Markov properties for mixed graphs

Markov properties for mixed graphs

[ [    [ [ Department of Statistics, Baker Hall, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
\printeade1
Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, United Kingdom. \printeade2
\smonth5 \syear2012\smonth10 \syear2012
\smonth5 \syear2012\smonth10 \syear2012
\smonth5 \syear2012\smonth10 \syear2012
Abstract

In this paper, we unify the Markov theory of a variety of different types of graphs used in graphical Markov models by introducing the class of loopless mixed graphs, and show that all independence models induced by -separation on such graphs are compositional graphoids. We focus in particular on the subclass of ribbonless graphs which as special cases include undirected graphs, bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and summary graphs. We define maximality of such graphs as well as a pairwise and a global Markov property. We prove that the global and pairwise Markov properties of a maximal ribbonless graph are equivalent for any independence model that is a compositional graphoid.

\kwd
\aid

0 \volume20 \issue2 2014 \firstpage676 \lastpage696 \doi10.3150/12-BEJ502

\runtitle

Markov properties for mixed graphs

{aug}

1]\fnmsKayvan \snmSadeghi\thanksref1label=e1]kayvans@andrew.cmu.edu and 2]\fnmsSteffen \snmLauritzen\corref\thanksref2label=e2]steffen@stats.ox.ac.uk

composition property \kwdglobal Markov property \kwdgraphoid \kwdindependence model \kwd-separation \kwdmaximality \kwdpairwise Markov property

1 Introduction

1.1 Introduction and motivation

Graphical Markov models have become widely used in recent years. The models use graphs to represent conditional independence relations for systems of random variables, with nodes of the graph corresponding to random variables and edges representing dependencies. Several classes of graphs with various independence interpretations have been described in the literature. These range from undirected graphs with simple separation for derivation of independencies lau96 () to various forms of mixed graphs kos02 (), ric02 (), wer11 (), including chain graphs with several different separation criteria fry90 (), cox93 (), koster97 (), and01 (), drt09 ().

In spite of the differences among these graphs, their structural similarities motivate an attempt to unify them. For this purpose, we introduce the class of loopless mixed graphs and let them entail independence models using the same separation criterion, -separation. This unification covers many graphical independence models in the literature with some independence models for chain graphs forming a notable exception; see Section 4 for further details. We show that any independence model generated by -separation in a loopless mixed graph is a compositional graphoid. This ensures that certain intuitive methods of reasoning are indeed valid for such graphs, as they in some sense behave as ordinary undirected graphs.

A common motivation for defining MC-graphs kos02 (), summary graphs wer11 (), and ancestral graphs ric02 (), is to represent independence relations implied by marginalisation over and conditioning on sets of variables satisfying the Markov property of a directed acyclic graph (DAG). The focus of our study is on a subclass of loopless mixed graphs which we shall term ribbonless. The class of ribbonless graphs is sufficiently rich to serve the same purpose: these graphs are obtained by a simple modification of MC graphs derived from a DAG after marginalisation and conditioning; and it contains summary graphs and ancestral graphs as special cases.

For ribbonless graphs, we define global and pairwise Markov properties, the latter being associated with interpreting missing edges in the graph as representing conditional independencies. We prove as our main result that a compositional graphoid independence model over a maximal ribbonless graph satisfies the global Markov property if and only if it satisfies the pairwise Markov property. This ensures that the independence models represented by such graphs are generated by their missing edges, which again supports the direct visual intuition.

1.2 Some early results on Markov properties

The concepts of pairwise and global Markov properties for undirected graphs were introduced in ham71 () in the context of random fields and shown to be equivalent for positive densities. Alternative proofs were later given independently by several authors, for example gri73 (), bes74 (); see also clifford90 (). An abstract variant of this theorem was proven in pea88 () for independence models satisfying graphoid axioms as these are satisfied by probabilistic distributions with positive densities; see also stu89 () and gei90 (). Independence models for undirected graphs were discussed comprehensively in Chapter 3 of lau96 ().

A global Markov property that uses the -separation criterion and a pairwise Markov property were defined in ric02 () for maximal ancestral graphs without considering conditions under which they are equivalent. We use a generalisation of these Markov properties for maximal ribbonless graphs, which contains maximal ancestral graphs as a subclass, and prove their equivalence for compositional graphoids. This has been mentioned as a conjecture in kan09 ().

