1 Introduction
Discovery of causal relationships from observational and experimental data is a central problem with applications across multiple areas of scientific endeavor. There has been considerable progress over the past decades on algorithms for eliciting causal relationships from data under a broad range of assumptions (Pearl, 2000; Spirtes et al., 2000; Shimizu et al., 2006). Most algorithms for causal discovery assume propositional data where instances are independent and identically distributed. However, in many real world applications, these assumptions are violated because the underlying data has a relational structure of the sort that is modeled in practice by an entityrelationship model (Chen, 1976). There has been considerable work on learning predictive models from relational data (Getoor and Taskar, 2007). Furthermore, researchers from different disciplines have studied causal relationships and resulting phenomena on relational world, e.g., peer effects (Sacerdote, 2000; Ogburn and VanderWeele, 2014), social contagion (Christakis and Fowler, 2007; Shalizi and Thomas, 2011), viral marketing (Leskovec et al., 2007), and information diffusion (Gruhl et al., 2004).
Motivated by the limitations of traditional approaches to learning causal relationships from relational data, Maier and his colleagues introduced the relational causal model (RCM) (Maier et al., 2010) and provided a sound and complete causal structure learning algorithm, called the relational causal discovery (RCD) algorithm (Maier et al., 2013a), for inferring causal relationships from relational data. The key idea behind RCM is that a cause and its effects are in a direct or indirect relationship that is reflected in the relational data. Traditional approaches for reasoning on and learning of a causal model cannot be trivially applied for relational causal model (Maier et al., 2013a). Reasoning on an RCM to infer a relational version of conditional independence (CI) makes use of a lifted representation, called abstract ground graphs (AGGs), in which traditional graphical criteria can be used to answer relational CI queries. The lifted representation is employed as an internal learning structure in RCD to reflect the inferred CI results among relational version of variables. RCD makes use of a new orientation rule designed specifically for RCM.
Motivation and Contributions
RCM (Maier et al., 2010) offer an attractive model for representing, reasoning about, and learning causal relationships implicit in relational data. Arbour et al. (2014) proposed a relational version of propensity score matching method to infer (relational) causal effects from observational data. Marazopoulou et al. (2015) extended RCM to cope with temporal relational data. They generalized both RCM and RCD to Temporal RCM and Temporal RCD, respectively. A lifted representation, called abstract ground graph (AGG), plays a central role in reasoning with and learning of RCM. The correctness of the algorithms proposed by Maier et al. (2013a) for learning RCM and Marazopoulou et al. (2015) for Temporal RCM, respectively, from observational data rely on the soundness and completeness of AGG for relational dseparation to reduce the learning of an RCM to learning of an AGG. The main contributions of this paper are as follows: (i) We show that AGG, as defined in Maier et al. (2013b) does not correctly abstract all ground graphs; (ii) We revise the definition of AGG to ensure that it correctly abstracts all ground graphs; (iii) We further show that AGG representation is not complete for relational dseparation, that is, there can exist conditional independence relations in an RCM that are not entailed by AGG; and (iv) Based on a careful examination of the relationship between the lack of completeness of AGG for relational dseparation and faithfulness conditions suggests that weaker notions of completeness, namely adjacency faithfulness and orientation faithfulness between an RCM and its AGG, can be used to learn an RCM from data.
2 Preliminaries
We follow notational conventions introduced in (Maier et al., 2013a, b; Maier, 2014). An entityrelationship model (Chen, 1976) abstracts the entities (e.g., employee, product) and relationships (e.g., develops) between entities in a domain using a relational schema. The instantiation of the schema is called a skeleton where entities form a network of relationships (e.g., QuinndevelopsLaptop, RogerdevelopsLaptop). Entities and relationships have attributes (e.g., salary of employees, success of products). Cardinality constraints specify the cardinality of relationships that an entity can participate in (e.g., many employees can develop a product.).^{1}^{1}1The examples are taken from Maier (2014). The following definitions are taken from Maier (2014):
Definition 1.
A relational schema is a tuple : a set of entity classes ; a set of relationship classes where and is arity for ; attribute classes where is a set of attribute classes of ; and cardinalities .
Every relationship class have two or more distinct entity classes.^{2}^{2}2In general, the same entity class can participate in a relationship class in two or more different roles. For simplicity, we only consider relationship classes only with distinct entity classes. We denote by all item classes . We denote by an item class that has an attribute class assuming, without loss of generality, that the attributes of different item classes are disjoint. Participation of an entity class in a relationship class is denoted by if .
Definition 2.
A relational skeleton is an instantiation of relational schema , represented by a graph of entities and relationships. Let denote a set of items of item class in . Let such that , , and , then we denote if there exists an edge between and in .
2.1 Relational Causal Model
Relational causal model (RCM, Maier et al., 2010) is a causal model where causes and their effects are related given an underlying relational schema. For example, the success of a product depends on the competence of employees who develop the product (see fig:rcm_example).
An RCM models relational dependencies; each relational dependency has a cause and its effect, which are represented by relational variables; a relational variable is a pair consisting of a relational path and an attribute.
Definition 3.
A relational path is an alternating sequence of entity class and relationship class . An item class is called base class or perspective and is called a terminal class. A relational path should satisfy:

