#!/usr/bin/env python3
import itertools
from collections import defaultdict
import networkx as nx
import numpy as np
from networkx.algorithms.components import connected_components
from pgmpy.base import UndirectedGraph
from pgmpy.factors import factor_product
from pgmpy.factors.discrete import DiscreteFactor
from pgmpy.independencies import Independencies
[docs]
class MarkovNetwork(UndirectedGraph):
"""
Base class for Markov Model.
A MarkovNetwork stores nodes and edges with potentials
MarkovNetwork holds undirected edges.
Parameters
----------
data : input graph
Data to initialize graph. If data=None (default) an empty
graph is created. The data can be an edge list, or any
NetworkX graph object.
Examples
--------
Create an empty Markov Model with no nodes and no edges.
>>> from pgmpy.models import MarkovNetwork
>>> G = MarkovNetwork()
G can be grown in several ways.
**Nodes:**
Add one node at a time:
>>> G.add_node('a')
Add the nodes from any container (a list, set or tuple or the nodes
from another graph).
>>> G.add_nodes_from(['a', 'b'])
**Edges:**
G can also be grown by adding edges.
Add one edge,
>>> G.add_edge('a', 'b')
a list of edges,
>>> G.add_edges_from([('a', 'b'), ('b', 'c')])
If some edges connect nodes not yet in the model, the nodes
are added automatically. There are no errors when adding
nodes or edges that already exist.
**Shortcuts:**
Many common graph features allow python syntax for speed reporting.
>>> 'a' in G # check if node in graph
True
>>> len(G) # number of nodes in graph
3
"""
def __init__(self, ebunch=None, latents=[]):
super(MarkovNetwork, self).__init__()
if ebunch:
self.add_edges_from(ebunch)
self.factors = []
self.latents = latents
[docs]
def add_edge(self, u, v, **kwargs):
"""
Add an edge between u and v.
The nodes u and v will be automatically added if they are
not already in the graph
Parameters
----------
u,v : nodes
Nodes can be any hashable Python object.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> G = MarkovNetwork()
>>> G.add_nodes_from(['Alice', 'Bob', 'Charles'])
>>> G.add_edge('Alice', 'Bob')
"""
# check that there is no self loop.
if u != v:
super(MarkovNetwork, self).add_edge(u, v, **kwargs)
else:
raise ValueError("Self loops are not allowed")
[docs]
def add_factors(self, *factors):
"""
Associate a factor to the graph.
See factors class for the order of potential values
Parameters
----------
*factor: pgmpy.factors.factors object
A factor object on any subset of the variables of the model which
is to be associated with the model.
Returns
-------
None
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> student = MarkovNetwork([('Alice', 'Bob'), ('Bob', 'Charles'),
... ('Charles', 'Debbie'), ('Debbie', 'Alice')])
>>> factor = DiscreteFactor(['Alice', 'Bob'], cardinality=[3, 2],
... values=np.random.rand(6))
>>> student.add_factors(factor)
"""
for factor in factors:
if set(factor.variables) - set(factor.variables).intersection(
set(self.nodes())
):
raise ValueError("Factors defined on variable not in the model", factor)
self.factors.append(factor)
[docs]
def get_factors(self, node=None):
"""
Returns all the factors containing the node. If node is not specified
returns all the factors that have been added till now to the graph.
Parameters
----------
node: any hashable python object (optional)
The node whose factor we want. If node is not specified
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> student = MarkovNetwork([('Alice', 'Bob'), ('Bob', 'Charles')])
>>> factor1 = DiscreteFactor(['Alice', 'Bob'], cardinality=[2, 2],
... values=np.random.rand(4))
>>> factor2 = DiscreteFactor(['Bob', 'Charles'], cardinality=[2, 3],
... values=np.ones(6))
>>> student.add_factors(factor1,factor2)
>>> student.get_factors()
[<DiscreteFactor representing phi(Alice:2, Bob:2) at 0x7f8a0e9bf630>,
<DiscreteFactor representing phi(Bob:2, Charles:3) at 0x7f8a0e9bf5f8>]
>>> student.get_factors('Alice')
[<DiscreteFactor representing phi(Alice:2, Bob:2) at 0x7f8a0e9bf630>]
"""
if node:
if node not in self.nodes():
raise ValueError("Node not present in the Undirected Graph")
node_factors = []
for factor in self.factors:
if node in factor.scope():
node_factors.append(factor)
return node_factors
else:
return self.factors
[docs]
def remove_factors(self, *factors):
"""
Removes the given factors from the added factors.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> student = MarkovNetwork([('Alice', 'Bob'), ('Bob', 'Charles')])
>>> factor = DiscreteFactor(['Alice', 'Bob'], cardinality=[2, 2],
... values=np.random.rand(4))
>>> student.add_factors(factor)
>>> student.remove_factors(factor)
"""
for factor in factors:
self.factors.remove(factor)
[docs]
def get_cardinality(self, node=None):
"""
Returns the cardinality of the node. If node is not specified returns
a dictionary with the given variable as keys and their respective cardinality
as values.