1.3 Structure of the paper

In the next section, we introduce the basic concepts of graph theory, general and probabilistic independence models, and compositional graphoids.

In Section 3, we introduce the class of loopless mixed graphs and additional graph theoretical definitions special to mixed graphs. We also associate the -separation criterion to this class, and prove for any loopless mixed graph that the independence model induced by -separation is a compositional graphoid.

In Section 4, we introduce the class of ribbonless graphs and the concept of anterior graphs. We describe the relations between these as well as subclasses of loopless mixed graphs that have been discussed in the literature.

In Section 5, we introduce the concept of maximality by demanding that any additional edge will change the independence model. It is shown that ribbonless graphs are not necessarily maximal, and conditions for maximality are given.

In Section 6, we define a pairwise and a global Markov property for independence models for ribbonless graphs, and prove our main result: that pairwise and global Markov properties are equivalent for compositional graphoid independence models over maximal ribbonless graphs.

2 Basic definitions and concepts

In this section, we introduce basic definitions and notation for independence models, graphs, and compositional graphoids.

2.1 Basic graph theoretical definitions

A graph is a triple consisting of a node set or vertex set , an edge set , and a relation that with each edge associates two nodes (not necessarily distinct), called its endpoints. When nodes and are the endpoints of an edge, they are adjacent and we write . We say the edge is between its two endpoints. We usually refer to a graph as an ordered pair . Graphs and are called equal if . In this case we write .

Notice that our graphs are labeled, that is, every node is considered as a different object. Hence, for example, graph is not equal to .

A loop is an edge with the same endpoints. Multiple edges are edges with the same pair of endpoints. A simple graph has neither loops nor multiple edges.

A subgraph of a graph is a graph such that and and the assignment of endpoints to edges in is the same as in . An induced subgraph by nodes is a subgraph that contains all and only nodes in and all edges between two nodes in . A subgraph induced by edges is a subgraph that contains all and only edges in and all nodes that are endpoints of edges in .

A walk is a list of nodes and edges such that for , the edge has endpoints and . A path is a walk with no repeated node or edge. If the graph is simple then the path can be uniquely determined by an ordered sequence of node sets. Throughout this paper, we use node sequences to describe paths even in graphs with multiple edges, as it usually is apparent from the context which of multiple edges belong to the path. We say a path is between the first and the last nodes of the list in . We call the first and the last nodes endpoints of the path and all other nodes inner nodes.

If and are paths, their combination is the path , where is the first node of which is on both paths. If then is simply the concatenation of the two paths. In general, the concatenation of two paths will be a walk and not a path as the paths may intersect in more than one point.

A subpath of a path is a path that can be considered a subgraph of with the ordering associated with . A cycle in a graph is a simple subgraph whose nodes can be placed around a circle so that two nodes are adjacent if they appear consecutively along the circle.

2.2 Independence models

An independence model over a set is a set of triples (called independence statements), where , , and are disjoint subsets of and can be empty, and and always being included in . The independence statement is interpreted as “ is independent of given ”.

An independence model over a set is a semi-graphoid if for disjoint subsets , , , and of , it satisfies the four following properties:

  1. [3.]

  2. if and only if (symmetry);

  3. if then and (decomposition);

  4. if then and (weak union);

  5. and if and only if (contraction).

A semi-graphoid for which the reverse implication of the weak union property holds is said to be a graphoid, that is

  1. [5.]

  2. if and then (intersection).

Furthermore, a graphoid or semi-graphoid for which the reverse implication of the decomposition property holds is said to be compositional, that is

  1. [6.]

  2. if and then (composition).

Notice that simple separation in an undirected graph will trivially satisfy all of these properties, and hence compositional graphoids are direct generalisations of independence models given by separation in undirected graphs.

2.3 Probabilistic conditional independence models

The most common independence models are induced by probability distributions. Consider a set and a collection of random variables with state spaces and joint distribution . We let etc. for each subset of . For disjoint subsets , , and of we use the short notation to denote that is conditionally independent of given daw79 (), lau96 (), that is, that for any measurable and -almost all and ,

We can now induce an independence model by letting

We say that an independence model is probabilistic if there is a distribution such that . We then also say that is faithful to .

