Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and new necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification. While our results do not completely close the gap between existing sufficient and necessary conditions we find, empirically, that our results allow us to prove the generic identifiability of many more mixed graphs than the prior state-of-the-art.
In a linear structural equation model (L-SEM) the joint distribution of a random vector obeys noisy linear interdependencies. These interdependencies can be expressed with a matrix equation of the form
where and are unknown parameters, and is a random vector of error terms with positive definite covariance matrix . Then has mean vector and covariance matrix
Following an approach that dates back to Wright [4, 5], we may view and as (weighted) adjacency matrices corresponding to directed and bidirected graphs, respectively. This yields a natural correspondence between L-SEMs and mixed graphs, that is, graphs with both directed edges, , and bidirected edges, . More precisely, the mixed graph is associated to the L-SEM in which is assumed to be zero if and, similarly, when . We write for the map obtained by restricting the map from (2) to pairs that satisfy the conditions encoded by the graph . We note that mixed graphs used to represent L-SEMs are often also called path diagrams.
where has 0 mean and covariance matrix
In this model, the random vector has covariance matrix
A first question that arises when specifying an L-SEM via a mixed graph is whether the map is injective, that is, whether any in the domain of can be uniquely recovered from the covariance matrix . When this injectivity holds we say that the model and also simply the graph is globally identifiable. Whether or not global identifiability holds can be decided in polynomial time [7, 8, 9]. However, in many cases global identifiability is too strong a condition. Indeed, the canonical instrumental variables model is not globally identifiable.
We will be instead interested in generic identifiability, that is, whether can be recovered from with probability 1 when choosing from any continuous distribution on the domain of . A current state-of-the-art, polynomial time verifiable, criterion for checking generic identifiability of a given mixed graph is the half-trek criterion (HTC) of , with generalizations by [11, 12, 13]. The sufficient condition that is part of the HTC operates by iteratively discovering invertible linear equation systems in the parameters which it uses to prove generic identifiability. A necessary condition given by the HTC detects cases in which the Jacobian matrix of fails to attain full column rank which implies that the parameterization is generically infinite-to-one. However, there remain a considerable number of cases in which the HTC remains inconclusive, that is, the graph satisfies the necessary but not the sufficient condition for generic identifiability.
We extend the applicability of the HTC in two ways. First, we show how the theorems on trek separation in  can be used to discover determinantal relations that in turn can be used to prove the generic identifiability of individual edge coefficients in L-SEMs. This method generalizes the use of conditional independence in known instrumental variable techniques; compare e.g. . Once we have shown that individual edges are generically identifiable with this new method, it would be ideal if identified edges could be integrated into the equation systems discovered by the HTC to prove that even more edges are generically identifiable. Unfortunately, the HTC is not well suited to integrate single edge identifications as it operates simultaneously on all edges incoming to a given node. Our second contribution resolves this issue by providing an edgewise half-trek criterion which operates on subsets of a node’s parents, rather than all parents at once. This edgewise criterion often identifies many more coefficients than the usual HTC. We note that, in the process of preparing this manuscript we discovered independent work of Chen ; some of our results can be seen as a generalization of results in his work.
The rest of this paper is organized as follows. In Section 2, we give a brief overview of the necessary background on mixed graphs, L-SEMs, and the half-trek criterion. In Section 3, we show how trek-separation allows the generic identification of edge coefficients as quotients of subdeterminants. We introduce the edgewise half-trek criterion in Section 4 and we discuss necessary conditions for the generic identifiability of edge coefficients in Section 5. Computational experiments showing the applicability of our sufficient conditions follow in Section 6, and we finish with a brief conclusion in Section 7. Some longer proofs are deferred to the appendix.
We assume some familiarity with the graphical representation of structural equation models and only give a brief overview of our objects of study. A more in-depth introduction can be found, for example, in  or, with a focus on the linear case considered here, in .
2.1 Mixed graphs and covariance matrices
Nonzero covariances in an L-SEM may arise through direct or through confounding effects. Mixed graphs with two types of edges have been used to represent these two sources of dependences.
Definition 2.1 (Mixed Graph).
A mixed graph on vertices is a triple where is the vertex set, are the directed edges, and are the bidirected edges. We require that there be no self-loops, so for all . If , we will write and if , we will write . As bidirected edges are symmetric we will also require that is symmetric, so that .
