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.


>>> # HMM graph with five states and observation nodes
... g = nx.DiGraph()
>>> g.add_edges_from(
...     [
...         ("S1", "S2"),
...         ("S2", "S3"),
...         ("S3", "S4"),
...         ("S4", "S5"),
...         ("S1", "O1"),
...         ("S2", "O2"),
...         ("S3", "O3"),
...         ("S4", "O4"),
...         ("S5", "O5"),
...     ]
... )
>>> # states/obs before 'S3' are d-separated from states/obs after 'S3'
... nx.d_separated(g, {"S1", "S2", "O1", "O2"}, {"S4", "S5", "O4", "O5"}, {"S3"})



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


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


Shachter, R. D. (1998). Bayes-ball: 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 (pp. 480–487). San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.


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

d_separated(G, x, y, z)

Return whether node sets x and y are d-separated by z.