This paper investigates the intersection property of conditional independence. For continuous random variables and X this property states that and implies . Here, “” stands for statistical independence and “” for statistical dependence (see Section 1.2 for precise definitions). The intersection property does not necessarily hold if the joint distribution does not have a density (e.g. Dawid ). Dawid  provides measure-theoretic necessary and sufficient conditions for the intersection property. In this work we assume the existence of a density (A0), see below.
It is well known that the intersection property holds if the joint distribution has a strictly positive density (e.g. Pearl , 1.1.5). Proposition 1 shows that if the density is not strictly positive, a weaker condition than the intersection property still holds. Corollary 1 states necessary and sufficient conditions for the intersection property. The result about strictly positive densities is contained as a special case. Drton et al. (, exercise 6.6) and Fink  develop analogous results for the discrete case.
1.1 Main contributions
In Section 3 we provide a sufficient and necessary condition on the density for the intersection property to hold (Corollary 1). This result is of interest in itself since the developed condition is weaker than strict positivity.
Studying the intersection property has direct applications to causal inference. Inferring causal relationships is a major challenge in science. In the last decades considerable effort has been made in order to learn causal statements from observational data. As a first step, causal discovery methods therefore estimate graphs from observational data and attach a causal meaning to these graphs (the terminology of causal inference is introduced in Section 4.1). Some causal discovery methods based on structural equation models (SEMs) require the intersection property for identification; they therefore rely on the strict positivity of the density. This is satisfied if the noise variables have full support, for example. Using the new characterization of the intersection property we can now replace the condition of strict positivity. In fact, we show in Section 4 that noise variables with a path-connected support are sufficient for identifiability of the graph (Proposition 3). This is already known for linear SEMs  but not for non-linear models. As an alternative, we provide a condition that excludes a specific kind of constant functions and leads to identifiability, too (Proposition 4).
In Section 2, we provide an example of an SEM that violates the intersection property. Its corresponding graph is not identifiable from the joint distribution. In correspondence to the theoretical results of this work, some noise densities in the example do not have a path-connected support and the functions are partially constant. We are not aware of any causal discovery method that is able to infer the correct graph or the correct Markov equivalence class; the example therefore shows current limits of causal inference techniques. It is non-generic in the case that it violates all sufficient assumptions mentioned in Section 4.
All proofs are provided in Appendix A.
1.2 Conditional independence and the intersection property
We now formally introduce the concept of conditional independence in the presence of densities and the intersection property. Let therefore and X be (possibly multi-dimensional) random variables that take values in metric spaces and , respectively. We first introduce assumptions regarding the existence of a density and some of its properties that appear in different parts of this paper.
The distribution is absolutely continuous with respect to a product measure of a metric space. We denote the density by . This can be a probability mass function or a probability density function, for example.
The density is continuous. If there is no variable C (or C is deterministic), then is continuous.
For each c with the set contains only one path-connected component (see Section 3).
The density is strictly positive.
Condition (A2′) implies (A2). We assume (A0) throughout the whole work.
In this paper we work with the following definition of conditional independence.
Definition 1 (Conditional (In)dependence). We call X independent of A conditional on B and write if and only if
for all such that . Otherwise, X and A are dependent conditional on B and we write .
The intersection property of conditional independence is defined as follows (e.g. Pearl , 1.1.5).
Definition 2 (Intersection Property). We say that the joint distribution of satisfies the intersection property if
We now give an example of a distribution that does not satisfy the intersection property (2). Since the joint distribution has a continuous density, the example shows that the intersection property requires further restrictions on the density apart from its existence. We will later use the same idea to prove Proposition 2 that shows the necessity of our new condition.
Example 1. Consider a so-called additive noise model (ANM; see Section 4.1) for random variables : (3)where are jointly independent, have continuous densities and satisfy and . Let the function f be of the form (4)where the function g can be chosen to make f arbitrarily smooth. Some parts of this structural equation model (SEM) are summarized in Figure 1. The distribution satisfies and but and . The (intuitive) reason for this as follows: we see from eq. (3). Further, if we know that A (or B) is positive, X has to take values close to ten and thus (); but when knowing that A is positive, the knowledge of B does not provide any additional information about X (). This means that the intersection property is violated. A formal proof is provided in the more general setting of Proposition 2. Within each component, however, that is if we consider the areas and separately, we do have the independence statement ; therefore the intersection property holds “locally”. This observation will be formalized as the weak intersection property in Proposition 1.
