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 pgm.models import MarkovModel
>>> from pgm.factors.discrete import DiscreteFactor
>>> G = MarkovModel()
>>> 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)
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 pgm.models import FactorGraph
>>> from pgm.factors.discrete import DiscreteFactor
>>> G = FactorGraph()
>>> G.add_nodes_from(['a', 'b', 'c'])
>>> phi1 = DiscreteFactor(['a', 'b'], [2, 2], np.random.rand(4))
>>> phi2 = DiscreteFactor(['b', 'c'], [2, 2], np.random.rand(4))
>>> G.add_factors(phi1, phi2)
>>> G.add_nodes_from([phi1, phi2])
>>> G.add_edges_from([('a', phi1), ('b', phi1),
... ('b', phi2), ('c', phi2)])
>>> G.get_factors()
>>> G.get_partition_function()
"""
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.get_variable_nodes()):
raise ValueError(
'DiscreteFactor for all the random variables not defined.')
return np.sum(factor.values)
def _get_kernel_from_bayesian_model(self, model):
"""
Computes the Gibbs transition models from a Bayesian Network.
'Probabilistic Graphical Model Principles and Techniques', Koller and
Friedman, Section 12.3.3 pp 512-513.
Parameters:
-----------
model: BayesianModel
The model from which probabilities will be computed.
"""
self.variables = np.array(model.nodes())
self.cardinalities = {
var: model.get_cpds(var).variable_card
for var in self.variables
}
for var in self.variables:
other_vars = [v for v in self.variables if var != v]
other_cards = [self.cardinalities[v] for v in other_vars]
cpds = [cpd for cpd in model.cpds if var in cpd.scope()]
prod_cpd = factor_product(*cpds)
kernel = {}
scope = set(prod_cpd.scope())
for tup in itertools.product(
*[range(card) for card in other_cards]):
states = [
State(v, s) for v, s in zip(other_vars, tup) if v in scope
]
prod_cpd_reduced = prod_cpd.reduce(states, inplace=False)
kernel[tup] = prod_cpd_reduced.values / sum(
prod_cpd_reduced.values)
self.transition_models[var] = kernel
def get_partition_function(self):
r"""
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 pgm.models import ClusterGraph
>>> from pgm.factors.discrete import DiscreteFactor
>>> G = ClusterGraph()
>>> G.add_nodes_from([('a', 'b', 'c'), ('a', 'b'), ('a', 'c')])
>>> G.add_edges_from([(('a', 'b', 'c'), ('a', 'b')),
... (('a', 'b', 'c'), ('a', 'c'))])
>>> phi1 = DiscreteFactor(['a', 'b', 'c'], [2, 2, 2], np.random.rand(8))
>>> phi2 = DiscreteFactor(['a', 'b'], [2, 2], np.random.rand(4))
>>> phi3 = DiscreteFactor(['a', 'c'], [2, 2], np.random.rand(4))
>>> G.add_factors(phi1, phi2, phi3)
>>> G.get_partition_function()
"""
if self.check_model():
factor = self.factors[0]
factor = factor_product(
factor,
*[self.factors[i] for i in range(1, len(self.factors))])
return np.sum(factor.values)
def map_query(self, variables=None, evidence=None, elimination_order=None):
"""
Computes the MAP Query over the variables given the evidence.
Parameters
----------
variables: list
list of variables over which we want to compute the max-marginal.
evidence: dict
a dict key, value pair as {var: state_of_var_observed}
None if no evidence
elimination_order: list
order of variable eliminations (if nothing is provided) order is
computed automatically
Examples
--------
>>> from pgm.inference import VariableElimination
>>> from pgm.models import BayesianModel
>>> import numpy as np
>>> import pandas as pd
>>> values = pd.DataFrame(np.random.randint(low=0, high=2, size=(1000, 5)),
... columns=['A', 'B', 'C', 'D', 'E'])
>>> model = BayesianModel([('A', 'B'), ('C', 'B'), ('C', 'D'), ('B', 'E')])
>>> model.fit(values)
>>> inference = VariableElimination(model)
>>> phi_query = inference.map_query(['A', 'B'])
"""
elimination_variables = set(self.variables) - set(
evidence.keys()) if evidence else set()
final_distribution = self._variable_elimination(
elimination_variables,
'maximize',
evidence=evidence,
elimination_order=elimination_order)
# To handle the case when no argument is passed then
# _variable_elimination returns a dict.
if isinstance(final_distribution, dict):
final_distribution = final_distribution.values()
distribution = factor_product(*final_distribution)
argmax = np.argmax(distribution.values)
assignment = distribution.assignment([argmax])[0]
map_query_results = {}
for var_assignment in assignment:
var, value = var_assignment
map_query_results[var] = value
if not variables:
return map_query_results
else:
return_dict = {}
for var in variables:
return_dict[var] = map_query_results[var]
return return_dict
def test_junction_tree_single_clique(self):
self.graph.add_edges_from([('x1', 'x2'), ('x2', 'x3'), ('x1', 'x3')])
phi = [
DiscreteFactor(edge, [2, 2], np.random.rand(4))
for edge in self.graph.edges()
]
self.graph.add_factors(*phi)
junction_tree = self.graph.to_junction_tree()
self.assertListEqual(hf.recursive_sorted(junction_tree.nodes()),
[['x1', 'x2', 'x3']])
factors = junction_tree.get_factors()
self.assertEqual(factors[0], factor_product(*phi))
def _get_factor(self, belief_prop, evidence):
"""
Extracts the required factor from the junction tree.
