Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables

1 Introduction . A standard problem setting in causal inference is to estimate the causal effect between two variables given a causal graphical model that specifies qualitative causal relations among observed variables [ Pearl , 2009 ] , including a possible presence of hidden confounding variables . The graphical model then allows to employ graphical criteria to identify valid adjustment sets , the most well-known being the backdoor criterion [ Pearl , 1993 ] and the generalized adjustment criterion [ Shpitser et al. , 2010 , Perković et al. , 2015 , 2018 ] , providing a complete identification of all valid adjustment sets . Estimators of causal effects based on such a valid adjustment set as a covariate are then unbiased , but for different adjustment sets the estimation variance may strongly vary . An optimal adjustment set may be characterized as one that has minimal asymptotic estimation variance . In current work , following Kuroki and Cai [ 2004 ] and Kuroki and Miyakawa [ 2003 ] , Henckel et al . [ 2019 ] ( abbreviated 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . HPM19 in the following ) showed that graphical optimality always holds for linear models in the causally sufficient case where all relevant variables are observed . In Witte et al . [ 2020 ] an alternative characterization of the optimal adjustment set is discussed and the approach was integrated into the IDA algorithm [ Maathuis et al. , 2009 , 2010 ] that does not require the causal graph to be known . Rotnitzky and Smucler [ 2019 ] extended the results in HPM19 to asymptotically linear non-parametric graphical models . HPM19 ’ s optimal adjustment set holds for the causally sufficient case ( no hidden variables ) and the authors gave an example with hidden variables where optimality does not hold in general , i.e. , the optimal adjustment set depends on the coefficients and noise terms ( more generally , the distribution ) , rather than just the graph . Most recently , Smucler et al . [ 2021 ] ( SSR20 ) partially extended these results to the non-parametric hidden variables case together with dynamic treatment regimes , i.e. , conditional causal effects . SSR20 provide a sufficient criterion for an optimal set to exist and a definition based on a certain undirected graph-construction using a result by van der Zander et al . [ 2019 ] . However , their sufficient criterion is very restrictive and a current major open problem is a necessary and sufficient condition for an optimal adjustment set to exist in the hidden variable case and a corresponding definition of an optimal set . My main theoretical contribution is a solution to this problem . Optimality for conditional causal effects in the hidden variables case is fully characterized by an information-theoretic approach involving a certain difference of conditional mutual informations among the observed variables termed the adjustment information . Maximizing the adjustment information formalizes the common intuition to choose adjustment sets that maximally constrain the effect variable and minimally constrain the cause variable . This allows to derive a necessary and sufficient graphical criterion for the existence of an optimal adjustment set . The derived optimal adjustment set also has the property of minimum cardinality , i.e. , no node can be removed without sacrificing optimality . Further , the optimal set is valid if and only if a valid adjustment set exists and has higher ( or equal ) adjustment information than the Adjust-set proposed in Perković et al . [ 2018 ] for any graph , whether graphical optimality holds or not . The results translate to minimal asymptotic estimation variance for a class of estimators whose asymptotic variance follows a certain information-theoretic relation that , at present , I could only verify theoretically for the linear case . As practical contributions the paper provides extensive numerical experiments that corroborate the theoretical results and show that the optimal adjustment set or minimized variants thereof often yield better variance also beyond the theoretically analyzed estimator class . Code is available in the python package https : //github.com/jakobrunge/tigramite . More detailed preliminaries , proofs , algorithms , and further numerical experiments are given in the Supplementary Material . 1.1 Preliminaries and problem setting . We consider causal effects in causal graphical models over a set of variables V with a joint distribution P = P ( V ) that is consistent with an acyclic directed mixed graph ( ADMG ) G = ( V , E ) . Two nodes can have possibly more than one edge which can be directed ( ← ) or bi-directed ( ↔ ) . See Fig . 