for every or , ;

for every , ; and

for every , if , then .
All valid relational paths on the given schema are denoted by . We denote the length of by , a subpath by or for , and the reversed path by . Note that all subpaths of a relational path as well as the corresponding reverse paths are valid. A relational variable is a pair of a relational path and an attribute class for the terminal class of . A relational variable is said to be canonical if its relational path has a length equal to 1. A relational dependency is of the form such that its cause and effect share the same base class and its effect is canonical.
Given a relational schema , a relational (causal) model is a pair of a structure , where is the set of relational dependencies, and is a set of parameters. We assume acyclicity of the model so that the attribute classes can be partially ordered based on . The parameters define conditional distributions, , for each pair where , , and is a set of causes of , i.e., . This paper focuses on the structure of RCM. Hence we often omit parameters from .
Terminal Set and Ground Graph
Because a skeleton is an instantiation of an underlying schema, a ground graph is an instantiation of the underlying RCM given a skeleton translating relational dependencies to every entity and relationship in the skeleton. It is obtained by interpreting the dependencies defined by the RCM on the skeleton using the terminal sets of each of the instances in the skeleton.
Given a relational skeleton , the terminal set of a relational path given a base , denoted by , is the set of terminal items reachable from when we traverse the skeleton along . Formally, a terminal set is defined recursively, and
This implies that and will be disjoint for . Restricting the traversals so as not to revisit any previously visited items corresponds to the bridge burning semantics (hereinafter, BBS) (Maier et al., 2013b). The instantiation of an RCM for a skeleton yields a ground graph which we denote by . The vertices of are labeled by pairs of items and its attribute, . There exists an edge in such that , , , and if and only if there exists a dependency such that .
In essence, RCM models dependencies on relational domain as follows: Causal relationships are described from the perspective of each item class; and are interpreted for each items to determine its causes in a skeleton yielding a ground graph. Since an RCM is defined on a given schema, RCM is interpreted on a skeleton so that every ground graph is an instantiation of the RCM.
Throughout this paper, unless specified otherwise, we assume a relational schema , a set of relational dependencies , and an RCM .
3 Reasoning With an Rcm
An RCM can be seen as a meta causal model or a template whose instantiation, a ground graph, corresponds to a traditional
causal model (e.g., a causal Bayesian network). Reasoning with causal models relies on
conditional independence (CI) relations among variables. Graphical criteria such as dseparation (Pearl, 2000) are often exploited to test CI given a model. Hence, the traditional definitions and methods for reasoning with causal models need to be “lifted” to the relational setting in order to be applicable to relational causal models.Definition 4 (Relational dseparation (Maier, 2014)).
Let , , and be three disjoint sets of relational variables with the same perspective defined over relational schema . Then, for relational model structure and are dseparated by if and only if, for all skeletons , and are dseparated by in ground graph for all .
There are two things implicit in this definition: (i) allgroundgraphs semantics which implies that dseparation must be hold over all instantiations of the model; (ii) the terminal set items of two different relational variables may overlap (which we refer to as intersectability). In other words, two relational variables and of the same perspective and the same attribute, are said to be intersectable if and only if:
(1) 
In order to allow testing of conditional independence on all ground graphs, Maier et al. (2013a) introduced an abstract ground graph (AGG), which abstracts all ground graphs and is able to cope with the intersectability of relational variables. We first recapitulate the original definition of AGGs.
3.1 Original Abstract Ground Graphs
An abstract ground graph is defined for a given relational model and a perspective (Maier et al., 2013a), Since we fix the model, we omit the subscript and denote the abstract ground graph for perspective by . The resulting graph consists of two types of vertices: and ; and two types of edges: and .
We denote by the set of all relational variables (RV) whose paths originate in . We denote by the set of all edges between the relational variables in . A relational variable edge (RVE) implies direct influence arising from one or more dependencies in . There is an RVE if there exists a dependency that can be interpreted as a direct influence from to from perspective . Such an interpretation is implemented by an function, which takes two relational paths and produces a set of relational paths: If , then there exists an RVE where
(2) 
, and ‘’ is a concatenation operator. We will use a binary join operator ‘’ for and denote by for a pivot . A schematic overview of is shown in fig:extendmethod.
