Identification of Strong Edges in AMP Chain Graphs

In most practical applications, the data available consists of observations. Therefore, it can rarely single out the true causal model. At best, it identifies the Markov equivalence class that contains the true causal model. In this paper, we represent causal models with the help of AMP chain graphs (Andersson et al.,, 2001). As argued by Peña, (2016), these graphs are suitable for representing causal linear models with additive Gaussian noise. Intuitively, the directed subgraph of a chain graph represents the causal relations in the domain, and the undirected subgraph represents the dependence structure of the noise terms. Additive noise is a rather common assumption in causal discovery (Peters et al.,, 2017), mainly because it produces tractable models which are useful for gaining insight into the system under study. Note also that linear structural equation models, which have extensively been studied for causal effect identification (Pearl,, 2009), are additive noise models.

In order to represent the equivalence class of chain graphs identified from the observations at hand, we typically use a distinguished member of it. In the literature, there are two distinguished members: The essential graph (Andersson and Perlman,, 2006), and the largest deflagged graph (Roverato and Studený,, 2006). In general, they do not coincide: The essential graph is a deflagged graph (Andersson and Perlman,, 2006, Lemma 3.2) but not necessarily the largest in the equivalence class (Andersson et al.,, 2001, p. 57). Unfortunately, the directed edges in either of the two representatives are not necessarily strong,¹¹1The term invariant or essential is also used in the literature. i.e. they may not be shared by every member of the equivalence class. Likewise for the undirected edges. In this paper, we use essential graphs to represent equivalence classes of chain graphs. And we develop a procedure for identifying which edges in an essential graph are strong. Note that while we assume that the true chain graph is unknown, its corresponding essential graph can be obtained from observational data as follows. First, learn a chain graph as shown by Peña, (2014, 2016) and Peña and Gómez-Olmedo, (2016) and, then, transform it into an essential graph as shown by Sonntag and Peña, (2015, Section 3).

Identifying the strong edges in an essential graph is important because it makes it possible to identify causal paths from data even though the data may not be able to single out the true chain graph: Simply output every directed path in the essential graph that consists of only strong edges. Of course, the true chain graph may have additional causal paths. Identifying the strong edges in an essential graph is also important because it allows to efficiently bound some causal effects of the form $p(y|do(x))$ where $X$ and $Y$ are singletons. The simplest way to bound such a causal effect consists in enumerating all the chain graphs that are equivalent to the essential graph and, then, computing the causal effect for each of them from the observational data by adjusting for the appropriate variables. Although we know how to enumerate the equivalent chain graphs (Sonntag and Peña,, 2015, Theorem 3), this method may be inefficient for all but small domains. Instead, we show in this paper how the knowledge of the strong edges in an essential graph allows to enumerate the adjusting sets without enumerating the equivalent chain graphs explicitly.

The rest of the paper is organized as follows. Section 2 introduces some preliminaries. Section 3 presents our algorithm to identify strong edges in an essential graph. Section 4 presents our procedure to bound causal effects when the true chain graph is unknown but its corresponding essential graph is known. Section 5 closes the paper with some discussion and lines of future research.

All the graphs and probability distributions in this paper are defined over a finite set $V$ unless otherwise stated. All the graphs contain at most one edge between a pair of nodes. The elements of $V$ are not distinguished from singletons.

The parents of a set of nodes $X$ of a graph $G$ is the set $Pa_{G}(X)=\{A|A\rightarrow B$ is in $G$ with $B\in X\}$. The children of $X$ is the set $Ch_{G}(X)=\{A|B\rightarrow A$ is in $G$ with $B\in X\}$. The neighbors of $X$ is the set $Ne_{G}(X)=\{A|A-B$ is in $G$ with $B\in X\}$. The adjacents of $X$ is the set $Ad_{G}(X)=\{A|A\rightarrow B$, $B\rightarrow A$ or $A-B$ is in $G$ with $B\in X\}$. The descendants of $X$ is the set $De_{G}(X)=\{A|B\rightarrow\cdots\rightarrow A$ is in $G$ with $B\in X\}$. A route from a node $V_{1}$ to a node $V_{n}$ in $G$ is a sequence of (not necessarily distinct) nodes $V_{1},\ldots,V_{n}$ such that $V_{i}\in Ad_{G}(V_{i+1})$ for all $1\leq i<n$. A route is called a cycle if $V_{n}=V_{1}$. A cycle has a chord if two non-consecutive nodes of the cycle are adjacent in $G$. A cycle is called semidirected if it is of the form $V_{1}\rightarrow V_{2}\multimap\cdots\multimap V_{n}$ where $\multimap$ is a short for $\rightarrow$ or $-$. A chain graph (CG) is a graph with (possibly) directed and undirected edges, and without semidirected cycles. A set of nodes of a CG $G$ is connected if there exists a route in $G$ between every pair of nodes in the set and such that all the edges in the route are undirected. A chain component of $G$ is a maximal connected set. Note that the chain components of $G$ can be sorted topologically, i.e. for every edge $A\rightarrow B$ in $G$, the component containing $A$ precedes the component containing $B$. A set of nodes of $G$ is complete if there is an undirected edge between every pair of nodes in the set. Moreover, a node is called simplicial if its neighbors are a complete set.

