Causal Identification with Matrix Equations

1 Introduction . Cause and effect relations are one of the most common types of knowledge sought after throughout the empirical sciences . These relations are one of the main ingredients in the construction of stable explanations , and usually underpin robust and generalizable decision-making strategies [ 19 , 26 ] . There is a growing literature that aims to systematically find causal relations by fusing observations , experiments , and substantive knowledge about the phenomenon under investigation [ 19 , 3 , 20 ] . Formally , the inferential target usually appears as the effect of a set of variables do ( X = x ) on another set of variables Y , which is written as P ( y|do ( x ) ) or Px ( y ) . Assumptions about the underlying data-generating processes are commonly expressed as a causal graph G over endogenous variables V. There are different lines of investigation that aims to establish the quantity Px ( y ) . First , one line of investigation attempts to exploit the non-parametric constraints encoded in G to determine whether the quantity Px ( y ) ( i.e. , query ) is uniquely computable from the different available distributions . For instance , Pearl ’ s celebrated method known as do-calculus provides a symbolic way of determining whether a query can be identified from G and observational data P ( V ) [ 18 ] . A number of necessary and sufficient conditions were developed for systematically determining the identifiability status of the query from observational data [ 18 , 29 , 23 , 9 ] . For example , the causal effect Px ( y ) in Fig . 1a is identified by a back-door criterion as Px ( y ) = P u P ( y|x , u ) P ( u ) . Further methods were developed to identify the query from a combination of observational and experimental distributions [ 2 , 14 , 12 , 11 ] . The key ideas behind these methods are ( i ) to decompose the given causal query into standard factors ( following Tian ’ s c-factorization , [ 29 ] ) , leveraging the graphical 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . constraints , and ( ii ) to identify these factors individually , matching one of the available distributions . We call this the factorization-based approach . Another line of investigation attempts to exploit assumptions about the relationship between unobserved confounders and the observable variables through the idea of proxy variables [ 5 , 8 , 10 , 15 , 32 ] . These methods rely on the cardinality and complexity of these relationships , which will possibly lead to the invertibility of certain matrices . The focus is usually on local conditions involving the treatment X , the outcome Y , and the proxies W , Z for the set of unobserved confounders U ; for example , see Fig . 1b . Note that since U is unobserved , the effect of X on Y , i.e. , Px ( y ) , is provably not identifiable by standard graphical methods discussed earlier . Still , if the distribution of the proxy given the unobserved confounder is available P ( W |U ) , in addition to P ( X , Y , W ) , the quantity Px ( y ) can be computed under the invertibility of P ( W |U ) , a matrix representation of P ( W |U ) where P ( W |U ) i , j = P ( wi|uj ) ( see Eq . ( 3 ) for detail ) . In case of Fig . 1c , the causal effect Px ( y ) is identified from P ( X , Y , W , Z ) through the following,1 Px ( y ) = P ( y|Z , x ) P ( W |Z , x ) 1P ( W ) , ( 1 ) where P ( W |Z , x ) has rank matching the size of domain of the unmeasured confounder U . Although typically not framed in terms of identification with multiple datasets , this setting corresponds to identifying Px ( y ) with , e.g. , marginal distributions { P ( X , Y , Z ) , P ( W , X , Z ) } or conditional distributions { P ( Y |Z , x ) , P ( W |Z , x ) , P ( W ) } . We call this the proxy approach.2 Despite all the power and successes achieved by the factorization and proxy approaches , there exist still interesting cases not covered by any of them individually . To witness , consider the causal graph in Fig . 1d and first notice that the effect is not identified from each approach . On the other hand , if we combine both approaches , we are able to obtain the causal effect is identifiable and given by , Px ( y ) = P t Px ( t , y ) = P t Pt , x1 ( y ) Px2 ( t ) = P t ⇥ P ( y|Z1 , x1 , t ) P ( W1|Z1 , x1 , t ) 1P ( W1 ) ⇤⇥ P ( t|Z2 , x2 ) P ( W2|Z2 , x2 ) 1P ( W2 ) ⇤ , where Pt , x1 ( y ) and Px2 ( t ) , the two factors of the query , are individually identified with the help of the proxy approach , i.e. , Eq . ( 1 ) . Our goal in this paper is to explicate how this can be accomplished from first principles . More broadly , we will develop a flexible and general graphical identification approach that combines both the factorization and matrix equations ( the underpinning idea of proxy-based methods ) . More specifically , our contributions are as follows : 1 . We connect the graphical and matrical approaches by characterizing matrix equations of probability distributions driven by graphical constraints in a causal graph . This leads to a better understanding of the identification through solving systems of equations . 