1A for an example . Kinships are defined as usual : parents pa ( X ) for “ •→X ” , spouses sp ( X ) for “ X↔• ” , children ch ( X ) for “ X→• ” . These sets all exclude X . Correspondingly descendants des ( X ) and ancestors an ( X ) are defined , which , on the other hand , both include X . The mediator nodes on causal paths from X to Y are denoted M = M ( X , Y ) and exclude X and Y . For detailed preliminaries , including the definition of open and blocked paths , see Supplementary Section A . In this work we only consider a univariate intervention variable X and effect variable Y . We simplify set notation and denote unions of variables as { W } ∪M ∪A = WMA . A ( possibly empty ) set of adjustment variables Z for the total causal effect of X on Y in an ADMG is called valid relative to ( X , Y ) if the interventional distribution for setting do ( X = x ) [ Pearl , 2009 ] factorizes as p ( Y |do ( X = x ) ) = ∫ p ( Y |x , z ) p ( z ) dz for non-empty Z and as p ( Y |do ( X = x ) ) = p ( Y |x ) for empty Z . Valid adjustment sets , the set of which is here denoted Z , can be read off from a given causal graph using the generalized adjustment criterion [ Perković et al. , 2015 , 2018 ] which generalizes Pearl ’ s back-door criterion [ Pearl , 2009 ] . To this end define forb ( X , Y ) = X ∪ des ( YM ) ( 1 ) ( henceforth just denoted as forb ) . A set Z is valid if both of the following conditions hold : ( i ) Z ∩ forb = ∅ , and ( ii ) all non-causal paths from X to Y are blocked by Z . An adjustment set is called minimal if no strict subset of Z is still valid . The validity conditions can in principle be manually checked directly from the graph , but , more conveniently , Perković et al . [ 2018 ] define an adjustment set called ‘ Adjust ’ that is valid if and only if a valid adjustment set exist . In our setting including conditioning variables S we call this set the valid ancestors defined as vancs ( X , Y , S ) = an ( XY S ) \ forb ( 2 ) and refer to this set as vancs or Adjust-set . Our quantity of interest is the average total causal effect of an intervention to set X to x vs. x′ on the effect variable Y given a set of selected ( conditioned ) variables S = s ∆yxx′|s = E ( Y |do ( x ) , s ) − E ( Y |do ( x′ ) , s ) . ( 3 ) We denote an estimator given a valid adjustment set Z as ∆̂yxx′|s.z . In the linear case ∆yxx′|s for x = x′ + 1 corresponds to the regression coefficient βY X·ZS in the regression of Y on X , Z , and S. The ordinary least squares ( OLS ) estimator β̂Y X·ZS is a consistent estimator of βY X·ZS . Figure 1A illustrates the problem setting : We are interested in the total causal effect of ( here univariate ) X on Y ( conditioned on S ) , which is here due to a direct link and an indirect causal path through a mediator M . There are six valid backdoor adjustment sets Z = { Z1 , Z2 , Z1Z2 , Z2Z3 , Z1Z3 , Z1Z2Z3 } . Z4 ∈ forb can not be included in any set because it is a descendant of YM . Here vancs = Z1Z2S . All valid adjustment sets remove the bias due to confounding by their definition . The question is which of these valid adjustment sets is statistically optimal in that it minimizes the asymptotic estimation variance ? More formally , the task is , given a graph G and ( X , Y , S ) , to chose a valid optimal set Zoptimal ∈ Z such that the causal effect estimator ’ s asymptotic variance Var ( ∆̂yxx′|s.z ) = E [ ( ∆yxx′|s − ∆̂yxx′|s.z ) 2 ] is minimal : Zoptimal ∈ argminZ∈ZVar ( ∆̂yxx′|s.z ) . ( 4 ) My proposed approach to optimal adjustment sets is based on information theory [ Cover and Thomas , 2006 ] . The main quantity of interest there is the conditional mutual information ( CMI ) defined as a difference IX ; Y |Z = HY |Z − HY |ZX of two ( conditional ) Shannon entropies HY |X = − ∫ x , y p ( x , y ) ln p ( y|x ) dxdy . Its main properties are non-negativity , IX ; Y |Z = 0 if and only if X ⊥⊥ Y |Z , and the chain rule IXW ; Y |Z = IX ; Y |Z + IW ; Y |ZX . All random variables in a CMI can be multivariate . Throughout the present paper we will assume the following . Assumptions 1 ( General setting and assumptions ) . We assume a causal graphical model over a set of variables V with a joint distribution P = P ( V ) that is consistent with an ADMG G = ( V , E ) . We assume a non-zero causal effect from X on Y , potentially through a set of mediators M , and given selected conditioned variables S , where S ∩ forb = ∅ . We assume that at least one valid adjustment set ( given S ) exists and , hence , the causal effect is identifiable ( except when stated otherwise ) . Finally , we assume the usual Causal Markov Condition ( implicit in semi-Markovian models ) and Faithfulness . 