Let and be two vertices of a mixed graph . A path from to is any sequence of edges from or beginning at and ending at . Here, we allow that directed edges be traversed against their natural direction (i.e., from head to tail). We also allow repeated vertices on a path. Sometimes, such paths are referred to as walks or also semi-walks. A path from to is directed if all of its edges are directed and point in the same direction, away from and towards .
Definition 2.2 (Treks and half-treks).
(a) A path from a source to a target is a trek if it has no colliding arrowheads, that is, is of the form
where , , and is the top node. Each trek has a left-hand side and a right-hand side . In the former case, and . In the latter case, and , with a part of both sides.
(b) A trek is a half-trek if . In this case is of the form
In particular, a half-trek from to is a trek from to which is either empty, begins with a bidirected edge, or begins with a directed edge pointing away from .
Some terminology is needed to reference the local neighborhood structure of a vertex . For the directed part , it is standard to define the set of parents and the set of descendents of as
respectively. The nodes incident to a bidirected edge can be thought of as having a common (latent) parent and thus we refer to the bidirected neighbors as siblings and define
Finally, we denote the sets of nodes that are trek reachable or half-trek reachable from by
Two sets of matrices may be associated with a given mixed graph . First, is the set of real matrices with support , i.e., those matrices with implying and for which invertible. Second, is the set of positive definite matrices with support , i.e., if , then implies . Based on (2), the distributions in the L-SEM given by have a covariance matrix that is parameterized by the map
with domain .
Our focus is solely on covariance matrices. Indeed, in the traditional case where the errors in (1) follow a multivariate normal distribution the covariance matrix contains all available information about the parameters .
Subsequently, the matrices and will also be regarded as matrices of indeterminants. The entries of may then be interpreted as formal power series. Let and be matrices of indeterminants with zero pattern corresponding to . Then has entries that are formal power series whose form is described by the Trek Rule of , see also Spirtes, Glymour, and Scheines . The Trek rule states that for every the corresponding entry of is the sum of all trek monomials corresponding to treks from to .
Definition 2.4 (Trek Monomial).
Let be two, not necessarily distinct, vertices, and let be the set of all treks from to in . If contains no bidirected edge and has top node , its trek monomial is defined as
If contains a bidirected edge connecting , then its trek monomial is
[Trek Rule] The covariance matrix corresponding to a mixed graph satisfies
2.2 Generic identifiability
We now formally introduce our problem of interest and review some of the prior work our results build on. We recall that an algebraic set is the zero-set of a collection of polynomials. An algebraic set that is a proper subset of Euclidean space has measure zero; see, e.g., the lemma in .
Definition 2.6 (Generic Identifiability).
(a) The model given by a mixed graph is generically identifiable if there exists a proper algebraic subset such that the fiber is a singleton set, that is, it satisfies
for all . In this case we will say, for simplicity, that is generically identifiable.
(b) Let be the projection for . We say that the edge coefficient is generically identifiable if there exists a proper algebraic subset such that for all . In this case, we will say that the edge is generically identifiable.
In all examples we know of, if generic identifiability holds, then the parameters can in fact be recovered using rational formulas.
Definition 2.7 (Rational Identifiability).
(a) A mixed graph , or rather the model it defines, is rationally identifiable if there exists a rational map and a proper algebraic subset such that is the identity on .
(b) An edge , or rather the coefficient , is rationally identifiable if there exists a rational function and a proper algebraic subset such that for all .
Definition 2.8 (Trek and Half-Trek Systems).
Let be a collection of treks in and let be the set of sources and targets of the respectively. Then we say that is a system of treks from to . If each is a half-trek, then is a system of half-treks. A collection of treks is said to have no sided intersection if
As our focus will be on the identification of individual edges in we do not state the identifiability result of  in its usual form, instead we present a slightly modified version which is easily seen to be implied by the proof of Theorem 1 in .
A set of nodes satisfies the half-trek criterion with respect to a vertex if
there is a system of half-treks with no sided intersection from to .
[HTC-identifiability] Suppose that in the mixed graph the set satisfies the half-trek criterion with respect to . If all directed edges with head are generically (rationally) identifiable, then all directed edges with as a head are generically (rationally) identifiable.
The sufficient condition for rational identifiability of in  is obtained through iterative application of Theorem 2.10.
3 Trek separation and identification by ratios of determinants
Let and be matrices of indeterminants corresponding to a mixed graph as specified in Section 2.1. Let , and let be the submatrix of obtained by retaining only the rows and columns indexed by and , respectively. The (generic) rank of such a submatrix can be completely characterized by considering the trek systems between the vertices in and . The formal statement of this result follows.
