def naive_bayes(data, target, estimator='mle'):
"""
Learn naive bayes model from data.
The Naive Bayes model is a Tree-based
model where all random variables have
the same parent (the "target" variable).
From a probabilistic standpoint, the implication
of this model is that all random variables
(i.e. features) are assumed to be
conditionally independent of any other random variable,
conditioned upon the single parent (target) variable.
It turns out that this model performs quite well
as a classifier, and can be used as such. Moreover,
this model is quite fast and simple to learn/create
from a computational standpoint.
Note that this function not only learns the structure,
but ALSO learns the parameters.
Arguments
---------
*data* : a nested numpy array
*target* : an integer
The target variable column in *data*
Returns
-------
*bn* : a BayesNet object,
with the structure instantiated.
Effects
-------
None
Notes
-----
"""
value_dict = dict(
zip(range(data.shape[1]), [list(np.unique(col)) for col in data.T]))
edge_dict = {target: [v for v in value_dict if v != target]}
edge_dict.update(dict([(rv, []) for rv in value_dict if rv != target]))
bn = BayesNet(edge_dict, value_dict)
if estimator == 'bayes':
bayes_estimator(bn, data)
else:
mle_estimator(bn, data)
return bn
def K2(bn, data, ed=None):
"""
K2 is bayesian posterior probability of structure given the data,
where N'ijk = 1.
"""
counts_dict = mle_estimator(bn, data)
a_ijk = []
k2 = 1
for rv, value in counts_dict.items():
nijk = value['cpt']
nijk_prime = 1
k2 *= gamma(nijk + nijk_prime) / gamma(nijk_prime)
nij_prime = nijk_prime * (len(nijk) / bn.card(rv))
nij = np.mean(nijk.reshape(-1, bn.card(rv)),
axis=1) # sum along parents
k2 *= gamma(nij_prime) / gamma(nij + nij_prime)
return k2
def hc_rr(data,
M=5,
R=8,
metric='AIC',
max_iter=300,
debug=False,
init_nodes=None,
restriction=None):
"""
Arguments
---------
*data* : a nested numpy array
The data from which the Bayesian network
structure will be learned.
*metric* : a string
Which score metric to use.
Options:
- AIC
- BIC / MDL
- LL (log-likelihood)
*max_iter* : an integer
The maximum number of iterations of the
hill-climbing algorithm to run. Note that
the algorithm will terminate on its own if no
improvement is made in a given iteration.
*debug* : boolean
Whether to print(the scores/moves of the)
algorithm as its happening.
*init_nodes* : a list of initialize nodes (number of nodes according to the dataset)
*restriction* : a list of 2-tuples
For MMHC algorithm, the list of allowable edge additions.
Returns
-------
*bn* : a BayesNet object
"""
nrow = data.shape[0]
ncol = data.shape[1]
names = range(ncol)
# INITIALIZE NETWORK W/ NO EDGES
# maintain children and parents dict for fast lookups
c_dict = dict([(n, []) for n in names])
p_dict = dict([(n, []) for n in names])
# COMPUTE INITIAL LIKELIHOOD SCORE
value_dict = dict([(n, np.unique(data[:, i]))
for i, n in enumerate(names)])
bn = BayesNet(c_dict)
mle_estimator(bn, data)
max_score = info_score(bn, data, metric)
_iter = 0
improvement = True
_restarts = 0
while improvement:
improvement = False
max_delta = 0
if debug:
print('ITERATION: ', _iter)
### TEST ARC ADDITIONS ###
for u in bn.nodes():
for v in bn.nodes():
if v not in c_dict[u] and u != v and not would_cause_cycle(
c_dict, u, v):
# FOR MMHC ALGORITHM -> Edge Restrictions
if (init_nodes is None or not (v in init_nodes)):
if restriction is None or (u, v) in restriction:
# SCORE FOR 'V' -> gaining a parent
old_cols = (v, ) + tuple(
p_dict[v]) # without 'u' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = old_cols + (u, ) # with'u' as parent
mi_new = mutual_information(data[:, new_cols])
delta_score = nrow * (mi_old - mi_new)
if delta_score > max_delta:
if debug:
print('Improved Arc Addition: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Addition'
max_arc = (u, v)
### TEST ARC DELETIONS ###
for u in bn.nodes():