We denote by the set of intersection variables (IVs), i.e., unordered pairs of intersectable relational variables in . Given two RVs and that are intersectable with each other, we denote the resulting intersection variable by (Here, the intersection symbol ‘’ denotes intersectability of the two relational variables). By the definition (Maier et al., 2013b), if there exists an RVE , then there exist edges and for every and intersectable with and , respectively. The IVs and the edges that connect them with RVs (IVEs) correspond to indirect influences (arising from intersectability) as opposed to direct influence due to dependencies (which are covered by RVs and RVEs). We denote by the set of all such edges that connect RVs with IVs.
Two AGGs with different perspectives share no vertices nor edges. Hence, we view all AGGs, , as a collection or a single multicomponent graph . We similarly define , , , and as the unions of their perspectivebased counterparts.
For any mutually disjoint sets of relational variables , and , one can test
, conditional independence admitted by the underlying probability distribution, by checking
(traditional) dseparation on an AGG^{3}^{3}3We denote conditional independence by ‘’ in general. We use ‘’ to represent (traditional) dseparation on a directed acyclic graph., e.g., or . Furthermore, we parenthesize conditional independence and use a subscript to specify the scope of the conditional independence, if necessary., where includes and their related IVs, fig:agg_example illustrates relational dseparation on an AGG.We later show that the preceding definition of AGG does not properly abstract all ground graphs; nor does it guarantee the correctness of reasoning about relational dseparation in an RCM. We revise the definition of AGG (sub:RevisionofAbstract) so as to ensure that the resulting AGG abstracts all ground graphs. However, we find that even with the revised definition of AGG, the AGG representation is not complete for relational dseparation, that is, there can exist conditional independence relations in an RCM that are not entailed by AGG (sub:COUNTEREXAMPLE). A careful examination of the lack of completeness of AGG for relational dseparation with respect to causal faithfulness yields useful insights that allow us to make use of weaker notions of faithfulness to learn RCM from data (sub:RELATINGNONCOMPLETENESSWITH).
3.2 Abstract Ground Graphs  a Revised Definition
Because of the importance of and in in reasoning about relational dseparation, it is possible that errors in abstracting all ground graphs could lead to errors in CI relations inferred from an . We proceed to show that 1) the criteria for determining intersectability (Maier, 2014) are not sufficient, and 2) the definition of , as it stands, does not guarantee the soundness of AGG as an abstract representation of the all ground graphs of an RCM. We provide the necessary and sufficient criteria for determining IVs and a sound definition for IVEs.
3.2.1 Intersectability and IV
The declarative characterization of intersectability (eq:intersectabilityformula) does not offer practical procedural criteria to determine intersectability. Based on the criteria (Maier, 2014), two different relational paths and are intersectable if and only if 1) they share the same perspective, say , and 2) they share the common terminal class, and 3) one path is not a prefix of the other. We will prove that the preceding criteria are not sufficient. In essence, we will show the conditions under which nonemptiness of for any in any skeleton always contradicts the BBS. For the proof, we define (the length of the longest required shared path) for two relational paths and of the common perspective as
is computed as follows. Initially set since . Repeat incrementing by if and either or with .
Lemma 1.
Given a relational schema , let and be two different relational paths satisfying the (necessary) criteria of Maier (2014) and . Let and be and , respectively. Then, and are intersectable if and only if .
Proof.
See Appendix. ∎
The lemma demonstrates the criteria by Maier (2014) do not rule out the case of where and cannot be intersectable.
3.2.2 Cointersectability and IVE
Based on the definition (Maier, 2014), an IVE exists between an IV, , and an RV, , if and only if there exists an RVE between and or and . It would indeed be appealing to define IV, , such that it inherits properties of the corresponding RVs, and . However, the abstract ground graph resulting from such a definition turns out to be not a sound representation of the underlying ground graphs. We proceed to prove this result.
Definition 5 (Cointersectability).
Given a relational schema , let , , , and be valid relational paths of the same perspective where and and are intersectable. Then, a tuple is said to be cointersectable if and only if
(3) 
We relate cointersectability with the definition of IVE. Let an RVE is due to some dependencies where . This implies
(4) 
and there are edges from of to of in . In order for the intersectability of with translates into an influence between and , it is necessary that there exists a skeleton that admits such influence. However, we can construct a counterexample that satisfies the necessary conditions for the existence of an RVE and the conditions for intersectability but does not satisfy the conditions for cointersectability (see fig:Comparisonofconditions for a comparison of eq:directinfluence, 1, and 3).
Example 1.
Let be a relational schema where , such that , , , , with the cardinality of each relationship and each entity in the relationship being . Let