2 . Building on this new characterization , we generalize proxy-based criteria and devise novel intermediary pseudoinverse criteria so as to identify a causal effect by utilizing the general inverse of a matrix and diverse collection of distributions . 1We focus on discrete variables . Detailed discussion on a continuous case can be found in [ 15 ] . We refer this identification condition MGT criterion where the acronym MGT comes from the surnames of the authors in [ 15 ] . 2Recent work in this line of research use the term proximal causal inference ( PCI ) [ 25 , 27 , 33 , 6 ] . [ 27 ] generalizes the results in [ 15 ] to allow observed confounders as well . [ 25 ] considered combining graphical and proximal approaches , similar to the motivational example . 3 . We develop a general identification algorithm that amalgamates graphical and matrical approaches , returning an identification formula for a causal query given a causal diagram and a set of marginal , experimental , and conditional distributions . We show that this method subsumes current state of the art in the literature . 2 Preliminaries . We follow notational conventions from literature on causal inference . We denote a variable by an upper case letter Y , and its value is denoted by its corresponding lower case letter y in the domain XY . A set of variables will be denoted by a bold capital letter Y with its value y . We may use [ ̇ , instead of [ , to emphasize the union of two disjoint sets . Given Z ✓W , w\Z denotes the value of W\Z consistent with w. Let a/B = ( A \B , a\B ) which retains B as a set of variables and values of a excluding B . Without loss of generality , we refer PZ ( V0|W ) a distribution . We may employ conditional to emphasize W ◆ ; , experimental or interventional Z ◆ ; compared to observational Z = ; , and marginal if Z [ W [ V0 ( V. We employ structural causal models ( SCMs ) [ 19 , Ch . 7 ] as the semantical framework to represent a domain of interest . An SCM M is a quadruple hU , V , P ( U ) , Fi . A set of exogenous variables U , which follows P ( U ) , is determined by factors outside the model . V is a set of endogenous variables whose values are determined by functions F = { fi } Vi2V such that Vi fi ( pai , ui ) where PAi ✓ V\ { Vi } and Ui ✓ U . Further , do ( x ) represents the operation of holding a set X to a constant x regardless of their original mechanisms . Such intervention induces a submodel Mx , which is M with fX replaced to x for X 2 X . The distribution over V induced by the submodel is denoted by Px ( V ) . We may employ letter Q to denote an interventional distribution , e.g. , Q = Pr . Each SCM ( model , for short ) induces a causal diagram ( or causal graph ) G = hV , Ei , where each type of edge represents a different causal relationship among the variables : ( i ) X ! Y if X is used as an argument of fY ; and ( ii ) X $ Y if UX and UY are correlated . Given a causal diagram G , familial relationships among its vertices are denoted by pa and an for parents and ancestors , respectively . Further , An is a set of ancestors including its argument as well . We denote by GXZ an edge subgraph of G which removes edges incoming to X and outgoing from Z. Causal relationships among other variables are captured in G\X , which is the subgraph of G over V\X . A vertex induced subgraph is denoted by G [ V0 ] where V0 ✓ V. Causal effect identification relies heavily on standard graphical constraints imposed by a causal diagram such as d-separation ( reading off conditional independence from the graph , [ 30 , 7 ] ) and do-calculus ( equivalence among interventional probabilities ) [ 18 ] , which we present in the Appendix for completeness . The latent projection ( or projection , for short ) of a causal diagram is a causal diagram retaining the causal relationships among a subset of variables . We denote by GhV0i the latent projection of G onto V0 ✓ V , the causal graph over V0 [ 31 ] . Conditional independence ( CI ) statements and do-calculus [ 18 ] on a projection are valid in G , vice versa . We formally define a latent projection in the supplementary material . The omitted proofs and derivations are also provided in [ 13 ] . 