Probabilistic independence models are always semi-graphoids pea88 (), whereas the converse is not necessarily true; see stu89 (). If has strictly positive density, the induced independence model is also a graphoid; see, for example, Proposition 3.1 in lau96 (). If the distribution is a regular multivariate Gaussian distribution, is a compositional graphoid. This follows from the fact that for such a distribution

where is the entry in the concentration matrix of the distribution of and hence setwise conditional independence is directly determined by nodewise conditional independence.

Probabilistic independence models with positive densities are not in general compositional graphoids; this only holds for special types of multivariate distributions such as the Gaussian mentioned above and, say, the symmetric binary distributions used in wer09 ().

3 Independence models for mixed graphs

3.1 Mixed graphs

A mixed graph is a graph containing three types of edges denoted by arrows, arcs (bi-directed edges), and lines (full lines). Notice that we allow multiple edges of the same type. A loopless mixed graph (LMG) is a mixed graph that does not contain any loops (a loop may be line, arrow, or arc). For an arrow , we say that the arrow is from to . We also call a parent of ,  a child of and we use the notation for the set of all parents of in the graph. In the cases of or , we say that there is an arrowhead at or pointing to .

A path is direction-preserving from to if all edges are arrows pointing from to . If there is a direction-preserving path from to then is an ancestor of and is a descendant of . We denote the set of ancestors of by . Notice that we do not include in its set of anteriors or descendants.

A tripath is a path with three nodes. Note that sad12 () used the term V-configuration for such a path. However, here we follow kii84 () and most texts by letting a V-configuration be a tripath with non-adjacent endpoints.

In a mixed graph the inner node of three tripaths , , and is a collider (or a collider node) and the inner node of any other tripath is a non-collider (or a non-collider node) on the tripath or more generally on any path of which the tripath is a subpath. We shall also say that the tripath itself with inner collider or non-collider node is a collider or non-collider. We may speak of a collider or non-collider without mentioning the relevant tripath or path when this is apparent from the context. Notice that a node may be a collider on one tripath and a non-collider on another.

Two paths and (including tripaths or edges) between and are called endpoint-identical if there is an arrowhead pointing to in if and only if there is an arrowhead pointing to in and similarly for . For example, the paths , , and are all endpoint-identical as they have an arrowhead pointing to but no arrowhead pointing to on the paths.

Figure 1: (a) A mixed graph . (b) The anterior graph of .

3.2 Anterior graphs and sets

The anterior graph of a loopless mixed graph , denoted by , is the graph obtained from by recursively removing arrowheads pointing to nodes that are the endpoints of a line, that is, by obtaining and from and respectively. Hence, it holds that if and only if there are no arrowheads pointing to lines in . Notice also that since removing an arrowhead pointing to a line does not affect other arrowheads pointing to lines, it does not matter which arrowhead is removed first; therefore, the order of removing arrowheads pointing to lines does not affect the final graph obtained.

A path from to () in is an anterior path if it has the form . Notice that this path may only contain lines or arrows. We shall say that is anterior of in if there is an anterior path from to in . Notice that although the anterior path is defined in we may from time to time refer to an anterior path in as the path corresponding to the anterior path in .

We use the notation for the set of all anteriors of . Notice that, since ancestral graphs have no arrowheads pointing to lines, we have for an ancestral graph. Thus, our definition of anterior extends the notion of anterior used in ric02 () for ancestral graphs with the minor difference that we do not include a node in its anterior set. However, it is different from and inconsistent with the definition of anteriors in fry90 () and and97 ().

For example, in the graph in Figure 1(a), and . This can be seen by looking at the anterior paths from to and from to (as well as from to ) in Figure 1(b).

We first show that transitivity holds for anteriors.

Lemma 1

For any loopless mixed graph it holds that if and then .

{pf}

If and , has anterior paths from to and from to . As no arrowhead meets a line in their combination is an anterior path from to in .

Here we also introduce a lemma that is used in several proofs of this paper.

Lemma 2

Let be a loopless mixed graph. If , then either or a descendant of is the endpoint of a line in .

{pf}

The proof uses induction on the number of arrowheads removed from to obtain . For the base, if it follows immediately from the definition of an anterior path that must be the endpoint of a line or we would have .

Next, suppose that is obtained from by removing arrowheads and let be obtained from by removing a single arrowhead pointing to a line from . Then is also the anterior graph of , but with only arrowheads needing removal. Thus, if in , it is also anterior to in . Consider now two cases:

Case I. Assume is an ancestor of in . Since is not an ancestor of in , must have been obtained by turning an arc into an arrow. Say this arrowhead points to . Then is an endpoint of a line and it is a descendant of in .

