D-Separation#

Algorithm for testing d-separation in DAGs.

d-separation is a test for conditional independence in probability distributions that can be factorized using DAGs. It is a purely graphical test that uses the underlying graph and makes no reference to the actual distribution parameters. See [1] for a formal definition.

The implementation is based on the conceptually simple linear time algorithm presented in [2]. Refer to [3], [4] for a couple of alternative algorithms.

The functional interface in NetworkX consists of three functions:

D-separators#

Here, we provide a brief overview of d-separation and related concepts that are relevant for understanding it:

The ideas of d-separation and d-connection relate to paths being open or blocked.

• A “path” is a sequence of nodes connected in order by edges. Unlike for most graph theory analysis, the direction of the edges is ignored. Thus the path can be thought of as a traditional path on the undirected version of the graph.

• A “candidate d-separator” z is a set of nodes being considered as possibly blocking all paths between two prescribed sets x and y of nodes. We refer to each node in the candidate d-separator as “known”.

• A “collider” node on a path is a node that is a successor of its two neighbor nodes on the path. That is, c is a collider if the edge directions along the path look like ... u -> c <- v ....

• If a collider node or any of its descendants are “known”, the collider is called an “open collider”. Otherwise it is a “blocking collider”.

• Any path can be “blocked” in two ways. If the path contains a “known” node that is not a collider, the path is blocked. Also, if the path contains a collider that is not a “known” node, the path is blocked.

• A path is “open” if it is not blocked. That is, it is open if every node is either an open collider or not a “known”. Said another way, every “known” in the path is a collider and every collider is open (has a “known” as a inclusive descendant). The concept of “open path” is meant to demonstrate a probabilistic conditional dependence between two nodes given prescribed knowledge (“known” nodes).

• Two sets x and y of nodes are “d-separated” by a set of nodes z if all paths between nodes in x and nodes in y are blocked. That is, if there are no open paths from any node in x to any node in y. Such a set z is a “d-separator” of x and y.

• A “minimal d-separator” is a d-separator z for which no node or subset of nodes can be removed with it still being a d-separator.

The d-separator blocks some paths between x and y but opens others. Nodes in the d-separator block paths if the nodes are not colliders. But if a collider or its descendant nodes are in the d-separation set, the colliders are open, allowing a path through that collider.

Illustration of D-separation with examples#

A pair of two nodes, u and v, are d-connected if there is a path from u to v that is not blocked. That means, there is an open path from u to v.

For example, if the d-separating set is the empty set, then the following paths are open between u and v:

• u <- n -> v

• u -> w -> … -> n -> v

If on the other hand, n is in the d-separating set, then n blocks those paths between u and v.

Colliders block a path if they and their descendants are not included in the d-separating set. An example of a path that is blocked when the d-separating set is empty is:

• u -> w -> … -> n <- v

The node n is a collider in this path and is not in the d-separating set. So n blocks this path. However, if n or a descendant of n is included in the d-separating set, then the path through the collider at n (… -> n <- …) is “open”.

D-separation is concerned with blocking all paths between nodes from x to y. A d-separating set between x and y is one where all paths are blocked.

D-separation and its applications in probability#

D-separation is commonly used in probabilistic causal-graph models. D-separation connects the idea of probabilistic “dependence” with separation in a graph. If one assumes the causal Markov condition [5], (every node is conditionally independent of its non-descendants, given its parents) then d-separation implies conditional independence in probability distributions. Symmetrically, d-connection implies dependence.

The intuition is as follows. The edges on a causal graph indicate which nodes influence the outcome of other nodes directly. An edge from u to v implies that the outcome of event u influences the probabilities for the outcome of event v. Certainly knowing u changes predictions for v. But also knowing v changes predictions for u. The outcomes are dependent. Furthermore, an edge from v to w would mean that w and v are dependent and thus that u could indirectly influence w.