3 Characterization of Matrix Equations of Graphical Constraints . In this section , we present characterizations of graphical constraints in a given causal diagram G , leading to equations expressed as the multiplication of matrices . The characterizations will further advance our understanding on the constraints imposed over the distributions generated by the underlying system compared to simple equivalence relationships such as conditional independence and do-calculus . To begin with , we denote by P the matrix representation of a distribution P where free outcome variables are rows and free condition or intervention variables correspond to columns . For instance , Pa , B ( C , d ) is a |XC |⇥ |XB | matrix and Pr ( A , b|C , d ) is a |XA|⇥ |XC | matrix . Further , we may use 0 and 00 to represent two disjoint subsets such that B = B0 [ ̇ B00 . Chain Rule with Conditional Independence The definitions of conditional and marginal distributions naturally lead to a sum of a product of probabilities . Let Q = Pr be an arbitrary interventional distribution . Let A , B , C , and R be disjoint . A marginal probability over chain-rule-induced multi- plication is expressed as Q ( a , b0|c ) = X b00 Q ( a|b , c ) Q ( b|c ) = Q ( a|b0 , B00 , c ) Q ( B00 , b0|c ) . Considering conditional independence , we can further enrich such a characterization . Lemma 1 . Given a causal diagram G , let Q = Pr for some r 2 XR where R ( V. Let A , B , C , D , E be disjoint subsets of V\R . If ( D ? A | B , C , E ) and ( E ? B | C , D ) in G\R , then , Q ( A , b0|c , D , e ) = Q ( A|b0 , B00 , c , e ) Q ( B00 , b0|c , D ) . The lemma emphasizes the condition under which the result of multiplication is a matrix not just a row or column . A special case of the lemma is appeared in MGT criterion where the inverse of a matrix multiplication P ( W |Z , x ) = P ( W |U , x ) P ( U |Z , x ) is utilized so as to cancel out an unknown distribution P ( W |U ) = P ( W |U , x ) . Adjustment Criterion Given a graph G and a causal effect of interest Px ( y ) , the adjustment criterion [ 24 ] seeks a set of covariates Z ✓ V\X\Y , called an adjustment set for a causal effect Px ( y ) , which grants the following expression , Px ( y ) = P z P ( y|x , z ) P ( z ) . Adjustment criterion generalizes back-door criterion [ 19 ] . Its matricized expression with employing Q = Pr is Qx ( y ) = X z Q ( y|x , z ) Q ( z ) = Q ( y|x , Z ) Q ( Z ) . This simple expression plays a central role in the identification with proxy variables . In many settings , the left hand side ( LHS ) is the query of interest and two terms in the RHS are usually available or to be inferred using other available quantities . However , substituting value y with Y , we can further yield ( under invertibility assumption ) Q ( Z ) = Q ( Y|x , Z ) 1Qx ( Y ) , which restores the covariate distribution of interest given a causal effect and conditional distribution . C-Factorization C-factorization [ 28 ] decomposes a causal effect Px ( y ) into the sum-product of c-factors ( simply , factors ) with respect to the given causal diagram G. Without loss of generality , let X be minimal such that no X0 ( X satisfies Px0 ( y ) 6= Px ( y ) ( i.e. , overriding X by anGX ( Y ) \X ) . For any projection H of G that preserves X [ Y , the following holds : Px ( y ) = X y+\y Px ( y + ) = X y+\y Y Yi2C ( H [ Y+ ] ) PpaH ( yi ) \yi ( yi ) , ( 2 ) where Y+ = AnHX ( Y ) and C ( · ) is the c-component decomposition ( partitioning the variables in the graph based on their connectivity through bidirected edges ) . Given a query Px ( y ) , we let c-factors FH = { hXi , Yii } i where { Yi } i form the c-component decomposition of H [ Y+ ] and Xi = paH ( Yi ) \Yi . The general identification method under partial-observability [ 12 ] can be summarily described as finding H such that each factor is identified by one of the available distributions . Let us denote Xij = Xi [ Xj . We focus here on the sum-product of a pair of factors at some latent projection H , which satisfies Pxi ( yi ) Pxj ( yj ) = Pxij\yij ( yij ) where the values are consistent [ 12 ] . Now consider the summation over Z ✓ Xi\Yj . Then , X z Pxi\z , z ( yi ) Pxj ( yj\z , z ) = Pxij\yij ( yij\Z ) . To properly represent this as a matrix multiplication , we should decide which variables will become rows and columns . Two matrices to be multiplied should share only Z , and other common variables , appearing in the two terms , or non-matching variables , appearing as intervention in left term and outcome in right term , need to be set to constants . To help understand , we illustrate in Fig . 