Definition 3.1 (t-separation).
A pair of sets with t-separates the sets if every trek between a vertex and a vertex intersects on the left or on the right.
In this definition, the symbols and are chosen to suggest left and right. Similarly, and are chosen to indicate sources and targets, respectively.
Theorem 3.2 (, )
Let be a non-negative integer. The submatrix has generic rank if and only if there exist sets with such that t-separates and .
Theorem 2.7 from  established this result for acyclic mixed graphs while  extended the result to all mixed graphs and even gave an explicit representation of the rational form of the subdeterminant , for . An immediate corollary to the above theorem, considering the proof of Theorem 2.17 in , rephrases its statement in terms of maximum flows in a special graph. For an introduction to maximum flow, and the well-known Max-flow Min-cut Theorem, see the book by Cormen et al. . Note that standard max-flow min-cut framework does not allow vertices to have maximum capacities or for there to be multiple sources and targets, introducing these modifications is, however, trivial and the resulting theorem is sometimes called the Generalized Max-flow Min-cut Theorem.
Let be the directed graph with and containing the following edges:
Turn into a network by giving all vertices and edges capacity 1. Let . Then has generic rank if and only if the max-flow from to in is .
Add vertices , with infinite capacity, to the graph along with edges, all with capacity 1, , for , and , for . Let be such that they -separate the sets and is minimal. By Theorem 3.2, has rank generically. Note that gives the minimal size cut (of size ). By the (generalized) Max-flow Min-cut theorem the max-flow from to is , and it is also the max flow from to . Hence has generic rank equal to the found max-flow.
Note that the maximum flow between vertex sets in a graph can be computed in polynomial time. Indeed, in our case, the conditions of Corollary 3.3 can be checked in time [20, page 725]. As the following example shows, Corollary 3.3 can be used to find determinantal constraints on . These constraints can then be leveraged to identify edges in .
Consider the mixed graph in Figure 2a, which is taken from Figure 3c in . The corresponding flow network is shown in Figure 2b. From Gröbner basis computations, is known to be rationally identifiable but the half-trek criterion fails to certify that any edge of is generically identifiable. Let and . Corollary 3.3 implies that has generically full rank as there is a flow of size 3 from to in , via the paths , , and . Now suppose that we remove the edge from , call the resulting graph , and let be the covariance matrix corresponding to . Then one may check that the max-flow from to in is . Thus where denotes the determinant. Now note that is the sum of all monomials given by treks from 1 to 5 that end in the edge . Hence, is obtained by summing over all treks from 1 to 5 that do not end in the edge . But in our graph this is just the sum over treks from 1 to 5 that do not use the edge at all. Therefore, . Similarly, it is straightforward to check that
By the multilinearity of the determinant, we deduce that
Applying Corollary 3.3 a final time, we recognize that is generically non-zero and, thus, the equation
generically and rationally identifies . In this case, the same strategy can be used to identify the edges and (but not ) in .
In the above example, there is a correspondence between trek systems in and trek systems in , the graph that has the edge to be identified removed. This allowed us to leverage Corollary 3.3 directly to show that (8) has determinant 0. Such a correspondence cannot always be obtained but exists in the following case.
Let be a mixed graph. Let be an edge in , and suppose that the edges are known to be generically (rationally) identifiable. Let be the subgraph of with the edges removed. Suppose there are sets , with such that:
the max-flow from to in equals , and
the max-flow from to in is smaller than .
Then is generically (rationally) identifiable by the equation
Let and be the covariance matrices corresponding to and , respectively. Since , we have that for all and . This holds because if a trek from to uses an edge then either or , violating our assumptions.
Now let and . Suppose that is a trek from to that uses the edge . Then since we must have that is used only on the right-hand side of . With it follows that is the last edge used in the trek because may only use directed edges after using and must end at . Hence, all treks from to which use must have this edge as their last edge on the right. But is obtained by summing over all treks from to which end in the edge and, thus, is the sum of the monomials for all treks from to that do not use the edge at all.
As the above argument holds for all , it follows that . Since this is true for all it follows, similarly as in Example 3.4, that
Using assumption (c) and applying Corollary 3.3, we have . Similarly, by assumption (b), generically. The desired result follows.