Case II. If is not an ancestor of in , the inductive hypothesis yields that is either adjacent to a line in or has a descendant in which is the endpoint of a line in . Let be the node adjacent to a line in . If the arrowhead removed is not on the direction-preserving path from to the conclusion obviously follows. Else, there must be node on which is adjacent to a line in and can be used instead of .

3.3 The -separation criterion

Here we define a separation criterion for LMGs. We use this criterion to induce independencies on LMGs and its subclasses defined in Section 3.

We first define an -connecting path: Let be a subset of the node set of an LMG. A path is -connecting given if all its collider nodes are in and all its non-collider nodes are outside . For two disjoint subsets of the node set and , we say that -separates and if there is no -connecting path between and given . In this case, we use the notation . Notice that the -separation criterion induces an independence model on by .

We note that -separation is unaffected if we replace multiple edges of the same type with a single edge of that type. The -separation criterion for LMGs is the same as the separation criterion defined in ric02 (). It is an extension of the -separation criterion introduced in pea88 (). Clearly, -separation is also an extension of simple separation in an undirected graph, as then all edges are lines.

For example, in graph in Figure 2 it holds that and, thus, is an -connecting path given . Therefore, .

Figure 2: A loopless mixed graph for which .

We now have the following theorem. A similar result for the induced independence model for MC graphs was given in Proposition 2.10 of kos02 ().

Theorem 1

For any loopless mixed graph , the independence model is a compositional graphoid.

{pf}

For and disjoint subsets , , , and of , we prove that satisfies the six compositional graphoid axioms: {longlist}[(6)]

Symmetry: If , then : If there is no -connecting path between and given , then there is no -connecting path between and given .

Decomposition: If , then : If there is no -connecting path between and given , then there is no -connecting path between and given .

Weak union: If then : From (2) we know that and . Suppose, for contradiction, that there exist -connecting paths between and given . Consider a shortest path of this type and call it . If there is no inner collider node on , then there is an -connecting path between and given , a contradiction. On all collider nodes are in . If all collider nodes are in , then there is an -connecting path between and given , again a contradiction. Hence, consider the closest collider node to on . Now since the nodes between and are not in , there is an -connecting path between and given . If , then this is obviously a contradiction. Otherwise there is a node , for which and thus an -connecting path between and given , a contradiction again. Therefore, there is no -connecting path between and given .

Contraction: If and , then : Suppose, for contradiction, that there exists an -connecting path between and given . Consider a shortest path of this type and call it . The path is either between and or between and . The path being between and contradicts . Therefore, is between and . In addition, since all inner collider nodes on are in and because , an inner non-collider node should be in . This contradicts the fact that is a shortest -connecting path between and given .

Intersection: If and , then : Suppose, for contradiction, that there exists an -connecting path between and given . Consider a shortest path of this type and call it . The path is either between and or between and . Because of symmetry between and in the formulation it is enough to suppose that is between and . Since all inner collider nodes on are in and because , an inner non-collider node should be in . This contradicts the fact that is a shortest -connecting path between and given .

Composition: If and , then : Suppose, for contradiction, that there exist -connecting paths between and given . Consider a path of this type and call it . Path is either between and or between and . Because of symmetry between and in the formula it is enough to suppose that is between and . But this contradicts .\qed\noqed

Theorem 1 implies that we can focus on establishing conditional independence for pairs of nodes, formulated in the corollary below.

Corollary 1

For a loopless mixed graph and disjoint subsets of its node set , , and , it holds that if and only if for every nodes and .

{pf}

The result follows from the fact that satisfies the decomposition and the composition properties.

4 Subclasses of loopless mixed graphs

LMGs and their associated independence models induced by -separation unify a variety of previously discussed graphical independence models.

4.1 Chain graphs

Important exceptions include certain independence models for chain graphs. Chain graphs themselves are LMGs, but at least four different Markov properties for chain graphs have been discussed in the literature. Drton drt09 () has classified them into (i) the LWF or block concentration Markov property, (ii) the AMP or concentration regression Markov property, (iii) a Markov property that is dual to the AMP Markov property, and (iv) and the multivariate regression Markov property. When the chain components consist entirely of arcs, the multivariate regression property is identical to the one induced by -separation. However, the independence model induced by -separation in a chain graph is typically different from any of the other chain graph interpretations; see also richardson98 (), ric01 () and lauritzenrichardson02 ().