2a the set relationships among Xi , Yi , Xj , and Yj in the case of Z = Xi \Yj . Therefore , Yi\Xj ( the shared variables other than Z ) and ( Xi [ Yj ) \Z ( the non-matching variables ) are fixed . These fixed variables are not colored in Fig . 2a . As a result , we can obtain a submatrix of PXij\Yij ( Yij\Z ) as the multiplication of the submatrices of PXi ( Yi ) and PXj ( Yj ) obtained via fixing variables . Lemma 2 ( Matrix Equation of C-Factorization with Two Factors ) . Given a causal diagram G and an experimental distribution Q = Pr where a causal effect is c-factorized as Qx ( y ) =P z Qxi ( yi ) Qxj ( yj ) in a projection H of G\R , the effect can be represented as a matrix multiplication , if Z ✓ Xi \Yj . Further , the corresponding matrix equation is Q ( xij\yij ) / ( Xj\Xi\Yi ) ( ( yij\Z ) / ( Yi\Xj ) ) = Qxi/Z ( yi/ ( Yi\Xj ) ) Qxj/ ( Xj\Xi\Yi ) ( yj/Z ) . We illustrate a causal graph in Fig . 2b where each variable matches to each region in the Euler diagram ( Fig . 2a ) . With Z = { 6 } and for an arbitrary instantiation of variables V2 , V4 , V5 , and V7 , PV3 , v4 , v5 ( V1 , v2 , v7 ) = Pv4 , v5 , V6 ( V1 , v2 ) Pv2 , V3 , v4 ( V6 , v7 ) . In this section , we connected different graphical constraints—from chain-rule with conditional independence to c-factorization—induced identification formulae to matrix equations . Results presented in this section are by no means complete yet cover , to the best of our knowledge , every sum-product expression appeared in both graphical and matrical identification approaches . Nevertheless , this suite of characterizations will provide a fundamental understanding of the mathematical structures involved in the identification methods with matrical expressions . 4 Generalized Proxy-based Criteria Equipped with the characterization from the previous section , we revisit single- and double-proxy settings [ 5 , 8 , 10 , 15 ] more formally , which identify a causal effect through the combination of chain rule , adjustment criterion ( c-factorization ) , and inverses of matrices given an observational distribution and additional external study involving an unobserved confounder . We investigate its extension capable of utilizing other types of available distributions . Results presented in this section is crucial in adopting matrical approaches into a factorization-based identification algorithm . Single-Proxy Setting We illustrate in Fig . 4 the available distributions and unknown distributions , considered in Fig . 1b , that can lead to Px ( y ) . In the figure , a matrix multiplication A = B · C is represented as B L ! A R C with positions annotated . First , note that ( X , Y ? W | U ) in a causal graph G is central , which grants P ( W |U , x , y ) = P ( W |U , x ) = P ( W |U ) . If P ( W |U ) is invertible,3 the three distributions P ( y , U |x ) , P ( U |x ) , and P ( U ) are obtained where P ( y|U , x ) is computed by chain rule ( Lemma 1 ) . Then , Px ( y ) is , with ↵ denoting elementwise division , P ( y|U , x ) z } | { ( P ( W |U ) 1P ( y , W |x ) ) | { z } P ( y , U |x ) ↵ ( P ( W |U ) 1P ( W |x ) ) | { z } P ( U |x ) > P ( W |U ) 1P ( W ) | { z } P ( U ) . ( 3 ) However , this scheme would not work in more restrictive conditions such as Fig . 3a or 3b . This challenging scenario can be handled if a different external study is available altogether with a surrogate 3In case of W has more categorical values than U , one can coarsen W to W 0 ensuring that P ( W 0|U ) is full rank . In other words , P ( W |U ) has full column rank . experiment . For example , when only Y is independent to W given U and X , the availability of an external study P ( W |U , x ) , instead of P ( W |U ) , and a surrogate experiment Px ( W ) , in addition to the observational distribution P ( Y , W , X ) , suffices to identify the causal effect ( see [ 13 ] for the derivations for Fig . 3a , 3b ) . These examples suggest that we can generalize the single proxy-based criterion to take advantage of a diverse collection of external studies and surrogate experiments . Theorem 1 . Given G , let X , Y , W , U , and R be disjoint subsets of V , Q = Pr for some r 2 XR , and H = G\R . A causal effect Pr , x ( y ) = Qx ( y ) is identifiable if ( 1 ) U is an adjustment admissible set for Qx ( y ) in H ; ( 2a ) ( Y ? W | U , X ) H and Q ( W|U , x ) , Q ( y , W|x ) and Q ( W|x ) are available where Q ( W|U , x ) has full column rank ; or ( 2b ) ( Y 6 ? W | U , X ) H and Q ( W|U , x , Y ) and Q ( Y , W|x ) are available where every Q ( W|U , x , y0 ) has full column rank for y0 2 XY ; ( 3 ) Qz0 ( W ) is available for Z ✓ X [ Y and Z0 = ( X [ Y ) \Z with