Parameters
----------
node: any hashable python object (optional)
The node whose cardinality we want. If node is not specified returns a
dictionary with the given variable as keys and their respective cardinality
as values.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> student = MarkovNetwork([('Alice', 'Bob'), ('Bob', 'Charles')])
>>> factor = DiscreteFactor(['Alice', 'Bob'], cardinality=[2, 2],
... values=np.random.rand(4))
>>> student.add_factors(factor)
>>> student.get_cardinality(node='Alice')
2
>>> student.get_cardinality()
defaultdict(<class 'int'>, {'Bob': 2, 'Alice': 2})
"""
if node:
for factor in self.factors:
for variable, cardinality in zip(factor.scope(), factor.cardinality):
if node == variable:
return cardinality
else:
cardinalities = defaultdict(int)
for factor in self.factors:
for variable, cardinality in zip(factor.scope(), factor.cardinality):
cardinalities[variable] = cardinality
return cardinalities
@property
def states(self):
"""
Returns a dictionary mapping each node to its list of possible states.
Returns
-------
state_dict: dict
Dictionary of nodes to possible states
"""
state_names_list = [phi.state_names for phi in self.factors]
state_dict = {
node: states for d in state_names_list for node, states in d.items()
}
return state_dict
[docs]
def check_model(self):
"""
Check the model for various errors. This method checks for the following
errors -
* Checks if the cardinalities of all the variables are consistent across all the factors.
* Factors are defined for all the random variables.
Returns
-------
check: boolean
True if all the checks are passed
"""
cardinalities = self.get_cardinality()
for factor in self.factors:
for variable, cardinality in zip(factor.scope(), factor.cardinality):
if cardinalities[variable] != cardinality:
raise ValueError(
f"Cardinality of variable {variable} not matching among factors"
)
if len(self.nodes()) != len(cardinalities):
raise ValueError("Factors for all the variables not defined")
for var1, var2 in itertools.combinations(factor.variables, 2):
if var2 not in self.neighbors(var1):
raise ValueError("DiscreteFactor inconsistent with the model.")
return True
[docs]
def to_factor_graph(self):
"""
Converts the Markov Model into Factor Graph.
A Factor Graph contains two types of nodes. One type corresponds to
random variables whereas the second type corresponds to factors over
these variables. The graph only contains edges between variables and
factor nodes. Each factor node is associated with one factor whose
scope is the set of variables that are its neighbors.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> student = MarkovNetwork([('Alice', 'Bob'), ('Bob', 'Charles')])
>>> factor1 = DiscreteFactor(['Alice', 'Bob'], [3, 2], np.random.rand(6))
>>> factor2 = DiscreteFactor(['Bob', 'Charles'], [2, 2], np.random.rand(4))
>>> student.add_factors(factor1, factor2)
>>> factor_graph = student.to_factor_graph()
"""
from pgmpy.models import FactorGraph
factor_graph = FactorGraph()
if not self.factors:
raise ValueError("Factors not associated with the random variables.")
factor_graph.add_nodes_from(self.nodes())
for factor in self.factors:
scope = factor.scope()
factor_node = "phi_" + "_".join(scope)
factor_graph.add_edges_from(itertools.product(scope, [factor_node]))
factor_graph.add_factors(factor)
return factor_graph
[docs]
def triangulate(self, heuristic="H6", order=None, inplace=False):
"""
Triangulate the graph.
If order of deletion is given heuristic algorithm will not be used.
Parameters
----------
heuristic: H1 | H2 | H3 | H4 | H5 | H6
The heuristic algorithm to use to decide the deletion order of
the variables to compute the triangulated graph.
Let X be the set of variables and X(i) denotes the i-th variable.
* S(i) - The size of the clique created by deleting the variable.
* E(i) - Cardinality of variable X(i).
* M(i) - Maximum size of cliques given by X(i) and its adjacent nodes.
* C(i) - Sum of size of cliques given by X(i) and its adjacent nodes.
The heuristic algorithm decide the deletion order if this way:
* H1 - Delete the variable with minimal S(i).