Without any knowledge about the system (candidate d-separating set is empty) a causal graph u -> v -> w allows all three nodes to be dependent. But if we know the outcome of v, the conditional probabilities of outcomes for u and w are independent of each other. That is, once we know the outcome for v, the probabilities for w do not depend on the outcome for u. This is the idea behind v blocking the path if it is “known” (in the candidate d-separating set).

The same argument works whether the direction of the edges are both left-going and when both arrows head out from the middle. Having a “known” node on a path blocks the collider-free path because those relationships make the conditional probabilities independent.

The direction of the causal edges does impact dependence precisely in the case of a collider e.g. u -> v <- w. In that situation, both u and w influence v. But they do not directly influence each other. So without any knowledge of any outcomes, u and w are independent. That is the idea behind colliders blocking the path. But, if v is known, the conditional probabilities of u and w can be dependent. This is the heart of Berkson’s Paradox [6]. For example, suppose u and w are boolean events (they either happen or do not) and v represents the outcome “at least one of u and w occur”. Then knowing v is true makes the conditional probabilities of u and w dependent. Essentially, knowing that at least one of them is true raises the probability of each. But further knowledge that w is true (or false) change the conditional probability of u to either the original value or 1. So the conditional probability of u depends on the outcome of w even though there is no causal relationship between them. When a collider is known, dependence can occur across paths through that collider. This is the reason open colliders do not block paths.

Furthermore, even if v is not “known”, if one of its descendants is “known” we can use that information to know more about v which again makes u and w potentially dependent. Suppose the chance of n occurring is much higher when v occurs (“at least one of u and w occur”). Then if we know n occurred, it is more likely that v occurred and that makes the chance of u and w dependent. This is the idea behind why a collider does no block a path if any descendant of the collider is “known”.

When two sets of nodes x and y are d-separated by a set z, it means that given the outcomes of the nodes in z, the probabilities of outcomes of the nodes in x are independent of the outcomes of the nodes in y and vice versa.

Examples#

A Hidden Markov Model with 5 observed states and 5 hidden states where the hidden states have causal relationships resulting in a path results in the following causal network. We check that early states along the path are separated from late state in the path by the d-separator of the middle hidden state. Thus if we condition on the middle hidden state, the early state probabilities are independent of the late state outcomes.

>>> G = nx.DiGraph()
...     [
...         ("H1", "H2"),
...         ("H2", "H3"),
...         ("H3", "H4"),
...         ("H4", "H5"),
...         ("H1", "O1"),
...         ("H2", "O2"),
...         ("H3", "O3"),
...         ("H4", "O4"),
...         ("H5", "O5"),
...     ]
... )
>>> x, y, z = ({"H1", "O1"}, {"H5", "O5"}, {"H3"})
>>> nx.is_d_separator(G, x, y, z)
True
>>> nx.is_minimal_d_separator(G, x, y, z)
True
>>> nx.is_minimal_d_separator(G, x, y, z | {"O3"})
False
>>> z = nx.find_minimal_d_separator(G, x | y, {"O2", "O3", "O4"})
>>> z == {"H2", "H4"}
True


If no minimal_d_separator exists, None is returned

>>> other_z = nx.find_minimal_d_separator(G, x | y, {"H2", "H3"})
>>> other_z is None
True


References#

[1]

Pearl, J. (2009). Causality. Cambridge: Cambridge University Press.

[2]

Darwiche, A. (2009). Modeling and reasoning with Bayesian networks. Cambridge: Cambridge University Press.

[3]

Shachter, Ross D. “Bayes-ball: The rational pastime (for determining irrelevance and requisite information in belief networks and influence diagrams).” In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence (UAI), (pp. 480–487). 1998.

[4]

Koller, D., & Friedman, N. (2009). Probabilistic graphical models: principles and techniques. The MIT Press.

 is_d_separator(G, x, y, z) Return whether node sets x and y are d-separated by z. is_minimal_d_separator(G, x, y, z, *[, ...]) Determine if z is a minimal d-separator for x and y. find_minimal_d_separator(G, x, y, *[, ...]) Returns a minimal d-separating set between x and y` if possible