2 Optimal adjustment sets . 2.1 Information-theoretic characterization . Figure 1B illustrates two causal effect estimates for a linear Gaussian model consistent with the graph in Fig . 1A . With Z = Z1Z2 ( blue ) the error is much larger than with O = Z2Z3 ( orange ) for two reasons : Z constrains the residual variance V ar ( Y |ZS ) of the effect variable Y less than O and , on the other hand , Z constrains the residual variance V ar ( X|ZS ) of the cause variable X more than O . Smaller estimator variance also holds for O compared to any other valid set in Z here . We information-theoretically formalize the resulting intuition to choose an adjustment set Z that maximally constrains the effect variable Y and minimally constrains the cause variable X . In terms of CMIs and given selected fixed conditions S the quantity to maximize can be stated as follows . Definition 1 ( Adjustment information ) . Consider a causal effect of X on Y for an adjustment set Z given a condition set S. The ( conditional ) adjustment ( set ) information , abbreviated JZ , is defined as JXY |S.Z ≡ IZ ; Y |XS − IX ; Z|S ( 5 ) = HY |XS −HX|S︸ ︷︷ ︸ not related to Z − ( HY |XZS −HX|ZS ) ︸ ︷︷ ︸ adjustment entropy ( 6 ) JZ is not necessarily positive if the dependence between X and Z ( given S ) is larger than that between Z and Y given XS . Equation ( 6 ) follows from the CMI definition . Fig . 1C illustrates the two CMIs in Eq . ( 5 ) in a Venn diagram . Before discussing the range of estimators for which maximizing the adjustment information JZ leads to a minimal asymptotic estimation variance in Sect . 2.2 , we characterize graphical optimality in an information-theoretic framework . Our goal is to provide graphical criteria for optimal adjustment sets , i.e. , criteria that depend only on the structure of the graph G and not on the distribution . Definition 2 ( Information-theoretical graphical optimality ) . Given Assumptions 1 we say that ( information-theoretical ) graphical optimality holds if there is a Z ∈ Z such that either there is no other Z′ ̸= Z ∈ Z or for all other Z′ ̸= Z ∈ Z and all distributions P consistent with G we have JZ ≥ JZ′ . My main result builds on the following lemma which relates graphical optimality to informationtheoretic inequalities in a necessary and sufficient comparison condition for an optimal set to exist . Lemma 1 ( Necessary and sufficient comparison criterion for existence of an optimal set ) . Given Assumptions 1 , if and only if there is a Z ∈ Z such that either there is no other Z′ ̸= Z ∈ Z or for all other Z′ ̸= Z ∈ Z and all distributions P consistent with G it holds that IZ\Z′ ; Y |Z′XS︸ ︷︷ ︸ ( i ) ≥ IZ′\Z ; Y |ZXS︸ ︷︷ ︸ ( iii ) and IX ; Z′\Z|ZS︸ ︷︷ ︸ ( ii ) ≥ IX ; Z\Z′|Z′S︸ ︷︷ ︸ ( iv ) , ( 7 ) then graphical optimality holds and Z is optimal implying JZ ≥ JZ′ . In SSR20 and HPM19 the corresponding conditional independence statements to the terms ( iii ) and ( iv ) in the inequalities ( 7 ) are used as a sufficient pairwise comparison criterion . However , Lemma 1 shows that for graphical optimality it is not necessary that terms ( iii ) and ( iv ) vanish , they just need to fulfill the inequalities ( 7 ) for a necessary and sufficient criterion . In principle , Lemma 1 can be used to cross-compare all pairs of sets , but firstly , it is difficult to explicitly evaluate ( 7 ) for all distributions P consistent with G and , secondly , iterating through all valid adjustment sets is computationally prohibitive even for small graph sizes . As an example , consider a confounding path consisting of 5 nodes . Then this path can be blocked by 25 − 1 different subsets . In the main result of this work ( Thm . 3 ) a necessary and sufficient criterion based purely on graphical properties is given .

Summary: The paper defines the optimality of a (valid) adjustment set in terms of a notion called 'adjustment information.'  Then, it focuses on estimators for which maximizing adjustment information equates to minimizing asymptotic estimation variance. The authors provide a sufficient and necessary graphical criterion for the existence of an optimal adjustment set under this notion. They also give an explicit and efficient construction of an adjustment set called the O set, which is proven to be optimal whenever graph optimality holds (the constraints in the graph are sufficient to determine an optimal adjustment set).