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from itertools import combinations
import pytest
import networkx as nx
def path_graph():
"""Return a path graph of length three."""
G = nx.path_graph(3, create_using=nx.DiGraph)
G.graph["name"] = "path"
nx.freeze(G)
return G
def fork_graph():
"""Return a three node fork graph."""
G = nx.DiGraph(name="fork")
G.add_edges_from([(0, 1), (0, 2)])
nx.freeze(G)
return G
def collider_graph():
"""Return a collider/v-structure graph with three nodes."""
G = nx.DiGraph(name="collider")
G.add_edges_from([(0, 2), (1, 2)])
nx.freeze(G)
return G
def naive_bayes_graph():
"""Return a simply Naive Bayes PGM graph."""
G = nx.DiGraph(name="naive_bayes")
G.add_edges_from([(0, 1), (0, 2), (0, 3), (0, 4)])
nx.freeze(G)
return G
def asia_graph():
"""Return the 'Asia' PGM graph."""
G = nx.DiGraph(name="asia")
G.add_edges_from(
[
("asia", "tuberculosis"),
("smoking", "cancer"),
("smoking", "bronchitis"),
("tuberculosis", "either"),
("cancer", "either"),
("either", "xray"),
("either", "dyspnea"),
("bronchitis", "dyspnea"),
]
)
nx.freeze(G)
return G
@pytest.fixture(name="path_graph")
def path_graph_fixture():
return path_graph()
@pytest.fixture(name="fork_graph")
def fork_graph_fixture():
return fork_graph()
@pytest.fixture(name="collider_graph")
def collider_graph_fixture():
return collider_graph()
@pytest.fixture(name="naive_bayes_graph")
def naive_bayes_graph_fixture():
return naive_bayes_graph()
@pytest.fixture(name="asia_graph")
def asia_graph_fixture():
return asia_graph()
@pytest.mark.parametrize(
"graph",
[path_graph(), fork_graph(), collider_graph(), naive_bayes_graph(), asia_graph()],
)
def test_markov_condition(graph):
"""Test that the Markov condition holds for each PGM graph."""
for node in graph.nodes:
parents = set(graph.predecessors(node))
non_descendants = graph.nodes - nx.descendants(graph, node) - {node} - parents
assert nx.d_separated(graph, {node}, non_descendants, parents)
def test_path_graph_dsep(path_graph):
"""Example-based test of d-separation for path_graph."""
assert nx.d_separated(path_graph, {0}, {2}, {1})
assert not nx.d_separated(path_graph, {0}, {2}, {})
def test_fork_graph_dsep(fork_graph):
"""Example-based test of d-separation for fork_graph."""
assert nx.d_separated(fork_graph, {1}, {2}, {0})
assert not nx.d_separated(fork_graph, {1}, {2}, {})
def test_collider_graph_dsep(collider_graph):
"""Example-based test of d-separation for collider_graph."""
assert nx.d_separated(collider_graph, {0}, {1}, {})
assert not nx.d_separated(collider_graph, {0}, {1}, {2})
def test_naive_bayes_dsep(naive_bayes_graph):
"""Example-based test of d-separation for naive_bayes_graph."""
for u, v in combinations(range(1, 5), 2):
assert nx.d_separated(naive_bayes_graph, {u}, {v}, {0})
assert not nx.d_separated(naive_bayes_graph, {u}, {v}, {})
def test_asia_graph_dsep(asia_graph):
"""Example-based test of d-separation for asia_graph."""
assert nx.d_separated(
asia_graph, {"asia", "smoking"}, {"dyspnea", "xray"}, {"bronchitis", "either"}
)
assert nx.d_separated(
asia_graph, {"tuberculosis", "cancer"}, {"bronchitis"}, {"smoking", "xray"}
)
def test_undirected_graphs_are_not_supported():
"""
Test that undirected graphs are not supported.
d-separation and its related algorithms do not apply in
the case of undirected graphs.
"""
g = nx.path_graph(3, nx.Graph)
with pytest.raises(nx.NetworkXNotImplemented):
nx.d_separated(g, {0}, {1}, {2})
with pytest.raises(nx.NetworkXNotImplemented):
nx.is_minimal_d_separator(g, {0}, {1}, {2})
with pytest.raises(nx.NetworkXNotImplemented):
nx.minimal_d_separator(g, {0}, {1})
def test_cyclic_graphs_raise_error():
"""
Test that cycle graphs should cause erroring.
This is because PGMs assume a directed acyclic graph.
"""
g = nx.cycle_graph(3, nx.DiGraph)
with pytest.raises(nx.NetworkXError):
nx.d_separated(g, {0}, {1}, {2})
with pytest.raises(nx.NetworkXError):
nx.minimal_d_separator(g, {0}, {1})
with pytest.raises(nx.NetworkXError):
nx.is_minimal_d_separator(g, {0}, {1}, {2})
def test_invalid_nodes_raise_error(asia_graph):
"""
Test that graphs that have invalid nodes passed in raise errors.
"""
with pytest.raises(nx.NodeNotFound):
nx.d_separated(asia_graph, {0}, {1}, {2})
with pytest.raises(nx.NodeNotFound):
nx.is_minimal_d_separator(asia_graph, 0, 1, {2})
with pytest.raises(nx.NodeNotFound):
nx.minimal_d_separator(asia_graph, 0, 1)
def test_minimal_d_separator():
# Case 1:
# create a graph A -> B <- C
# B -> D -> E;
# B -> F;
# G -> E;
edge_list = [("A", "B"), ("C", "B"), ("B", "D"), ("D", "E"), ("B", "F"), ("G", "E")]
G = nx.DiGraph(edge_list)
assert not nx.d_separated(G, {"B"}, {"E"}, set())
# minimal set of the corresponding graph
# for B and E should be (D,)
Zmin = nx.minimal_d_separator(G, "B", "E")
# the minimal separating set should pass the test for minimality
assert nx.is_minimal_d_separator(G, "B", "E", Zmin)
assert Zmin == {"D"}
# Case 2:
# create a graph A -> B -> C
# B -> D -> C;
edge_list = [("A", "B"), ("B", "C"), ("B", "D"), ("D", "C")]
G = nx.DiGraph(edge_list)
assert not nx.d_separated(G, {"A"}, {"C"}, set())
Zmin = nx.minimal_d_separator(G, "A", "C")
# the minimal separating set should pass the test for minimality
assert nx.is_minimal_d_separator(G, "A", "C", Zmin)
assert Zmin == {"B"}
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