* H2 - Delete the variable with minimal S(i)/E(i).
* H3 - Delete the variable with minimal S(i) - M(i).
* H4 - Delete the variable with minimal S(i) - C(i).
* H5 - Delete the variable with minimal S(i)/M(i).
* H6 - Delete the variable with minimal S(i)/C(i).
order: list, tuple (array-like)
The order of deletion of the variables to compute the triagulated
graph. If order is given heuristic algorithm will not be used.
inplace: True | False
if inplace is true then adds the edges to the object from
which it is called else returns a new object.
References
----------
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.3607
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> G = MarkovNetwork()
>>> G.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> G.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
... ('x4', 'x7'), ('x5', 'x7')])
>>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in G.edges()]
>>> G.add_factors(*phi)
>>> G_chordal = G.triangulate()
"""
self.check_model()
if self.is_triangulated():
if inplace:
return
else:
return self
graph_copy = nx.Graph(self.edges())
edge_set = set()
def _find_common_cliques(cliques_list):
"""
Finds the common cliques among the given set of cliques for
corresponding node.
"""
common = set([tuple(x) for x in cliques_list[0]])
for i in range(1, len(cliques_list)):
common = common & set([tuple(x) for x in cliques_list[i]])
return list(common)
def _find_size_of_clique(clique, cardinalities):
"""
Computes the size of a clique.
Size of a clique is defined as product of cardinalities of all the
nodes present in the clique.
"""
return list(
map(lambda x: np.prod([cardinalities[node] for node in x]), clique)
)
def _get_cliques_dict(node):
"""
Returns a dictionary in the form of {node: cliques_formed} of the
node along with its neighboring nodes.
clique_dict_removed would be containing the cliques created
after deletion of the node
clique_dict_node would be containing the cliques created before
deletion of the node
"""
graph_working_copy = nx.Graph(graph_copy.edges())
neighbors = list(graph_working_copy.neighbors(node))
graph_working_copy.add_edges_from(itertools.combinations(neighbors, 2))
clique_dict = {var: [] for var in [node] + neighbors}
max_cliques = list(nx.find_cliques(graph_working_copy))
for var in [node] + neighbors:
for clique in max_cliques:
if var in clique:
clique_dict[var].append(clique)
graph_working_copy.remove_node(node)
clique_dict_removed = {var: [] for var in neighbors}
max_cliques = list(nx.find_cliques(graph_working_copy))
for var in neighbors:
for clique in max_cliques:
if var in clique:
clique_dict_removed[var].append(clique)
return clique_dict, clique_dict_removed
if not order:
order = []
cardinalities = self.get_cardinality()
for index in range(self.number_of_nodes()):
# S represents the size of clique created by deleting the
# node from the graph
S = {}
# M represents the size of maximum size of cliques given by
# the node and its adjacent node
M = {}
# C represents the sum of size of the cliques created by the
# node and its adjacent node
C = {}
for node in set(graph_copy.nodes()) - set(order):
clique_dict, clique_dict_removed = _get_cliques_dict(node)
S[node] = _find_size_of_clique(
_find_common_cliques(list(clique_dict_removed.values())),
cardinalities,
)[0]
common_clique_size = _find_size_of_clique(
_find_common_cliques(list(clique_dict.values())), cardinalities
)
M[node] = np.max(common_clique_size)
C[node] = np.sum(common_clique_size)
if heuristic == "H1":
node_to_delete = min(S, key=S.get)
elif heuristic == "H2":
S_by_E = {key: S[key] / cardinalities[key] for key in S}
node_to_delete = min(S_by_E, key=S_by_E.get)
elif heuristic == "H3":
S_minus_M = {key: S[key] - M[key] for key in S}
node_to_delete = min(S_minus_M, key=S_minus_M.get)
elif heuristic == "H4":
S_minus_C = {key: S[key] - C[key] for key in S}
node_to_delete = min(S_minus_C, key=S_minus_C.get)
elif heuristic == "H5":
S_by_M = {key: S[key] / M[key] for key in S}
node_to_delete = min(S_by_M, key=S_by_M.get)
else:
S_by_C = {key: S[key] / C[key] for key in S}
node_to_delete = min(S_by_C, key=S_by_C.get)
order.append(node_to_delete)
graph_copy = nx.Graph(self.edges())
for node in order:
for edge in itertools.combinations(graph_copy.neighbors(node), 2):
graph_copy.add_edge(edge[0], edge[1])
edge_set.add(edge)
graph_copy.remove_node(node)
if inplace:
for edge in edge_set:
self.add_edge(edge[0], edge[1])
return self
else:
graph_copy = MarkovNetwork(self.edges())
for edge in edge_set:
graph_copy.add_edge(edge[0], edge[1])
return graph_copy
[docs]
def to_junction_tree(self):
"""
Creates a junction tree (or clique tree) for a given markov model.