,

,

, and

.
Observe that

;

and are intersectable; and

is a subpath of .
This example satisfies eq:intersectabilityformula and eq:directinfluence. Assume for contradiction that there exists a skeleton satisfying eq:cointer. Since, in this example, the cardinality of each relationship and each entity in the relationship is , for each , there exists only one and only one . By the assumption, . Since is a subpath of , will end at (see fig:counterexampleforcointer). Due to BBS, , that is, . This contradicts the assumption that .
This counterexample clearly represents there is an interdependency between intersection variables and RVEs. Therefore, we revise the definition of accompanying cointersectability.
Definition 6 (Ive).
There exists an IVE edge, (or ), if and only if there exists a relational path such that , and (or ) is cointersectable.
To determine IVEs, cointersectability of a tuple can be computed by solving a constraint satisfaction problem involving four paths in the tuple.
Implications of Cointersectability
We investigated the necessary and sufficient criteria for intersectability and revised the definition of IVE so as to guarantee that AGG correctly abstracts all ground graphs as asserted (although incorrectly) by Theorem 4.5.2 (Maier, 2014). The new criterion, called cointersectability, is especially interesting since it describes the interdependency between intersection variables and related relational variable edges. Several of the key results (e.g., soundness and completeness of AGG for relational dseparation, Theorem 4.5.2) and concepts (e.g., (,)reachability) of Maier (2014) are based on independence between intersection variables and related relational variable edges. Hence, it is useful to carefully scrutinize the relationship between AGG and relational dseparation.
4 NonCompleteness of Agg for Relational DSeparation
We first revisit the definition of relational dseparation. Given three disjoint sets of relational variables , , and of a common perspective , and are relational dseparated given , denoted by , if and only if
From Theorem 4.5.4 of (Maier, 2014), the lifted representation is said to be sound (or complete) for relational dseparation of if (traditional) dseparation holds on the with a modified CI query only when (or whenever) relational dseparation holds true. Then, the completeness of AGG for relational dseparation can be represented as
The completeness can be proved by the construction of a skeleton demonstrating dconnection for some if . In other words, we might disprove the completeness by showing
4.1 A Counterexample
The following counterexample shows that AGG is not complete for relational dseparation.
Example.
Let be a relational schema such that: ; with , , and ; ; and . Let be a relational model with
such that and . Let , , , and be four relational variables of the same perspective where their relational paths are distinct where

