrees with the original (unknown) SCM on all counterfactual probabilities. Using this as the main idea, the authors attempt to bound the counterfactual probabilities by maximising/minimising the counterfactual probabilities over all discrete SCMs in this family.    The paper makes no assumptions about the missing exogenous variables in the SCM model under question. This makes for a very powerful idea which can be used to bound counterfactual queries.