For a given markov model (H) a junction tree (G) is a graph
1. where each node in G corresponds to a maximal clique in H
2. each sepset in G separates the variables strictly on one side of the
edge to other.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> mm = MarkovNetwork()
>>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
... ('x4', 'x7'), ('x5', 'x7')])
>>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in mm.edges()]
>>> mm.add_factors(*phi)
>>> junction_tree = mm.to_junction_tree()
"""
from pgmpy.models import JunctionTree
# Check whether the model is valid or not
self.check_model()
# Triangulate the graph to make it chordal
triangulated_graph = self.triangulate()
# Find maximal cliques in the chordal graph
cliques = list(map(tuple, nx.find_cliques(triangulated_graph)))
# If there is only 1 clique, then the junction tree formed is just a
# clique tree with that single clique as the node
if len(cliques) == 1:
clique_trees = JunctionTree()
clique_trees.add_node(cliques[0])
# Else if the number of cliques is more than 1 then create a complete
# graph with all the cliques as nodes and weight of the edges being
# the length of sepset between two cliques
elif len(cliques) >= 2:
complete_graph = UndirectedGraph()
edges = list(itertools.combinations(cliques, 2))
weights = list(map(lambda x: len(set(x[0]).intersection(set(x[1]))), edges))
for edge, weight in zip(edges, weights):
complete_graph.add_edge(*edge, weight=-weight)
# Create clique trees by minimum (or maximum) spanning tree method
clique_trees = JunctionTree(
nx.minimum_spanning_tree(complete_graph).edges()
)
# Check whether the factors are defined for all the random variables or not
all_vars = itertools.chain(*[factor.scope() for factor in self.factors])
if set(all_vars) != set(self.nodes()):
ValueError("DiscreteFactor for all the random variables not specified")
# Dictionary stating whether the factor is used to create clique
# potential or not
# If false, then it is not used to create any clique potential
is_used = {factor: False for factor in self.factors}
for node in clique_trees.nodes():
clique_factors = []
for factor in self.factors:
# If the factor is not used in creating any clique potential as
# well as has any variable of the given clique in its scope,
# then use it in creating clique potential
if not is_used[factor] and set(factor.scope()).issubset(node):
clique_factors.append(factor)
is_used[factor] = True
# To compute clique potential, initially set it as unity factor
var_card = [self.get_cardinality()[x] for x in node]
clique_potential = DiscreteFactor(
node, var_card, np.ones(np.prod(var_card))
)
# multiply it with the factors associated with the variables present
# in the clique (or node)
# Checking if there's clique_factors, to handle the case when clique_factors
# is empty, otherwise factor_product with throw an error [ref #889]
if clique_factors:
clique_potential *= factor_product(*clique_factors)
clique_trees.add_factors(clique_potential)
if not all(is_used.values()):
raise ValueError(
"All the factors were not used to create Junction Tree."
"Extra factors are defined."
)
return clique_trees
[docs]
def markov_blanket(self, node):
"""
Returns a markov blanket for a random variable.
Markov blanket is the neighboring nodes of the given node.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> mm = MarkovNetwork()
>>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
... ('x4', 'x7'), ('x5', 'x7')])
>>> mm.markov_blanket('x1')
"""
return self.neighbors(node)
[docs]
def get_local_independencies(self, latex=False):
"""
Returns all the local independencies present in the markov model.
Local independencies are the independence assertion in the form of
.. math:: {X \perp W - {X} - MB(X) | MB(X)}
where MB is the markov blanket of all the random variables in X
Parameters
----------
latex: boolean
If latex=True then latex string of the indepedence assertion would
be created
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> mm = MarkovNetwork()
>>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
... ('x4', 'x7'), ('x5', 'x7')])
>>> mm.get_local_independencies()
"""
local_independencies = Independencies()
all_vars = set(self.nodes())
for node in self.nodes():
markov_blanket = set(self.markov_blanket(node))
rest = all_vars - set([node]) - markov_blanket
try:
local_independencies.add_assertions(
[node, list(rest), list(markov_blanket)]
)
except ValueError:
pass
local_independencies.reduce()
if latex:
return local_independencies.latex_string()
else:
return local_independencies
[docs]
def to_bayesian_model(self):
"""
Creates a Bayesian Model which is a minimum I-Map for this Markov Model.