4.2 Ribbonless graphs

The class of MC graphs, defined in kos02 (), contains line loops and uses a different separation criterion for inducing an independence model. However, a small modification of any MC graph that is derived from a DAG after marginalisation and conditioning yields a so-called ribbonless graph, which is loopless and induces the same independence model as the MC graph, but by -separation sadthesis (). Any ribbonless graph can be generated from a DAG by marginalisation and conditioning and ribbonless graphs are stable under these operations sad12 (). The remaining part of this paper deals with such graphs. We first give a formal definition of a ribbon.

A ribbon is a collider tripath such that both of the following two conditions hold:

  1. [(2)]

  2. there is no endpoint-identical edge between and , that is, there is no -arc in the case of ; there is no -line in the case of ; and there is no arrow from to in the case of ;

  3. or a descendant of is the endpoint of a line or is on a direction-preserving cycle.

If or a descendant of is the endpoint of a line, then we say the ribbon is straight and if they are on a direction-preserving cycle we say the ribbon is cyclic. A ribbonless graph (RG) is an LMG that has no ribbons as induced subgraphs. Figure 3 illustrates a straight ribbon and the simplest cyclic ribbon.

Figure 3: (a) A straight ribbon with . (b) The simplest cyclic ribbon .

Figure 4(a) illustrates a graph containing a straight ribbon and Figure 4(b) illustrates a ribbonless graph. Notice that is not a ribbon here since there is a line between and and this is an endpoint-identical edge.

Figure 4: (a) A graph that is not ribbonless. (b) A ribbonless graph.

We proceed to establish that ribbonless graphs yield identical independence models to their anterior graphs and need the following lemma.

Lemma 3

Let be a ribbonless graph. If there is a collider tripath in that is non-collider in , then has an -edge that is endpoint-identical to .

{pf}

Suppose that is a sequence of graphs, where each graph has been generated by removing one arrowhead pointing to a full line from the previous graph starting from .

Consider the first intermediate graph where turns into a non-collider tripath. We prove by reverse induction that, for each , is a straight ribbon unless there is an endpoint-identical -edge to .

In , the node is obviously the endpoint of a line and the result holds. Thus, we assume that the result holds for . In , it is easy to observe that if the line that makes the ribbon is an arrow pointing to another line or if an arrow on the direction-preserving cycle pointing to a line is an arc then or a descendant of is still the endpoint of a line. Therefore, the result holds in . Therefore, by reverse induction, this result holds in , and since is ribbonless, in there is an endpoint-identical -edge to . For the graph in Figure 3(a), the anterior graph is the graph where all edges become undirected. Clearly there is no endpoint-identical edge and the conclusion of Lemma 3 does not hold. This illustrates the role of a graph being ribbonless.

Proposition 1

For a ribbonless graph , it holds that , that is, and are Markov equivalent.

{pf}

It is enough to prove that there is an -connecting path between and given in if and only if there is an -connecting path between and given in .

Suppose that there is an -connecting path between and given in . All non-colliders on the path in are preserved in . In addition, by Lemma 3, a collider tripath becomes non-collider if there is an endpoint-identical -edge to . In this case, the -edge can be used instead of to establish an -connecting path in .

Conversely, suppose that there is an -connecting path between and given in . Collider tripaths are collider tripaths in , and if a non-collider tripath has been collider in then, by Lemma 3, one can again use the -edge instead of . Thus the only thing that remains to be proven is that a direction-preserving path pointing to a member of in remains direction-preserving in .

In this case, by the same argument as in Lemma 3, if for the collider tripath , where , the arrowhead of an arrow on the direction-preserving path in is taken away then is a ribbon unless there is an endpoint-identical -edge to . Hence, we can use the -edge instead of to establish an -connecting path.

Thus, the absence of ribbons ensures that the Markov property is unchanged by forming the anterior graph . Again, as the anterior graph of the graph in Figure 3(a) is the graph with all edges becoming undirected, we have in but not in , illustrating that absence of ribbons is essential for the Markov equivalence of and .