It will turn out to be important that the two path-connected components of the support of A and B cannot be connected by an axis-parallel line. This motivates the notation introduced in Section 3. Remark 1 in Section 4 discusses the causal interpretation of Example 1.
3 Necessary and sufficient condition for the intersection property
This section characterizes the intersection property in terms of the joint density over the corresponding random variables. In particular, we state a weak intersection property (Proposition 1) that leads to a necessary and sufficient condition for the classical intersection property, see Corollary 1.
We will see that the intersection property fails in Example 1 because of the two “separated” components in Figure 1. In order to formulate our results we first require the notion of path-connectedness. A continuous mapping into a metric space is called a path between and in . A subset is called path-connected if every pair of points in can be connected by a path in . We can always decompose into its (disjoint) path-connected components. 1 The following definition provides a formalization of the intuition that the two components in Figure 1 are “separated”.
Definition 3. (i) For each c with we define the (not necessarily closed) support of A and B as We further write for all sets (ii) We denote the path-connected components of by , , with some index set . Two path-connected components and are said to be coordinate-wise connected if (The intuition is that we can draw an axis-parallel line from to .) We then say that and are equivalent if and only if there exists a sequence with all neighbours and being coordinate-wise connected. We represent the equivalence classes by the union of all its members. These unions we denote by , .
We further introduce a deterministic function of the variables A and B. We set We have that if and only if if and only if . Furthermore, the projections are disjoint for different i; the same holds for .
(iii) The case where there is no variable C can be treated as if C was deterministic: for some c.
Using Definition 3 we are now able to state the two main results, Propositions 1 and 2. As a direct consequence we obtain Corollary 1 which generalizes the condition of strictly positive densities.
Proposition 1 (Weak Intersection Property). Assume (A0), (A1) and that and . Consider now c with and the variable as defined in Definition 3(ii). We then have the weak intersection property:
This means that
for all with . The values of do not provide additional information if we already know .
We call this property the weak intersection property for the following reason: if , then by definition if and only if and therefore In this sense, eq. (5) is strictly weaker than .
Furthermore, Proposition 1 includes the intersection property for positive densities as a special case. If the density is indeed strictly positive, then there is only a single path-connected component and a single equivalence class . Therefore, is constant and it follows from eq. (5) and Lemma 1 (see “Proof of Proposition 1” in Appendix A.) that .
Proposition 2 (Failure of Intersection Property). Assume (A0), (A1) and that there exist two different sets for some with . Then there exists a random variable X such that the intersection property (2) does not hold for the joint distribution of .
As a direct corollary from these two propositions we obtain a characterization of the intersection property in the case of continuous densities.
Corollary 1 (Intersection Property). Assume (A0) and (A1).
The intersection property (2) holds for all variables X if and only if all components
In particular, this is the case if (A2) holds (there is only one path-connected component) or (A2’) holds (the density is strictly positive).
4 Application to causal discovery
We will first introduce some graph notation that we use for formulating the application to causal inference.
4.1 Notation and prerequisites
Standard graph definitions can be found in Lauritzen , Spirtes et al.  and many others. We follow the presentation of Section 1.1 in Peters et al. . A graph contains nodes (often identified with random variables ) and edges between nodes. A graph is called a proper subgraph of G if and with . A node i is called a parent of j if and a child if . The set of parents of j is denoted by , the set of its children by . Two nodes i and j are adjacent if either or . We say that there is an undirected edge between two adjacent nodes i and j if and . An edge between two adjacent nodes is directed if it is not undirected. We then write for . Three nodes are called a v-structure if one node is a child of the two others that themselves are not adjacent. A path in G is a sequence of (at least two) distinct vertices , such that there is an edge between and for all . If for all k we speak of a directed path from to and call a descendant of . We denote all descendants of by and all non-descendants of , excluding i, by . In this work, i is neither a descendant nor a non-descendant of itself. G is called a directed acyclic graph (DAG), if all edges are directed and there is no pair of nodes (j, k) such that there are directed paths from j to k and from k to j. In a DAG, a path between and is blocked by a set (with neither nor in this set) whenever there is a node , such that one of the following two possibilities hold: (1) and or or , or (2) and neither nor any of its descendants is in . We say that two disjoint subsets of vertices and are d-separated by a third (also disjoint) subset if every path between nodes in and is blocked by . A joint distribution is said to be Markov with respect to the DAG G if d-sep. by for all disjoint sets . It is said to be faithful to the DAG G if d-sep. by for all disjoint sets . Finally, a distribution satisfies causal minimality with respect to G if it is Markov with respect to G, but not to any proper subgraph of G.