Theorem 3.5 generalizes the ideas underlying instrumental variable methods such as those discussed in . Indeed, this prior work uses d-separation as opposed to t-separation. D-separation characterizes conditional independence which in the present context corresponds to the vanishing of particular almost principal determinants of the covariance matrix. In contrast, Theorem 3.5 allows us to leverage arbitrary determinantal relations; compare . The graph in Figure 2a is an example in which d-separation and traditional instrumental variable techniques cannot explain the rational identifiability of the coefficient for edge .
While assumption (a) in the above Theorem allows for the easy application of Corollary 3.3, this assumption can be relaxed by generalizing one direction of Corollary 3.3. We state this generalization as the following lemma, which is concerned with asymmetric treatment of edges that appear on the left versus right-hand side of treks. The lemma’s proof is deferred to Appendix A.
Let be a mixed graph, and let and be the matrices of indeterminants corresponding to the directed and the bidirected part of , respectively. Let and define matrices and with
Define a network with vertex set , edge set containing
with all edges and vertices having capacity 1. Let . Then, for any with , we have that if the max-flow from to in is .
We may now state our more general result.
Let be a mixed graph, , and suppose that the edges are known to be generically (rationally) identifiable. Recalling Equation (13), let be with the edges removed. Suppose there are sets and such that and
the max-flow from to in equals , and
the max-flow from to in is .
Then is rationally identifiable by the equation
By assumption (b) and Corollary 3.3, is generically non-zero. Therefore, equation (14) holds if
To show this we note that, by the multilinearity of the determinant, we have
Write for the matrix that appears on the right-hand side of this equation.
Consider any two indices and with and . If a trek from to uses one of the edges , for , on its right-hand side then , a contradiction since by assumption (a). Similarly, since the difference is obtained by summing the monomials for treks between and which do not use any edge on their right side. From this we may write
where equals but with its , , entries set to 0. The fact that under assumption (c) is the content of Lemma 3.7 (where we take and ). Given this lemma our desired result then follows.
Clearly Theorem 3.8 can be applied whenever Theorem 3.5 can. Moreover, as the next example shows, there are cases in which Theorem 3.8 can be used while Theorem 3.5 cannot.
Let be the mixed graph from Figure 3. Take and . Then Theorem 3.8 implies that is rationally identifiable. Theorem 3.5 cannot be applied in this case as .
For a fixed choice of and , the conditions (a)-(c) in Theorem 3.8 can be verified in polynomial time. Indeed, conditions (b) and (c) involve only max-flow computations that take time in general. Condition (a) can be checked by computing the descendants of , which can be done with any graph traversal algorithm (e.g., depth first search, see ), and then computing the intersection between the descendants and which can be done in time.
In order to apply Theorem 3.8 algorithmically, however, we have to consider all possible subsets , and check our condition for each pair. Naively done this operation takes exponential time. It remains an interesting problem for further study to determine whether or not the problem of finding suitable sets and is NP-hard. We note that a similar problem arises for instrumental variables/d-separation, where  were able to give a polynomial time algorithm for finding suitable sets in graphs that are acyclic. Given our results so far we will maintain polynomial time guarantees simply by considering only subsets of bounded size .
4 Edgewise generic identifiability
While our results from Section 3 can be used together with the HTC there is notable lack of synergy between the two methods as Theorem 3.8 requires that all directed edges incoming to a node be generically identifiable before that node can be used to prove the generic identifiability of other edges. Aiming to strengthen the HTC while allowing it to better use identifications produced by Theorem 3.8, the following theorem establishes a sufficient condition for the generic identifiability of any set of incoming edges to a fixed node. While in the process of preparing this manuscript we discovered the work of Chen ; our following theorem can be seen as a generalization of his Theorem 1, see Remark 4.2 for a discussion of the primary difference between our theorem and that in .
Let be a, non-empty, mixed graph and let . Let and suppose there exists with such that,
[(i)] there exists a half-trek system from to with no sided intersection,
[(ii)] for every trek from to we have that either
ends with an edge of the form where either or is known to be generically (rationally) identifiable, or
begins with an edge of the form where is known to be generically (rationally) identifiable.
Then for each we have that is generically (rationally) identifiable.
Let be the matrices of indeterminants corresponding to , and let be the covariance matrix. Recall our notation for the set of treks from to in . By the trek rule (Prop. 2.5), is the sum of monomials for treks from to .
Recalling that , enumerate and . Now, for , let be the set of all edges incoming to known to be generically (rationally) identifiable.