Independence models induced by -separation in a ribbonless graph can be induced by marginalisation over and conditioning on a DAG-independence model sad12 (). This implies that independence models corresponding to RGs are probabilistic, that is, any RG has a faithful probability distribution.

4.3 Other subclasses of loopless mixed graphs

Other subclasses of LMGs that use -separation and have been discussed in the literature are summary graphs wer11 (), ancestral graphs ric02 (), acyclic directed mixed graphs spi97 (), ric03 (), undirected or concentration graphs dar80 (), lau96 (), bidirected or covariance graphs cox93 (), kau96 (), wer98 (), drtonrichardson08 (), and the class of directed acyclic graphs kii84 (), pea88 (), gei90 (). In papers on summary graphs and regression chain graphs, dashed undirected edges (without arrowheads) have often been used in place of bi-directed edges. Using the latter as we have done here makes the idea of a collider more immediate so -separation can be used directly and the relation between the various types of graphs becomes transparent.

The use of some of the above graphs are motivated by representing independence models obtained by marginalisation over and conditioning on subsets of the node set of a DAG. For those graphs, arcs indicate marginalisation and lines indicate conditioning.

The diagram in Figure 5 illustrates the hierarchy of subclasses of LMGs and their associated independence models generated by -separation. For example, it can be seen from the diagram that bidirected graphs are also ancestral graphs, since they form a subclass of multivariate regression chain graphs, which again form a subclass of ancestral graphs. Notice that the associated classes of independence models are all distinct except for ancestral, summary, and ribbonless graphs, which are alternative representations of the same class of independence models.

Figure 5: The hierarchy of subclasses of LMGs.

5 Maximal ribbonless graphs

Among the independence models over the node set of a graph , those that are of interest to us conform with , meaning that in implies for any . Henceforth, we assume that independence models conform with , unless otherwise stated.

For example, the independence model conforms with the graph in Figure 6, whereas does not conform with because of the independence statement .

Figure 6: The independence model conforms with whereas does not.

A ribbonless graph is called maximal if by adding any edge to , the independence model induced by -separation changes. Note that in wer11 () a graph that is maximal is called an independence graph.

The independence models on RGs induced by -separation conform with the graphs; hence for maximal graphs, adding an edge to the graph makes the independence model smaller. Therefore, we have the lemma below.

Lemma 4

A graph is maximal if and only if for every pair of non-adjacent nodes and of , there exists a subset of such that .

{pf}

The result follows directly from the definition of maximality.

RGs are not maximal in general. To see this consider the RG in Figure 7. There is no such that . This is because if , the path is -connecting given , and if , is -connecting given .

Figure 7: A non-maximal RG.

To characterise maximal RGs, we need the following notion: A path is a primitive inducing path between and if and only if for every , ,

  1. [(ii)]

  2. is a collider on the path; and

  3. .

This definition is a trivial extension of a primitive inducing path as defined for ancestral graphs in ric02 (). Note in particular that we consider any edge between and to be a primitive inducing path. In Figure 7, is a primitive inducing path.

Next, we need the following lemmas. These also establish a pairwise Markov property for maximal RGs.

Lemma 5

A non-collider node on a path between and in a ribbonless graph is either in or an anterior of a collider node on . Moreover, the relevant subpath of between and , or is an anterior path in .

{pf}

Let be a non-collider node on a path in . Then from at least one side (say from ) there is no arrowhead on pointing to . By moving towards on the path as long as , , is non-collider on the path, we obtain that . This implies that if no is a collider then and hence the lemma follows.

Lemma 6

For nodes and in an RG that are not connected by any primitive inducing paths (and hence ), it holds that .

{pf}

Suppose, for contradiction, there is an -connecting path between and given and denote a shortest such path by . If there is a non-collider node on then, by Lemma 5, is either in or it is an anterior of a collider node on . But since is -connecting given , collider nodes are in themselves. Hence, , which contradicts the fact that is -connecting. Therefore, all inner nodes of must be colliders.

Now we know that all inner nodes of are in and . If, for a collider tripath on , then, by Lemma 2 and since the graph is ribbonless, there is an endpoint-identical -edge to the tripath, which contradicts being shortest. Therefore, , which implies that is primitive inducing, again a contradiction. Therefore, there is no -connecting path between and given , and hence . Next, in Theorem 2 we give a necessary and sufficient condition for an RG to be maximal. The analogous result for ancestral graphs was proved in Theorem 4.2 of ric02 ().