In order to infer graphs from distributions, one requires assumptions that relate the joint distribution with properties of the graph, which is often assumed to be a DAG. Constraint-based or independence-based methods [3, 8] and some score-based methods [10, 11] assume the Markov condition and faithfulness. These two assumptions make the Markov equivalence class of the correct graph identifiable from the joint distribution, i.e. the skeleton and the v-structures of the graph can be inferred from the joint distribution .
Alternatively [6, 9, 13], we can assume an additive noise models (ANMs). In these models, the joint distribution over is generated by an SEM (6)with continuous, non-constant functions , additive and jointly independent noise variables with mean zero and sets that are the parents of i in a DAG G. To simplify notation, we have identified variable with its index (or node) i. These models can be shown to satisfy the Markov condition (Pearl , theorem 1.4.1); the functions being non-constant correspond to causal minimality (Peters et al. , proposition 17), which is strictly weaker than faithfulness. We now define what we mean by identifiability of the DAG in continuous ANMs. Consider a certain class of SEMs and suppose that the distribution is generated from such an SEM. We say that G is identifiable from P if P cannot be generated by an SEM from the same class but with a different graph .
Loosely speaking, Peters et al. (, theorem 28) prove that
4.2 Intersection property and causal discovery
We first revisit Example 1 and interpret it from a causal point of view.
Remark 1 (Example 1 continued). Example 1 has the following important implication for causal inference. The distribution can be generated by two different DAGs, namely and , see Figure 1. The SEM (3) corresponds to the former DAG. A slightly modified version of eq. (3) where replaces the last equation in eq. (3) corresponds to the latter DAG. The distribution satisfies causal minimality with respect to both DAGs. Since it violates faithfulness and the intersection property, we are not aware of any causal inference method that is able to recover the correct graph structure based on observational data only. Recall that Peters et al.  assume strictly positive densities in order to assure the intersection property. More precisely, Example 1 shows that lemma 38 in Peters et al. , see Appendix B., does not hold anymore when the positivity is violated.
In order to prove , Peters et al.  require a strictly positive density. This is because the key results used in the proof is proposition 29 which is proved using lemma 38, which itself relies on the intersection property (proposition 29 and lemma 38 are provided in Appendix B.). But since Corollary 1 provides weaker assumption for the intersection property, we are now able to obtain new identifiability results.
Proposition 3. Assume that a joint distribution over is generated by an ANM (6). Assume further that the noise variables have continuous densities and that the support of each noise variable , is path-connected. Then, statement holds.
Example 1 violates the assumption of Proposition 3 since the support of A is not path-connected. It satisfies another important property, too: the function f is constant on some intervals. The following proposition shows that this is necessary to violate identifiability.
Proposition 4. Assume that a joint distribution over is generated by an ANM (6) with graph G. Let us denote the non-descendants of by . Assume that the structural equations are non-constant in the following way: for all , for all its parents and for all , there are such that and and . Here, represents the value of all parents of except . Then for any , it holds that . Therefore, statement follows.
Proposition 4 provides an alternative way to prove identifiability. The results are summarized in Table 1.
It is possible to prove the intersection property of conditional independence for variables whose distributions do not have a strictly positive density. A necessary and sufficient condition for the intersection property is that all path-connected components of the support of the density are equivalent, that is, they can be connected by axis-parallel lines. In particular, this condition is satisfied for densities whose support is path-connected. In the general case, the intersection property still holds after conditioning on an equivalence class of path-connected components; we call this the weak intersection property. We believe that the assumption of a density that is continuous in A, and C can be weakened even further.