Parameters:
----------
belief_prop: Belief Propagation
Belief Propagation which needs to be updated.
evidence: dict
a dict key, value pair as {var: state_of_var_observed}
"""
final_factor = factor_product(*belief_prop.junction_tree.get_factors())
if evidence:
for var in evidence:
if var in final_factor.scope():
final_factor.reduce([(var, evidence[var])])
return final_factor
def max_marginal(self,
variables=None,
evidence=None,
elimination_order=None):
"""
Computes the max-marginal over the variables given the evidence.
Parameters
----------
variables: list
list of variables over which we want to compute the max-marginal.
evidence: dict
a dict key, value pair as {var: state_of_var_observed}
None if no evidence
elimination_order: list
order of variable eliminations (if nothing is provided) order is
computed automatically
Examples
--------
>>> import numpy as np
>>> import pandas as pd
>>> from pgm.models import BayesianModel
>>> from pgm.inference import VariableElimination
>>> values = pd.DataFrame(np.random.randint(low=0, high=2, size=(1000, 5)),
... columns=['A', 'B', 'C', 'D', 'E'])
>>> model = BayesianModel([('A', 'B'), ('C', 'B'), ('C', 'D'), ('B', 'E')])
>>> model.fit(values)
>>> inference = VariableElimination(model)
>>> phi_query = inference.max_marginal(['A', 'B'])
"""
if not variables:
variables = []
final_distribution = self._variable_elimination(
variables,
'maximize',
evidence=evidence,
elimination_order=elimination_order)
# To handle the case when no argument is passed then
# _variable_elimination returns a dict.
if isinstance(final_distribution, dict):
final_distribution = final_distribution.values()
return np.max(factor_product(*final_distribution).values)
def _get_kernel_from_markov_model(self, model):
"""
Computes the Gibbs transition models from a Markov Network.
'Probabilistic Graphical Model Principles and Techniques', Koller and
Friedman, Section 12.3.3 pp 512-513.
Parameters:
-----------
model: MarkovModel
The model from which probabilities will be computed.
"""
self.variables = np.array(model.nodes())
factors_dict = {var: [] for var in self.variables}
for factor in model.get_factors():
for var in factor.scope():
factors_dict[var].append(factor)
# Take factor product
factors_dict = {
var: factor_product(*factors) if len(factors) > 1 else factors[0]
for var, factors in factors_dict.items()
}
self.cardinalities = {
var: factors_dict[var].get_cardinality([var])[var]
for var in self.variables
}
for var in self.variables:
other_vars = [v for v in self.variables if var != v]
other_cards = [self.cardinalities[v] for v in other_vars]
kernel = {}
factor = factors_dict[var]
scope = set(factor.scope())
for tup in itertools.product(
*[range(card) for card in other_cards]):
states = [
State(var, s) for var, s in zip(other_vars, tup)
if var in scope
]
reduced_factor = factor.reduce(states, inplace=False)
kernel[tup] = reduced_factor.values / sum(
reduced_factor.values)
self.transition_models[var] = kernel
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 pgm.models import MarkovModel
>>> from pgm.factors.discrete import DiscreteFactor
>>> mm = MarkovModel()
>>> 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 pgm.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.product(var_card)))
# multiply it with the factors associated with the variables present
# in the clique (or node)
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
def _variable_elimination(self,
variables,
operation,
evidence=None,
elimination_order=None):
"""
Implementation of a generalized variable elimination.
Parameters
----------
variables: list, array-like
variables that are not to be eliminated.
operation: str ('marginalize' | 'maximize')
The operation to do for eliminating the variable.
evidence: dict
a dict key, value pair as {var: state_of_var_observed}
None if no evidence
elimination_order: list, array-like
list of variables representing the order in which they
are to be eliminated. If None order is computed automatically.
"""
if isinstance(variables, string_types):
raise TypeError("variables must be a list of strings")
if isinstance(evidence, string_types):
raise TypeError("evidence must be a list of strings")
# Dealing with the case when variables is not provided.
if not variables:
all_factors = []
for factor_li in self.factors.values():
all_factors.extend(factor_li)
return set(all_factors)
eliminated_variables = set()
working_factors = {
node: {factor
for factor in self.factors[node]}
for node in self.factors
}
# Dealing with evidence. Reducing factors over it before VE is run.
if evidence:
for evidence_var in evidence:
for factor in working_factors[evidence_var]:
factor_reduced = factor.reduce(
[(evidence_var, evidence[evidence_var])],
inplace=False)
for var in factor_reduced.scope():
working_factors[var].remove(factor)
working_factors[var].add(factor_reduced)
del working_factors[evidence_var]
# TODO: Modify it to find the optimal elimination order
if not elimination_order:
elimination_order = list(
set(self.variables) - set(variables) -
set(evidence.keys() if evidence else []))
elif any(var in elimination_order for var in set(variables).union(
set(evidence.keys() if evidence else []))):
raise ValueError(
"Elimination order contains variables which are in"
" variables or evidence args")
for var in elimination_order:
# Removing all the factors containing the variables which are
# eliminated (as all the factors should be considered only once)
factors = [
factor for factor in working_factors[var]
if not set(factor.variables).intersection(eliminated_variables)
]
phi = factor_product(*factors)
phi = getattr(phi, operation)([var], inplace=False)
del working_factors[var]
for variable in phi.variables:
working_factors[variable].add(phi)
eliminated_variables.add(var)
final_distribution = set()
for node in working_factors:
factors = working_factors[node]
for factor in factors:
if not set(
factor.variables).intersection(eliminated_variables):
final_distribution.add(factor)
query_var_factor = {}
for query_var in variables:
phi = factor_product(*final_distribution)
query_var_factor[query_var] = phi.marginalize(
list(set(variables) - set([query_var])),
inplace=False).normalize(inplace=False)
return query_var_factor