Theorem 2

A ribbonless graph is maximal if and only if does not contain any primitive inducing paths between non-adjacent nodes.

{pf}

Let be a primitive inducing path between and in , and let be a subset , where is the node set of . We need to show that there is an -connecting path between and given .

This is immediate if each internal node, that is, each of , is in by just using , so assume that this is not the case. Thus there is an internal node of not in , and we may assume that there is one in . Pick such a node , , as far along the path to as possible. Consider a direction-preserving path from to , and let denote the reverse of this path. Note that no internal node in is in . Let be the part of from to . If each internal node in this path is in then we are done by taking the path followed by (note that no node can be repeated since each internal node in is in and each internal node in is outside ). So suppose not. Let be the first node in that is not in . Then (by the way was chosen), so . Let be the part of from to , and let be a direction-preserving path from to . Note that no internal node in is in . If and have no intersection, then much as above we obtain an -connecting path given by taking followed by , followed by . If and do intersect, then we obtain an -connecting path as required by following up to the first node on and then following .

By letting for every non-adjacent nodes and , the other direction follows from Lemmas 4 and 6. For other special types of graphs that are subclasses of RGs, the condition for maximality of RGs may get further simplified. Among the subclasses of RGs that have been mentioned in this paper, summary graphs, ancestral graphs, and acyclic directed mixed graphs are not necessarily maximal, while all others are maximal. This can be seen by checking whether primitive inducing paths are permissible in each subclass.

A Markov equivalent maximal graph can be generated from a non-maximal graph by adding endpoint-identical edges to a primitive inducing path between a pair of non-adjacent nodes. We refer the reader to sadthesis () for details. The following lemma establishes that anterior graphs of maximal graphs are themselves maximal.

Lemma 7

Let be a ribbonless graph and its anterior graph. Then if is maximal, so is .

{pf}

If, for contradiction, is not maximal, then Theorem 2 implies that there is a primitive inducing path in between non-adjacent nodes and . Consider a shortest primitive inducing path between and and denote it by . We know that all inner nodes of are colliders in . This trivially implies that all inner nodes of are colliders in too. In addition, each inner node on is in in . In , unless an arrow on the direction-preserving path from to or is an arc turning into an arrow in . In this case, is an ancestor of a node that is the endpoint of a line. Hence the tripath on is a ribbon unless there is an endpoint-identical -edge to the tripath, which contradicts the fact that is shortest. Therefore, is a primitive inducing path in , a contradiction. Hence, is maximal.

6 Markov properties for ribbonless graphs

In this section, we give a precise definition of the global and pairwise Markov properties for an independence model defined over the node set of a ribbonless graph. Further we show that these two Markov properties are equivalent for a maximal ribbonless graph if is also a compositional graphoid. This result is a direct generalisation of the similar result of pea88 () for undirected graphs and graphoids.

6.1 Global and pairwise Markov properties

For a ribbonless graph , an independence model defined over satisfies the global Markov property w.r.t. if it holds for , , and disjoint subsets of that

Similarly, an independence model defined over satisfies the pairwise Markov property w.r.t. if it holds for any nodes and that

For example, for the graph in Figure 8, the pairwise Markov property would imply that as and . It would also imply that .

Figure 8: The pairwise Markov property for this RG implies, for example, . The global Markov property would for example imply .

Clearly, the independence model induced by -separation always satisfies the global Markov property w.r.t. . By Lemma 4, Lemma 6, and Theorem 2, satisfies the pairwise Markov property if and only if is maximal.

6.2 Equivalence of pairwise and global Markov properties

Before establishing the main result of this section, we need two lemmas.

Lemma 8

Let be a compositional graphoid over a set and and be disjoint subsets of . It then holds that the marginal independence model

which is defined over , is a compositional graphoid.

{pf}

All the six compositional graphoid properties for follow trivially from the facts that for , , and such that , if and only if , and satisfies the six properties. Notice that the notion of a marginal independence model is identical to the notion formally defined in ric02 () with a different notation; it was also discussed in sad12 () with the same notation as in this paper.

The following lemma gives sufficient conditions for the combination of two -connecting paths in anterior graphs to be -connecting.