This insight has a direct application in causal inference (which is rather of theoretical nature than having implications for practical methods). In the context of continuous ANMs, we relax important conditions for identifiability of the graph from the joint distribution. Furthermore, there is some interest in uniform consistency in causal inference. For linear Gaussian SEMs, for example, the PC algorithm  exploits conditional independences, that is, vanishing partial correlations. Zhang and Spirtes  prove uniform consistency under the assumption that non-vanishing partial correlations cannot be arbitrarily close to zero (this condition is referred to as “strong faithfulness”). Our work suggests that in order to prove uniform consistency for continuous ANMs, one may need to be “bounded away” from Example 1.
The author thanks the anonymous reviewers for their insightful and constructive comments. He further thanks Thomas Kahle and Mathias Drton for pointing out the link to algebraic statistics for discrete variables. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no 326496.
Proof of Proposition 1
We require the following well-known lemma (e.g. Dawid ).
Lemma 1. We have if and only if for all such that .
Proof. (of Proposition 1) To simplify notation we write . We have by Lemma 1 (7)for all with . As the main argument we show that (8)for all with for the same i.
Step 1, we prove eq. (8) for . We first show that there is a path , such that for all , and and . Since the interval is compact and is continuous, the path is compact, too (for notational simplicity we identify the path with its image). Define for each point on the path an open ball with radius small enough such that all in the ball satisfy (this is possible because is assumed to be continuous). Because these balls are path-connected, they also lie in . They form an open cover of the path , and we can thus choose a finite subset of balls, of size n say, that still provides an open cover of the path. Without loss of generality let be the centre of ball 1 and be the centre of ball n. It suffices to show that eq. (8) holds for the centres of two neighbouring balls, say and . Choose a point from the non-empty intersection of those two balls. Since and for the Euclidean metric d, we have that , , , and are all greater than zero. Therefore, using eq. (7) several times, This shows eq. (8) for .
Step 2, we prove eq. (8) for and , where and are coordinate-wise connected (and thus equivalent). If , we know that from the argument given in step 1. If , then there is a such that and . By eq. (7) and the argument from step 1 we have
Consider now such that (which implies ) and consider , say. Observe further that for . We thus have with . It is the case, however, that for all there is a with . But since also we have by eq. (8). Ergo, This implies Together with eq. (7) this leads to □
Proof of Proposition 2
Proof. Define X according to where is uniformly distributed with independent of . Define g according to Fix a value c with . We then have for all with that because can be written as a function of A or of B. We therefore have that and . Depending on whether b is in or not we have or , respectively. Thus, This shows that . Note that is not necessarily continuous, see (A1). □
Proof of Proposition 3
Proof. Since the true structure corresponds to a DAG, we can find a causal ordering, i.e. a permutation such that In this ordering, is a source node and is a sink node. We can then rewrite the structural equation model in eq. (6) as
where the functions are the same as except they are constant in the additional input arguments.
The density of the random vector has path-connected support by the following argument: consider a one-dimensional random variable N with mean zero and a (possibly multivariate) random vector X both with path-connected support and a continuous function f. Then, the support of the random vector is path-connected, too. Indeed, consider two points and from the support of . The path can then be constructed by concatenating three sub-paths: (1) the path between and (N’s support is path-connected), (2) the path between and on the graph of f (which is path-connected due to the continuity of f) and (3) the path between and , analogously to (1).
Therefore, the intersection property (2) holds for any disjoint sets of variables by Proposition 1. Thus, the statements of lemma 38 and thus proposition 29 from Peters et al.  remain correct, which proves for noise variables with continuous densities and path-connected support. □
Proof of Proposition 4
Proof. The proof is immediate. Since (the means are not the same) the statement follows from Lemma 1.
In this case, lemma 38 might not hold but more importantly proposition 29 does (both from Peters et al. . This proves . □
Technical results for identifiability in additive noise models
We provide the two key results required for proving property in Section 4.1. The intersection property is used to prove the “only if” part of lemma 38, which itself is used to prove proposition 29.
Lemma 38  Consider the random vector and assume that the joint distribution has a (strictly) positive density. Then the joint distribution over satisfies causal minimality with respect to a DAG G if and only if and with we have that
Proposition 29  Let G and be two different DAGs over variables . Assume that the joint distribution over has a strictly positive density and satisfies the Markov condition and causal minimality with respect to G and . Then there are variables such that for the sets , and we have
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