We now recall the interpretation of CGs due to Andersson et al., (2001), also known as AMP CGs.²²2Andersson et al., (2001) interpret CGs via the so-called augmentation criterion. Levitz et al., (2001, Theorem 4.1) introduce the so-called p-separation criterion and prove its equivalence to the augmentation criterion. Peña, (2016, Theorem 2) introduce the route-based criterion that we use in this paper and prove its equivalence to the p-separation criterion. A node $B$ in a route $\rho$ in a CG $G$ is called a triplex node in $\rho$ if $A\rightarrow B\leftarrow C$, $A\rightarrow B-C$, or $A-B\leftarrow C$ is a subroute of $\rho$. Moreover, $\rho$ is said to be $Z$-open with $Z\subseteq V$ when (i) every triplex node in $\rho$ is in $Z$, and (ii) every non-triplex node in $\rho$ is outside $Z$. Let $X$, $Y$ and $Z$ denote three disjoint subsets of $V$. When there is no $Z$-open route in $G$ between a node in $X$ and a node in $Y$, we say that $X$ is separated from $Y$ given $Z$ in $G$ and denote it as $X\!\perp\!_{G}Y|Z$. The statistical independences represented by $G$ are the separations $X\!\perp\!_{G}Y|Z$. A probability distribution $p$ is Markovian with respect to $G$ if the independences represented by $G$ are a subset of those in $p$. If the two sets of independences coincide, then $p$ is faithful to $G$. Two CGs are Markov equivalent if the sets of distributions that are Markovian with respect to each CG are the same. If a CG has an induced subgraph of the form $A\rightarrow B\leftarrow C$, $A\rightarrow B-C$ or $A-B\leftarrow C$, then we say that the CG has a triplex $(A,B,C)$. Two CGs are Markov equivalent if and only if they have the same adjacencies and triplexes (Andersson et al.,, 2001, Theorem 5).

We say that a directed edge in a CG is strong if it appears in every equivalent CG. Likewise for undirected edges. Therefore, strong edges are features of a class of equivalent CGs. Clearly, strong directed edges correspond to directed edges in the EG of the equivalence class. However, the opposite is not true. Likewise for strong undirected edges. For an example, consider the EG $A\rightarrow B\leftarrow C-D$. The naive way to detect which edges in an EG are strong consists in generating all the CGs in the equivalence class and, then, recording the shared edges. Since there may be many CGs in the equivalence class, enumerating them in an efficient manner is paramount, but challenging. In truth, it suffices to enumerate what we call the minimally oriented CGs in order to identify the strong directed edges and, then, find one maximally oriented CG to identify the strong undirected edges. We prove these claims in Section 3.1. Although there are typically considerably fewer minimally oriented CGs, enumerating them in an efficient manner seems challenging too. That is why we present in Section 3.2 an algorithm that does not rely on enumerating CGs or minimally oriented CGs.

When the true CG is unknown, a causal effect of the form $p(y|do(x))$ with $X,Y\in V$ cannot be computed, but it can be bounded as follows:

1. 1.

   Obtain all the CGs that are Markov equivalent to the true one by running the learning algorithm developed by Peña, (2014, 2016) or Peña and Gómez-Olmedo, (2016).

2. 2.