The ordering of parents may not remain constant. It would depend on the
ordering of variable in the junction tree (which is not constant) all the
time. Also, if the model is not connected, the connected components are
treated as separate models, converted, and then joined together.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> mm = MarkovNetwork()
>>> mm.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> mm.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
... ('x4', 'x7'), ('x5', 'x7')])
>>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in mm.edges()]
>>> mm.add_factors(*phi)
>>> bm = mm.to_bayesian_model()
"""
from pgmpy.models import BayesianNetwork
# If the graph is not connected, treat them as separate models and join them together in the end.
bms = []
for node_set in connected_components(self):
bm = BayesianNetwork()
var_clique_dict = defaultdict(tuple)
var_order = []
subgraph = self.subgraph(node_set)
# Create a Junction Tree from the Markov Model.
# Creation of Clique Tree involves triangulation, finding maximal cliques
# and creating a tree from these cliques
junction_tree = MarkovNetwork(subgraph.edges()).to_junction_tree()
# create an ordering of the nodes based on the ordering of the clique
# in which it appeared first
root_node = next(iter(junction_tree.nodes()))
bfs_edges = nx.bfs_edges(junction_tree, root_node)
for node in root_node:
var_clique_dict[node] = root_node
var_order.append(node)
for edge in bfs_edges:
clique_node = edge[1]
for node in clique_node:
if not var_clique_dict[node]:
var_clique_dict[node] = clique_node
var_order.append(node)
# create a Bayesian Network by adding edges from parent of node to node as
# par(x_i) = (var(c_k) - x_i) \cap {x_1, ..., x_{i-1}}
for node_index in range(len(var_order)):
node = var_order[node_index]
node_parents = (set(var_clique_dict[node]) - set([node])).intersection(
set(var_order[:node_index])
)
bm.add_edges_from([(parent, node) for parent in node_parents])
# TODO : Convert factor into CPDs
bms.append(bm)
# Join the bms in a single model.
final_bm = BayesianNetwork()
for bm in bms:
final_bm.add_edges_from(bm.edges())
final_bm.add_nodes_from(bm.nodes())
return final_bm
[docs]
def get_partition_function(self):
"""
Returns the partition function for a given undirected graph.
A partition function is defined as
.. math:: \sum_{X}(\prod_{i=1}^{m} \phi_i)
where m is the number of factors present in the graph
and X are all the random variables present.
Examples
--------
>>> from pgmpy.models import MarkovNetwork
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> G = MarkovNetwork()
>>> G.add_nodes_from(['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7'])
>>> G.add_edges_from([('x1', 'x3'), ('x1', 'x4'), ('x2', 'x4'),
... ('x2', 'x5'), ('x3', 'x6'), ('x4', 'x6'),
... ('x4', 'x7'), ('x5', 'x7')])
>>> phi = [DiscreteFactor(edge, [2, 2], np.random.rand(4)) for edge in G.edges()]
>>> G.add_factors(*phi)
>>> G.get_partition_function()
"""
self.check_model()
factor = self.factors[0]
factor = factor_product(
factor, *[self.factors[i] for i in range(1, len(self.factors))]
)
if set(factor.scope()) != set(self.nodes()):
raise ValueError("DiscreteFactor for all the random variables not defined.")
return np.sum(factor.values)
[docs]
def copy(self):
"""
Returns a copy of this Markov Model.
Returns
-------
MarkovNetwork: Copy of this Markov model.
Examples
--------
>>> from pgmpy.factors.discrete import DiscreteFactor
>>> from pgmpy.models import MarkovNetwork
>>> G = MarkovNetwork()
>>> G.add_nodes_from([('a', 'b'), ('b', 'c')])
>>> G.add_edge(('a', 'b'), ('b', 'c'))
>>> G_copy = G.copy()
>>> G_copy.edges()
EdgeView([(('a', 'b'), ('b', 'c'))])
>>> G_copy.nodes()
[('a', 'b'), ('b', 'c')]
>>> factor = DiscreteFactor([('a', 'b')], cardinality=[3],
... values=np.random.rand(3))
>>> G.add_factors(factor)
>>> G.get_factors()
[<DiscreteFactor representing phi(('a', 'b'):3) at 0x...>]
>>> G_copy.get_factors()
[]
"""
clone_graph = MarkovNetwork(self.edges())
clone_graph.add_nodes_from(self.nodes())
if self.factors:
factors_copy = [factor.copy() for factor in self.factors]
clone_graph.add_factors(*factors_copy)
return clone_graph