Our approach is to build a linear system of equations in the unknowns , having a unique solution. Consider the set of all treks between and . Because of condition (ii) we have that and all treks from to either end in a directed edge of the form , with or known to be generically identifiable, or must start in a directed edge of the form for some . Now note that for any ,
equals the sum of the monomials for all treks from to that do not start with a directed edge of the form for . Hence we find that the sum of all monomials for treks from to that do not start with an edge of the form for equals
Now the sum over all treks between and that start with an edge of the form for is easily seen to be the quantity . Thus,
Rewriting this we have
Notice that, in the above equation, if and then it must be the case that there is a trek from to ending in the edge which does not start with an edge of the form where is known to be generically identifiable. It then follows, by condition (ii)(a), that since we must have that is known to be generically identifiable. It then follows that the only unknowns quantities (that is, those not assumed to be generically identifiable) in the above displayed equation are the which appear linearly on the left hand side. Thus we have exhibited one linear equation in the unknown parameters .
Repeating the above argument for each of the , we obtain linear equations in unknowns. It remains to show that the system of equations is generically non-singular. This amounts to showing generic invertibility for the matrix with entries
The invertibility of follows from the existence of the half-trek system from to with no sided intersection and Lemma 4.3 below. We conclude that each is generically (rationally) identifiable as claimed.
Our Theorem 4.1 generalizes Theorem 1 in  in two ways. Firstly, we make the trivial, but for our purposes important, modification to formulate our theorem in a fashion that is agnostic as to how prior generic identifications were obtained. For the presentation in  it was more natural to focus only on such identifications being obtained from prior applications of his theorem. Secondly, and more substantially, the results in  do not consider the possibility that, recalling the setting of Theorem 4.1, some of the edges incoming to may be known to be generically identifiable; failing to use this information makes the conditions on the set more restrictive. Indeed, but for our first modification, our theorem reduces to the result in  if we replace condition (ii)(a) by the condition “ ends with an edge of the form where .”
As an example of how the above difference can appear in practice consider Figure 4 and suppose we have restricted the size of edge sets we consider to be of size 1 (for larger graphs, this may be required for computational efficiency). Then, using and , one easily checks that is generically identifiable. But now, showing that is generically identifiable using is impossible using Theorem 1 in  because of the trek but this trek provides no problem for Theorem 4.1 as we have already shown that is generically identifiable.
The following lemma generalizes Lemma 2 from  and completes the proof of Theorem 4.1.
Let be a mixed graph on nodes with associated covariance matrix . Moreover, let . For every let . Suppose there exists a half-trek system from to with no sided intersection. Then the matrix defined by
is generically invertible.
The proof of this lemma is deferred to Appendix B. Note that if let and strengthen condition (ii)(b) to require that all edges incoming to be generically identifiable whenever there exists a half-trek from to , then Theorem 4.1 reduces to Theorem 2.10 of , the usual half-trek identifiability theorem.
The conditions of Theorem 4.1 can be easily checked in polynomial time using max-flow computations, just as with the standard half-trek criterion. Unfortunately, in general, we do not know for which subset we should be checking the conditions of Theorem 4.1. This, in practice, means that we will have to check all subsets . There are, of course, exponentially many such subsets in general. If we are in a setting where we may assume that all vertices have bounded in-degree, then checking all subsets requires only polynomial time. In the case that in-degrees are not bounded, we may also maintain polynomial time complexity by only considering subsets of sufficiently large or small size. We provide pseudocode for an algorithm to iteratively identify the coefficients of a mixed graph leveraging Theorem 4.1 in Algorithm Algorithm 1.
|Algorithm 1 Edgewise identification algorithm.|
|1:||Input: A mixed graph with and a set of edges, , known to be generically identifiable.|
|8:||Using max-flow computations, does there exist a half-trek system from to of size with no sided intersection?|
|9:||if is true true|
|11:||Break out of the current loop|
|15:||until No additional edges have been added to on the most recent loop.|
|16:||Output: , the set of edges found to be generically (rationally) identifiable.|
5 Edgewise generic nonidentifiability
In prior sections we have focused solely on sufficient conditions for demonstrating the generic identifiability of edges in a mixed graph. This, of course, begs the question of if there are any complementary necessary conditions. That is, if there exist conditions that, when failed, show that a given edge is generically many-to-one. To our knowledge, the following is the only known necessary condition for generic identifiability and considers all parameters of a mixed graph simultaneously.
(Theorem 2 of )Suppose is a mixed graph in which every family of subsets of the vertex set either contains a set that fails to satisfy the half-trek criterion with respect to or contains a pair of sets with and . Then the parameterization is generically infinite-to-one.