   Compute the causal effect for each CG obtained as follows. Like in a Bayesian network, any causal effect in a CG $G$ is computable uniquely from observed quantities (i.e. it is identifiable) by adjusting for the appropriate variables. Specifically,

   $$p(y|do(x))=\int p(y|x,z)p(z)dz$$

   where $Z=Ne_{G}(X)\cup Pa_{G}(X\cup Ne_{G}(X))$ and $Y\notin Z$. The role of $Z$ is to block every non-causal path in $G$ between $X$ and $Y$. We call $Z$ the adjusting set in $G$.

Unfortunately, the learning algorithm in step 1 may be too time consuming for all but small domains. At least, this is the conclusion that follows from the experimental results reported by Sonntag et al., (2015) for a similar algorithm for learning Lauritzen-Wermuth-Frydenberg CGs. Instead, we propose the following alternative approach:

1. 1’.

   Learn the EG $G^{*}$ corresponding to the true CG from data as follows. First, learn a CG from data as shown by Peña, (2014, 2016) and Peña and Gómez-Olmedo, (2016) and, then, transform it into an EG as shown by Sonntag and Peña, (2015, Section 3).

2. 2’.

   Enumerate all the CGs that are Markov equivalent to $G^{*}$ as shown by Sonntag and Peña, (2015, Theorem 3).

3. 3’.

   Compute the causal effect for each CG enumerated as shown above.

This approach has successfully been applied when the causal models are represented by other graphical models than CGs (Hyttinen et al.,, 2015; Malinsky and Spirtes,, 2016; Maathuis et al.,, 2009). The experimental results reported by Peña and Gómez-Olmedo, (2016) indicate that the learning algorithm in step 1’ scales to medium sized domains. However, the enumeration in step 2’ may be too time consuming for all but small domains. Alternatively, we may try to enumerate the adjusting sets in the equivalent CGs without enumerating these explicitly. Specifically, we know that all the adjusting sets are subsets of $Ad_{G^{*}}(X)\cup Ad_{G^{*}}(Ad_{G^{*}}(X))$, because all the equivalent CGs have the same adjacencies as $G^{*}$. Therefore, we can adjust for every subset of $Ad_{G^{*}}(X)\cup Ad_{G^{*}}(Ad_{G^{*}}(X))$ to obtain bounds for the causal effect of interest. True that some of these subsets are not valid adjusting sets in the sense that they do not correspond to any of the equivalent CGs. However, this does not make the bounds invalid, just more loose. The rest of the section studies a case where all and only the valid adjusting sets can be enumerated efficiently.

Assume that we believe a priori that the dependencies in the domain at hand are due to causal rather than non-causal relationships. Then, we believe a posteriori that the true CG is a maximally oriented CG, because such CGs have the fewest undirected edges in the equivalence class of the EG $G^{*}$ learned from the data in step 1’. Moreover, recall from Section 3.1 that all of them have the same undirected edges. Therefore, we can bound the causal effect $p(y|do(x))$ by modifying the latter framework above so that only maximally oriented CGs are enumerated in step 2’. A maximally oriented CG that is equivalent to $G^{*}$ can be obtained from $G^{*}$ by repeatedly performing feasible splits (Sonntag and Peña,, 2015, Theorem 3). Unfortunately, this enumeration method may be inefficient for all but small domains. Instead, we show below how to enumerate the adjusting sets in the maximally oriented CGs that are equivalent to $G^{*}$ without enumerating these explicitly.

Given a node $X\in V$, we define $St_{G^{*}}(X)=\{A|A-X$ is a strong edge in $G^{*}\}$ and $Nst_{G^{*}}(X)=\{A|A-X$ is a non-strong edge in $G^{*}\}$. Given a set $S\subseteq Nst_{G^{*}}(X)$, we let $G^{*}_{S\rightarrow X}$ denote the graph that is obtained from $G^{*}$ by replacing the edge $A-X$ with $A\rightarrow X$ for all $A\in S$, and replacing the edge $A-X$ with $A\leftarrow X$ for all $A\in Nst_{G^{*}}(X)\setminus S$. Moreover, we say that $G^{*}_{S\rightarrow X}$ is locally valid if $G^{*}_{S\rightarrow X}$ does not have any triplex $(A,X,B)$ that is not in $G^{*}$. The next theorem proves that producing the adjusting sets in the equivalent maximally oriented CGs simplifies to produce locally valid sets.

The procedure outlined above can be simplified as follows.