,

,

, and

.
Given the above example, we can make two claims.
Claim 1.
.
Proof.
See Appendix. ∎
Assuming that AGG is complete for relational dseparation, we can infer and there must exist a pair of a skeleton and a base that satisfies . However, we claim that such a skeleton and base may not exist.
Claim 2.
There is no and such that
Proof.
See Appendix. ∎
The counterexample demonstrates that a dconnection path captured in an might not have a corresponding dconnection path in any ground graph.
Corollary 1.
The revised (as well as the original) abstract ground graph for an RCM is not complete for relational dseparation.
It is possible that an additional test can be utilized to check whether there exists such a ground graph that can represent a dconnection path captured in . However, the efficiency of such an additional test is unknown and designing such a test is beyond the scope of this paper.
4.2 Relating NonCompleteness With Faithfulness
In light of the preceding result that AGG is not complete for relational dseparation, we proceed to examine the relationship between an RCM and its lifted representation in terms of the sets of conditional independence relationships that they admit. In RCM, there are several levels of relationship regarding the sets of conditional independence: between the underlying probability distributions and the ground graphs, between the ground graphs of an RCM and the RCM, and between the RCM and its AGG:
In RCM, the causal Markov condition and causal faithfulness condition (see below) can be applied between a ground graph and its underlying probability distribution . Both conditions are assumed for learning an RCM from relational data. Relational dseparation requires a set of conditional independence of using those deduced from every ground graph for every . In light of the lack of completeness of AGG for relational dseparation, the set of conditional independence relations admitted by and its lifted representation are not necessarily equivalent (see cor:noncompleteness).
We will relate and using an analogy of causal Markov condition and faithfulness (Spirtes et al., 2000; Ramsey and Spirtes, 2006) interpreting and as a DAG and a distribution , respectively. We first recapitulate the definitions for causal Markov condition and faithfulness.
Definition 7 (Causal Markov Condition (Ramsey and Spirtes, 2006)).
Given a set of variables whose causal structure can be represented by a DAG , every variable is probabilistically independent of its noneffects (nondescendants in ) conditional on its direct causes (parents in ).
The causal Markov condition (i.e., local Markov condition) is not directly translated into the relationship between and since they refer to different variables. However, the soundness of for relational dseparation of (i.e., global Markov condition) would be sufficient to interpret causal Markov condition between and . That is,
where , , and are distinct relational variables sharing a common perspective.
Definition 8 (Causal Faithfulness Condition (Ramsey and Spirtes, 2006)).
Given a set of variables whose causal structure can be represented by a DAG, no conditional independence holds unless entailed by the causal Markov condition.
By the counterexample above, is not strictly faithful to , because more conditional independences hold in than those entailed by .
4.2.1 Weaker Faithfulness Conditions
Ramsey and Spirtes (2006) showed that the two weaker types of faithfulness – adjacencyfaithfulness and orientationfaithfulness – are sufficient to retrieve a maximallyoriented causal structure from a data under the causal Markov condition. What we have showed is that there are more conditional independence hold in than those entailed by its corresponding . However, the two weaker faithfulness conditions hold true (if they are appropriately interpreted in an RCM and its lifted representation).
AdjacencyFaithfulness
Definition 9 (AdjacencyFaithfulness (Ramsey and Spirtes, 2006)).
Given a set of variables whose causal structure can be represented by a DAG , if two variables , are adjacent in , then they are dependent conditional on any subset of .
Let , be two distinct relational variables of the same perspective . We limit and to be nonintersectable to each other. Otherwise, they must not be adjacent to each other by the definition of RCM since an edge between intersectable relational variables yields a feedback in a ground graph. If there is an edge in ,
We can construct a skeleton where its corresponding ground graph satisfies that and are singletons and are disjoint to for . Lemma 4.4.1 by Maier (2014) describes a method to construct a minimal skeleton to represent and with a single . It guarantees that and are singletons and every relational variable satisfies and .
OrientationFaithfulness
Definition 10 (OrientationFaithfulness (Ramsey and Spirtes, 2006)).
Given a set of variables whose causal structure can be represented by a DAG , let be any unshielded triple in .