This theorem operates by showing that, given its conditions, the Jacobian of the map fails to have full column rank and thus must have infinite-to-one fibers. Unfortunately this theorem does not give any indication regarding which edges are, in particular, generically infinite-to-one. The theorem below gives a simple condition which guarantees that a directed edge is generically infinite-to-one.
Let be a mixed graph and let . Suppose that for every we have either or is not half-trek reachable from . Let be the projection for . Then is infinite for all .
Let and . We will show that for each matrix that agrees with in all but (possibly) the entry, we can find for which . The claim then follows by noting that the choices for allow for infinitely many values of .
Let be as above, and let be such that . Then
Whenever then and in the above equation. Thus
Next suppose, without loss of generality, that and . Then, since is a non-sibling of , we must have that is not half-trek reachable from , and hence . But then
Now since , we have that . Therefore,
Let . We have just shown that for every such that . To see that it remains to show that is positive definite. But this is obvious from its definition since is positive definite and is invertible. We conclude that which proves the claim.
Let be the graph in Figure 5a. Using the necessary condition of the HTC, Theorem 5.1, we find that is generically infinite-to-one. To identify which edges of are themselves infinite-to-one we use Theorem 5.2. Doing so, one easily finds that the edge of is generically infinite-to-one. Indeed, using the edgewise identification techniques of Section 4, we see that all other directed edges of are generically identifiable so we have completely characterized which directed edges of are, and are not, generically identifiable.
We stress, however, that Theorem 5.2 does not imply Theorem 5.1; that is, there are graphs for which Theorem 5.1 shows is infinite-to-one but Theorem 5.2 cannot verify that any edges of are infinite-to-one. For example, see Figure 5b.
6 Computational experiments
In this section we will provide some computational experiments that demonstrate the usefulness of our theorems in extending the applicability of the half-trek criterion. All of our following experiments are carried out in the R programming language and the following algorithms are implemented in our R package SEMID which is available on CRAN, the Comprehensive R Archive Network [22, 23], as well as on GitHub. We will be considering four different identification algorithms for checking generic identifiability:
The standard half-trek criterion (HTC) algorithm.
The edgewise identification (EID) algorithm, displayed in Algorithm 1, where the input set of is empty.
The trek-separation identification (TSID) algorithm. Similarly as for Algorithm 1 this algorithm iteratively applies Theorem 3.8 until it fails to identify any additional edges. (Since we are considering a small number of nodes there is no need to limit the size of sets and we are searching for in our computation.)
The EID TSID algorithm. This algorithm alternates between the EID and TSID algorithms until it fails to identify any additional edges.
We emphasize that when all of the directed edges, i.e., the matrix is generically (rationally) identifiable then we also have that is generically (rationally) identifiable.
In Table 1 from , the authors list all 112 acyclic non-isomorphic mixed graphs on 5 nodes which are generically identifiable but for which the half-trek criterion remains inconclusive even when using decomposition techniques. We run the EID, TSID, and EID TSID algorithms upon the 112 inconclusive graphs and find that 23 can be declared generically identifiable by the EID algorithm, 0 by the TSID algorithm, and 98 by the EID TSID algorithm. Thus it is only by using both the determinantal equations discovered by t-separation and the edgewise identification techniques that one sees a substantial increase in the number of graphs that can be declared generically identifiable.
We observe a similar trend to the above when allowing cyclic mixed graphs. In Table 2 of , the authors list 75 randomly chosen, cyclic (i.e., containing a loop in the directed part), mixed graphs that are known to be rationally identifiable but cannot be certified so by the half-trek criterion. Of these 75 graphs, 4 are certified to be generically identifiable by the EID algorithm, 0 by the TSID algorithm, and 34 by the EID TSID algorithm.
A listing of the 14 acyclic and 41 cyclic mixed graphs that could not be identified by the EID TSID algorithm are listed as integer pairs in Table 1. The algorithm to convert a pair in that table to a mixed graph on nodes is
For , for , do Add edge to if Replace with
For , for , do Add edge to if Replace with
See Figure 6 for an example of a cyclic and acyclic graph that the EIDTSID algorithm fails to correctly certify as generically identifiable.