Lemma 9

Let be the anterior graph of a ribbonless graph and suppose that there are paths between and and between and which are -connecting given . The combination is then an -connecting path between and given in each of the following mutually exclusive situations:

  1. [(b2)]

  2. is a collider and ;

  3. with an arrowhead pointing to on the -edge and ;

  4. is a non-collider and ;

  5. with no arrowhead pointing to on the -edge.

{pf}

Let be the combination of and . If and either (a1) or (b1) holds then the conclusion is obvious. The cases (a2) or (b2) are only relevant when .

Next consider the situation where . Since and are -connecting, for to be -connecting we only need to check the tripath . We have to deal with two cases:

Case 1: is a non-collider.

In this case there is no arrowhead pointing to from at least one of or . This means that on or on is a non-collider, and since and were both -connecting we have . Hence is -connecting.

Case 2: is a collider. We need to consider the following two subcases:

Case 2.1. If is a collider and any of or is also a collider then and is -connecting.

Case 2.2. If is a collider but and are both non-colliders then by Lemma 5, the subpath of from to a collider node or to is an anterior path and similarly for , , and . However, since is an anterior graph and there are arrowheads pointing to , these anterior paths must be direction-preserving and thus and . Now we have the two following further subcases:

Case 2.2.1: One of the subpaths of from to is direction-preserving. Because and are -connecting we must have or in . Thus, and is -connecting.

Case 2.2.2: Both subpaths of and from to are direction-preserving. Then is collider or with an arrowhead pointing to on the -edge and (b1) and (b2) are impossible. If (a1) or (a2) holds is -connecting since then .

We are now ready to establish the main result of this paper.

Theorem 3

Let be a maximal ribbonless graph. If an independence model over the node set of is a compositional graphoid, then satisfies the pairwise Markov property w.r.t. if and only if it satisfies the global Markov property w.r.t. .

{pf}

() If is a compositional graphoid and satisfies the global Markov property it follows from Theorem 2 and Lemma 6 that it satisfies the pairwise Markov property.

() Now suppose that satisfies the pairwise Markov property and compositional graphoid axioms. For subsets , , and of the node set of , we should prove that implies . By composition, it is sufficient to show this when and are singletons, that is, that implies .

Further we observe that it is sufficient to establish the result in the case when is itself an anterior graph. Proposition 1 gives that in , which implies in . In addition, by Lemma 7, is a maximal graph. Moreover, and have the same anterior sets, and therefore the same pairwise Markov property. Thus in the following, we assume that is an anterior graph.

We prove the result in two main parts. In part I, we prove the result for the case that . In part II, we use the result of part I to establish the general case.

Part I. Suppose that . We use induction on the number of nodes of the graph. The induction base for a graph with two nodes is trivial. Thus, suppose that the result holds for all anterior graphs with fewer than nodes and assume that has nodes.

Let and , where is the node set of the graph. First in case I.1 we suppose that , and then in case I.2 we suppose that .

Case I.1. Consider to be the subgraph induced by . Consider the marginal independence model and defined over . By Lemma 8, is a compositional graphoid. In addition, it satisfies the pairwise Markov property: This is because two non-adjacent nodes and in are non-adjacent in and by the pairwise Markov property for , , where is the anterior set in . We know that and hence . In addition, for a node in , . Therefore, .

We also know that in implies in since there is no -connecting path between and given in and by removing nodes and edges from no new -connecting paths are generated. Therefore, by the induction hypothesis . This implies that .

Case I.2. Now suppose that and thus the node set of is . We prove the result by reverse induction on : For the base, and the result follows trivially from the pairwise Markov property.

For the inductive step, consider a node . We want to show that is not simultaneously -connected to both and : Suppose, for contradiction, there are -connecting paths and given . If (b1) or (b2) of Lemma 9 hold then and are -connected given which contradicts . So we need only consider the cases where is collider or with an arrowhead pointing to on the -edge. However, we know that or . Because of symmetry between and suppose that . Since is an anterior graph and there is an arrowhead pointing to we have . Hence, there is a direction-preserving path from to . If no node on is in then (b1) or (b2) of Lemma 9 implies that the combination of and is an -connecting path between and , again a contradiction. If there is a node on that is in then and again, by (a1) and (a2) of Lemma 9, and are -connected given , again a contradiction.

We conclude that, given , is not -connected to both and . By symmetry, suppose that .

We also have that . Since is a compositional graphoid (Theorem 1) the composition property gives that