[label=(O1)]

if , then and are dependent given any subset of that contains ;

[label=(O2)]

otherwise, and are dependent conditional on any subset of that does not contain .
Let , , and be three distinct relational variables of the same perspective forming an unshielded triple in . Similarly, is not intersectable to both and . The condition (O1) can be written as
if edges are oriented as in . Otherwise,
for the condition (O2). Again, constructing a minimal skeleton for , , and guarantees that no can represent any item in . Thus, the existence of in the conditional determines (in)dependence in the ground graph induced from the minimal skeleton.
Learning RCM with Noncomplete AGG
RCD (Relational Causal Discovery, Maier et al. (2013a)) is an algorithm for learning the structure of an RCM from relational data. In learning RCM, AGG plays a key role: AGG is constructed using CI tests to obtain the relational dependencies of an RCM. The lack of completeness of AGG for relational dseparation in RCM raises questions about the correctness of RCD. A careful examination of AGG through the lens of faithfulness suggests that adjacencyfaithful and orientationfaithful conditions can be applied to to recover correct partiallyoriented dependencies for an RCM. However, it is still unclear whether RCD recovers maximallyoriented dependencies with the acyclicity of AGG (i.e., relational variables) not the acyclicity of RCM (i.e., attribute classes). This raises the possibility of an algorithm for learning the structure of an RCM from relational data that does not require the intermediate step of constructing a lifted representation.
5 Concluding Remarks
There is a growing interest in relational causal models (Maier et al., 2010, 2013a, 2013b; Maier, 2014; Arbour et al., 2014; Marazopoulou et al., 2015). A lifted representation, called abstract ground graph (AGG), plays a central role in reasoning with and learning of RCM. The correctness of the algorithm proposed by Maier et al. (2013a) for learning RCM from data relies on the soundness and completeness of AGG for relational dseparation to reduce the learning of an RCM to learning of an AGG. We showed that AGG, as defined in (Maier et al., 2013a), does not correctly abstract all ground graphs. We revised the definition of AGG to ensure that it correctly abstracts all ground graphs. We further showed that AGG representation is not complete for relational dseparation, that is, there can exist conditional independence relations in an RCM that are not entailed by AGG. Our examination of the relationship between the lack of completeness of AGG for relational dseparation and faithfulness suggests that weaker notions of completeness, namely adjacency faithfulness and orientation faithfulness between an RCM and its AGG can be used to learn an RCM from data. Work in progress is aimed at: 1) identifying the necessary and sufficient criteria for guaranteeing the completeness of AGG for relational dseparation; 2) establishing whether the RCD algorithm outputs a maximallyoriented RCM even when the completeness of AGG for relational dseparation does not hold; and 3) devising a structure learning algorithm that does not rely on a lifted representation.
Appendix
We first prove lem:intersectabilitynecsuf in sub:IntersectabilityandIV.
Lemma.
Given a relational schema , let and be two different relational paths satisfying the (necessary) criteria of Maier (2014) and . Let and be and , respectively. Then, and are intersectable if and only if .
Proof.
(If part) If , then we can construct a skeleton such that for some by adding unique items for and for and complete the skeleton in the same manner as shown in Lemma 3.4.1 (Maier, 2014). Note that if , then since and a relational path is an alternating sequence. This guarantees that there are at least two items for .
(Only if part) Let be in for some arbitrary skeleton and
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