|(4456, 113)||(345, 440)||(6629, 512)||(75321, 516)|
|(360, 117)||(71329, 18)||(74536, 788)||(75398, 20)|
|(6275, 172)||(81089, 0)||(5545, 96)||(70803, 896)|
|(6307, 172)||(4714, 41)||(75112, 72)||(4457, 592)|
|(6275, 188)||(70881, 80)||(74970, 4)||(74883, 522)|
|(360, 369)||(74963, 512)||(4579, 384)||(350, 112)|
|(4696, 401)||(74886, 268)||(70594, 65)||(74883, 2)|
|(4936, 401)||(5058, 304)||(74921, 66)||(74950, 260)|
|(4936, 402)||(70821, 513)||(70474, 640)||(74890, 38)|
|(4680, 403)||(74915, 6)||(74922, 66)||(81076, 0)|
|(840, 466)||(5267, 82)||(13160, 65)||(70851, 32)|
|(5257, 658)||(76852, 128)||(4938, 448)||(1430, 120)|
|(5257, 659)||(71075, 516)||(4730, 640)||(5251, 418)|
|(4680, 914)||(4397, 897)||(70358, 1)|
By exploiting the trek-separation characterization of the vanishing of subdeterminants of the covariance matrix corresponding to a mixed graph , we have shown that individual edge coefficients can be generically identified by quotients of subdeterminants. This constitutes a generalization of instrumental variable techniques that are derived from conditional independence. We have also shown how this information, in concert with a generalized half-trek criterion, allows us to prove that substantially more graphs have all or some subset of their parameters generically identifiable.
Our work on identification by ratios of determinants focuses on a single edge coefficient. However, it seems possible to give a generalization that is in the spirit of the generalized instrumental sets from ; see also . These leverage several conditional independencies to find a linear equation system that can be used to identify several edge coefficients simultaneously, under specific assumptions on the interplay of the conditional independencies and the edges to be identified. We illustrate the idea of how to do this using general determinants in the following example. However, a full exploration of this idea is beyond the scope of this paper. In particular, we are still lacking mathematical tools that, in suitable generality, could be used to certify that constructed linear equation systems have a unique solution.
Let be the graph in Figure 7 with corresponding covariance matrix . Then, by similar considerations to those in Example 3.4, one may show that
Using computer algebra we find that the matrix on the left hand side of the above equation has all non-zero polynomial entries, so that this is not equivalent to simply applying Theorem 3.8 for and separately, and has non-zero determinant. It follows that the above system is generically invertible and thus and are generically identifiable.
This material is based on work started in June 2016 at the Mathematics Research Communities (Week on Algebraic Statistics). The work was supported by the National Science Foundation under Grant Number DMS 1321794 and 1712535.
Proof of Lemma 3.7
We will require a known generalization of the Gessel-Viennot-Lindström lemma which we now state.
Let be a directed graph with vertices and corresponding matrix of indeterminants . Let be a directed path in . Then define the loop erased path corresponding to recursively as follows. If contains no loops then . Otherwise there exist indices such that . Then where . It can be shown that is well defined (i.e. is independent of the ordering of the above recursion).
[Gessel-Viennot-Lindström Generalization, Theorem 6.1 from ] Let be a directed graph with vertices and corresponding matrix of indeterminants . Define and for any directed path in define the path polynomial . Then for any we have that
here the above inner sum is over all directed path systems with going from to for all , where and share no vertices for . Hence if and only if every system of directed paths from to has two paths which share a vertex.
The remaining proof of Lemma 3.7 proceeds in several parts and closely follows similar results in  and . As such we will state several lemmas whose proofs require only small modifications of existing results (such as replacing the standard Gessel-Viennot-Lindström Lemma with its generalization above). In such cases we will simply direct the reader to the corresponding proof and sketch the necessary modifications.
Let be a mixed graph and let . We say a trek in avoids on the left (right) if the left (right) side of uses no edges from . Similarly we say a system of treks in avoids on the left (right) if every trek avoids on the left (right). If we say that a trek (or trek system) avoids if it avoids on the left and on the right.
Let be a mixed graph and let be matrices of indeterminants corresponding to the directed and bidirected parts of respectively. Suppose that so that is diagonal. Letting and be as in Lemma 3.7 we have that for any with , if and only if
the max-flow from to in is .
In the following, whenever we say “As in x,” we mean “As in the proof of x in .”
As in Lemma 3.2, we have if and only if for every set with we have or . As in Prop. 3.5, using the above result, and applying our version of the Gessel-Viennot-Lindström Lemma, we have that if and only if every system of (simple) treks avoiding has sided intersection.
