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: EDIT:  I thank the authors for their clarifications and corrections to my previous line of thought, and I am prepared to revise my score up to 5 as a result.  I take the point about polynomial vs linear programming, and I can see that Lemma 4 is a non-trivial result.  The authors, however, make no effort to point to Lemma 4 as being the primary source of the novelty in Theorem 1!  Overall, the content of the paper seems too rich for a conference with a page limit like this one, it might be better off submitted to a journal with much more explanation and more examples.  ----  This paper gives a Monte Carlo method for obtaining bounds from any DAG model with hidden variables.   The authors illustrate the algorithm on the front door model, as well as the instrumental variables model; they give several more examples in the supplementary material.    I think the paper could be improved by explaining more explicitly how the Bayesian algorithm for obtaining samples helps, when presumably one could solve the linear program (6).