z0 consistent with x [ y such that ( Z ? W | U , Z0 ) in H ; and U is an adjustment admissible set for Qz0 ( W ) in H. The theorem presents a sound condition to elicit the causal effect through combining various distributions especially when the given situation is more restrictive . Double-Proxy Setting We now generalize MGT criterion to utilize distributions other than the originally considered observational study . MGT criterion for a double-proxy setting relies on the following conditions to identify Px ( y ) with P ( X , Y , Z , W ) : ( C1 ) U is an adjustment set for Px ( y ) in G ; ( C2 ) Y ? Z | U , X in G ; ( C3 ) Z , X ? W | U in G ; and ( C4 ) P ( W |Z , x ) has rank |XU | . Under these conditions , algebraic relationships between a causal effect Px ( y ) and other distributions can be illustrated as in Fig . 5a where the distributions form a closed loop alternating between ( i ) distributions with an unmeasured confounder and ( ii ) given distributions and a query . Among the four multipli- cations , Px ( y ) corresponds to an adjustment criterion ( C1 ) so as to Px ( y ) = P ( y|U , x ) P ( U ) . Others are due to the chain-rule combined with CI , e.g. , ( C2 ) P ( y|U , x ) = P ( y|U , x , z ) and ( C3 ) P ( W |U ) = P ( W |U , x , z ) . With ( C4 ) , which implies that both P ( W |U ) and P ( U |Z , x ) are invertible under coarsening of W and Z if necessary [ 15 , 1 ] , the causal effect Px ( y ) can be expressed as Eq . ( 1 ) by subsequently rewriting P ( U |Z , x ) and P ( y|U , x ) , and contracting P ( W ) = P ( W |U ) P ( U ) ( see [ 13 ] Appendix C for an illustration ) . To examine the possible extension of the setting , we relax ( C3 ) , where the CI grants the use of matrix multiplication leading to a chain-rule P ( W ) = P ( W |U ) P ( U ) and P ( W |Z , x ) = P ( W |U ) P ( U |Z , x ) . One can consider three relaxed versions of ( C3 ) , namely , ( Z ? W | U , X ) , ( X ? W | U , Z ) , and , ultimately , dropping ( C3 ) illustrated respectively in Fig . 3c , 3d and 3e . For concreteness , consider the first relaxation ( Z ? W | U , X ) . The assumption grants P ( W |U , Z , x ) =P ( W |U , x ) . Given that a surrogate experiment Px ( W ) is accessible , and it can be decomposed as Px ( W ) = P ( W |U , x ) P ( U ) where U is also an adjustment admissible set for Px ( w ) , then , Px ( y ) is identified . Other two relaxations turned out to be much more challenging , yet the causal effect can be identified by exploiting different surrogate experiments or additional external study ( see [ 13 ] Appendix C for detail ) . Motivated by these examples , we present a theorem that extends MGT criterion with varying degrees of the assumption and data collection . Theorem 2 ( Generalized MGT Criterion ) . Given a causal graph G , let X , Y , Z , W , U , R ⇢ V be disjoint sets of variables where R can be empty . Let Q = Pr for some r 2 XR and H = G\R . Let S ✓ X [ Z . A causal effect Qx ( y ) = Px , r ( y ) is identifiable in G if , for some z , ( 1 ) U is an adjustment set for Qx ( y ) in H ; ( 2 ) ( Y ? Z | U , X ) H ; ( 3 ) ( W ? S0 | U , S ) H where S0 = ( X [ Z ) \S ; ( 4 ) U is an adjustment set for Qs0 ( W ) in H ; ( 5 ) Q ( W|U , s0 ) is invertible ; ( 6 ) Q ( U|z/ ( S [ Z0 ) , x ) is invertible for some Z0 ✓ Z\S and Q ( y|z/ ( S [ Z0 ) , x ) , Q ( W|z/ ( S [ Z0 ) , x ) , and Qs0 ( W ) are available . Additionally , Q ( W|U , s0/Z0 ) is available if Z0 6= ; . The theorem broadens the applicability of the MGT criterion as an identification template to a range of collection of distributions . Although the conditions involved in generalizing MGT criterion non-trivial , we can similarly draw a ( simplified ) big picture of the generalized criterion as in Fig . 5b . The key idea is relaxing the C3 condition for a set of variables while considering sets of variables ( explicitly ) and interventional distributions . We have established generalized identification criteria exploiting proxy variables.4 The purpose of both criteria aligns well with the philosophy of general identification [ 14 , 12 ] so that they can be smoothly integrated into graphical approaches .

Summary: This paper provides a graphical identification approach that combines both graphical and matrical approaches. In particular, they apply certain matrix chain-rule and inversion criteria, previously used in for example certain proxy-based approaches, to a matrix version of the C-factorization to arrive at a new causal effect identification algorithm. They show that their algorithm applies to cases that could not be identified with certain previous algorithms and thereby rendering them incomplete.