Now noticing that simplifies the definition of , we have as in Prop. 3.5 that the (simple) treks from to avoiding in are in bijective correspondence with directed paths from to in . Finally the result follows by noticing that max-flow systems from to in of size correspond to systems of treks from to avoiding with no-sided intersection (that is, if one exists so does the other). Combining the above if and only if statements, the result then follows.
We have now proven our desired result in the case , it remains to show that this implies the case . To this end, we say that is the bidirected subdivision of if it equals but where we have replaced every bidirected edge with a vertex and two edges and (with associated parameters . Note that we have subdivided every bidirected edge into two directed edges which motivates the naming convention. Let and be equal to and respectively but where we have also added in the new edges and for every . Let be matrices of indeterminants corresponding to and let , correspond to just as for . We now have the following result that relates and .
Let , be as in the prior paragraph. Then letting we have that, for any polynomial taking, as input, an matrix of variables, we have that if and only if . In particular, since the subdeterminant of a matrix is a polynomial in the entries of the matrix, we have that for any with , if and only if .
This proof follows, essentially exactly, as the first part of the proof of Prop. 2.5 in .
Now we show that the above subdivision trick produces a graph for which the max-flow between vertex sets is the same as for .
Consider the graphs from the Lemma 3.7 statement and let be corresponding flow graph for the bidirected subdivision of . Let . Then the maximum flow from to in equals the maximum flow from to in .
Recall that a flow system on a graph is an assignment of flow to the edges and vertices of the graph satisfying the usual flow constraints. Also recall that, for graphs with integral capacities, there always exists a max-flow system between subsets of nodes for which all flow assignments upon edges and vertices take values in . We will show that any (integral valued) max-flow system from to in corresponds to a unique flow system in with the same total flow and vice-versa. Our result then follows.
Let be a max-flow system from to on from to with integral flow assignments. Since and have all capacities equal to 1 it follows that assigns either 0 or 1 flow to all edges and vertices in the graph.
We now construct a flow system on with the same capacity. First let assign the same capacity to all edges and vertices that shares with . Note that if does not assign any flow to any of the edges incoming to the vertices then already corresponds to a flow system on with the same total flow. Suppose otherwise that assigns 1 unit of flow to the edges . Since and the have capacity 1 it follows that and for all . For each edge , since has two outgoing edges and , there are two possible cases:
Case 1: assigns 1 flow to .
In this case assign a flow of 1 to the edge in .
Case 2: assigns 1 flow to .
In this case assign a flow of 1 to the edge in .
It is easy to check that is indeed a valid flow system on with the same flow as .
To see the oppose direction let be a max-flow system from to on from to with integral flow assignments. We now construct a flow system on with the same capacity. As before, first let assign the same capacity to all edges and vertices that shares with . Note that if does not assign any flow to any of the edges for then already corresponds to a flow system on with the same total flow. Suppose otherwise that assigns 1 unit of flow to the edges with for all . Since all vertices in have capacity 1 we must have that and for all . There are two possible cases:
Case 1: and .
In this case assign a flow of 1 along the path in .
Case 2: and .
In this case assign a flow of 1 to the edges and in .
One may now check that is a valid flow system on with the same flow as .
Finally we are in a position to easily prove Lemma 3.7. Note that, by Lemma A.5 we have that if and only if . By Lemma A.4 we have that if and only if the max-flow from to in equals . Finally Lemma A.6 gives us that the max-flow from to in equals the max-flow from to in . Hence we have that if and only if the max-flow from to in equals , this was our desired statement.
B Proof of Lemma 4.3
The proof of this lemma follows almost identically as the proof of Lemma 2 in . We simply restate the arguments there in our setting. For any let be the set of half treks from to in . Also let be the set of all treks from to in which do not begin with an edge of the form for any . Then it is easy to see that . Now, by the Trek Rule (Proposition 2.5), we have that
Now for any system of treks define the monomial
Then, by Leibniz’s formula for the determinant, we have that
where the above sum is over all trek systems from to using treks only in the set ; here the is the sign of the permutation that writes in the order of their appearance as targets of the treks in .
By assumption, there exists a half-trek system from to with no-sided intersection. Since such a system exists, let be a half-trek system of minimum total length among all such half-trek systems. Since for all it follows that is included as one of the trek systems in the summation (15). Let be any system of treks from to such that . Lemma 1 from  proves that we must have so that is the unique system of treks from to with corresponding trek monomial . It thus follows that the coefficient of the monomial in is and thus is not the zero polynomial (or power series if the sum is infinite). Hence, for generic choices of , we have that so that is generically invertible.
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