143 lines
4.1 KiB
Python
143 lines
4.1 KiB
Python
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"""
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Algorithm for testing d-separation in DAGs.
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*d-separation* is a test for conditional independence in probability
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distributions that can be factorized using DAGs. It is a purely
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graphical test that uses the underlying graph and makes no reference
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to the actual distribution parameters. See [1]_ for a formal
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definition.
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The implementation is based on the conceptually simple linear time
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algorithm presented in [2]_. Refer to [3]_, [4]_ for a couple of
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alternative algorithms.
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Examples
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--------
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>>>
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>>> # HMM graph with five states and observation nodes
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... g = nx.DiGraph()
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>>> g.add_edges_from(
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... [
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... ("S1", "S2"),
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... ("S2", "S3"),
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... ("S3", "S4"),
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... ("S4", "S5"),
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... ("S1", "O1"),
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... ("S2", "O2"),
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... ("S3", "O3"),
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... ("S4", "O4"),
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... ("S5", "O5"),
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... ]
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... )
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>>>
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>>> # states/obs before 'S3' are d-separated from states/obs after 'S3'
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... nx.d_separated(g, {"S1", "S2", "O1", "O2"}, {"S4", "S5", "O4", "O5"}, {"S3"})
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True
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References
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----------
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.. [1] Pearl, J. (2009). Causality. Cambridge: Cambridge University Press.
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.. [2] Darwiche, A. (2009). Modeling and reasoning with Bayesian networks.
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Cambridge: Cambridge University Press.
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.. [3] Shachter, R. D. (1998).
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Bayes-ball: rational pastime (for determining irrelevance and requisite
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information in belief networks and influence diagrams).
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In , Proceedings of the Fourteenth Conference on Uncertainty in Artificial
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Intelligence (pp. 480–487).
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San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.
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.. [4] Koller, D., & Friedman, N. (2009).
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Probabilistic graphical models: principles and techniques. The MIT Press.
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"""
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from collections import deque
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import networkx as nx
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from networkx.utils import UnionFind, not_implemented_for
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__all__ = ["d_separated"]
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@not_implemented_for("undirected")
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def d_separated(G, x, y, z):
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"""
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Return whether node sets ``x`` and ``y`` are d-separated by ``z``.
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Parameters
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----------
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G : graph
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A NetworkX DAG.
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x : set
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First set of nodes in ``G``.
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y : set
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Second set of nodes in ``G``.
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z : set
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Set of conditioning nodes in ``G``. Can be empty set.
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Returns
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-------
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b : bool
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A boolean that is true if ``x`` is d-separated from ``y`` given ``z`` in ``G``.
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Raises
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------
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NetworkXError
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The *d-separation* test is commonly used with directed
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graphical models which are acyclic. Accordingly, the algorithm
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raises a :exc:`NetworkXError` if the input graph is not a DAG.
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NodeNotFound
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If any of the input nodes are not found in the graph,
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a :exc:`NodeNotFound` exception is raised.
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"""
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if not nx.is_directed_acyclic_graph(G):
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raise nx.NetworkXError("graph should be directed acyclic")
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union_xyz = x.union(y).union(z)
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if any(n not in G.nodes for n in union_xyz):
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raise nx.NodeNotFound("one or more specified nodes not found in the graph")
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G_copy = G.copy()
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# transform the graph by removing leaves that are not in x | y | z
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# until no more leaves can be removed.
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leaves = deque([n for n in G_copy.nodes if G_copy.out_degree[n] == 0])
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while len(leaves) > 0:
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leaf = leaves.popleft()
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if leaf not in union_xyz:
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for p in G_copy.predecessors(leaf):
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if G_copy.out_degree[p] == 1:
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leaves.append(p)
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G_copy.remove_node(leaf)
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# transform the graph by removing outgoing edges from the
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# conditioning set.
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edges_to_remove = list(G_copy.out_edges(z))
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G_copy.remove_edges_from(edges_to_remove)
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# use disjoint-set data structure to check if any node in `x`
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# occurs in the same weakly connected component as a node in `y`.
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disjoint_set = UnionFind(G_copy.nodes())
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for component in nx.weakly_connected_components(G_copy):
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disjoint_set.union(*component)
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disjoint_set.union(*x)
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disjoint_set.union(*y)
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if x and y and disjoint_set[next(iter(x))] == disjoint_set[next(iter(y))]:
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return False
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else:
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return True
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