for v in bn.nodes():
if v in c_dict[u]:
# SCORE FOR 'V' -> losing a parent
old_cols = (v, ) + tuple(p_dict[v]) # with 'u' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = tuple([i for i in old_cols
if i != u]) # without 'u' as parent
mi_new = mutual_information(data[:, new_cols])
delta_score = nrow * (mi_old - mi_new)
if delta_score > max_delta:
if debug:
print('Improved Arc Deletion: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Deletion'
max_arc = (u, v)
### TEST ARC REVERSALS ###
for u in bn.nodes():
for v in bn.nodes():
if v in c_dict[u] and not would_cause_cycle(
c_dict, v, u, reverse=True) and (
init_nodes is None or
not (u in init_nodes)) and (restriction is None or
(v, u) in restriction):
# SCORE FOR 'U' -> gaining 'v' as parent
old_cols = (u, ) + tuple(
p_dict[v]) # without 'v' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = old_cols + (v, ) # with 'v' as parent
mi_new = mutual_information(data[:, new_cols])
delta1 = nrow * (mi_old - mi_new)
# SCORE FOR 'V' -> losing 'u' as parent
old_cols = (v, ) + tuple(p_dict[v]) # with 'u' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = tuple([u for i in old_cols
if i != u]) # without 'u' as parent
mi_new = mutual_information(data[:, new_cols])
delta2 = nrow * (mi_old - mi_new)
# COMBINED DELTA-SCORES
delta_score = delta1 + delta2
if delta_score > max_delta:
if debug:
print('Improved Arc Reversal: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Reversal'
max_arc = (u, v)
### DETERMINE IF/WHERE IMPROVEMENT WAS MADE ###
if max_delta != 0:
improvement = True
u, v = max_arc
if max_operation == 'Addition':
if debug:
print('ADDING: ', max_arc, '\n')
c_dict[u].append(v)
p_dict[v].append(u)
elif max_operation == 'Deletion':
if debug:
print('DELETING: ', max_arc, '\n')
c_dict[u].remove(v)
p_dict[v].remove(u)
elif max_operation == 'Reversal':
if debug:
print('REVERSING: ', max_arc, '\n')
c_dict[u].remove(v)
p_dict[v].remove(u)
c_dict[v].append(u)
p_dict[u].append(v)
else:
if debug:
print('No Improvement on Iter: ', _iter)
#### RESTART WITH RANDOM MOVES ####
if _restarts < R:
improvement = True # make another pass of hill climbing
_iter = 0 # reset iterations
if debug:
print('Restart - ', _restarts)
_restarts += 1
for _ in range(M):
# 0 = Addition, 1 = Deletion, 2 = Reversal
operation = np.random.choice([0, 1, 2])
if operation == 0:
u, v = np.random.choice(list(bn.nodes()),
size=2,
replace=False)
# IF EDGE DOESN'T EXIST, ADD IT
if u not in p_dict[
v] and u != v and not would_cause_cycle(
c_dict, u,
v) and (init_nodes is None
or not (v in init_nodes)) and (
restriction is None or
(u, v) in restriction):
if debug:
print('RESTART - ADDING: ', (u, v))
c_dict[u].append(v)
p_dict[v].append(u)
elif operation == 1:
u, v = np.random.choice(list(bn.nodes()),
size=2,
replace=False)
# IF EDGE EXISTS, DELETE IT
if u in p_dict[v]:
if debug:
print('RESTART - DELETING: ', (u, v))
c_dict[u].remove(v)
p_dict[v].remove(u)
elif operation == 2:
u, v = np.random.choice(list(bn.nodes()),
size=2,
replace=False)
# IF EDGE EXISTS, REVERSE IT
if u in p_dict[v] and not would_cause_cycle(
c_dict, v, u, reverse=True) and (
init_nodes is None or not (u in init_nodes)
) and (restriction is None or
(v, u) in restriction):
if debug:
print('RESTART - REVERSING: ', (u, v))
c_dict[u].remove(v)
p_dict[v].remove(u)
c_dict[v].append(u)
p_dict[u].append(v)
break
### TEST FOR MAX ITERATION ###
_iter += 1
if _iter > max_iter:
if debug:
print('Max Iteration Reached')
break
bn = BayesNet(c_dict)
return bn
def tabu(data, k=5, metric='AIC', max_iter=100, debug=False, restriction=None):
"""
Tabu search for score-based structure learning.
The algorithm maintains a list called "tabu_list",
which consists of 3-tuples, where the first two
elements constitute the edge which is tabued, and
the third element is a string - either 'Addition',
'Deletion', or 'Reversal' denoting the operation
associated with the edge.
Arguments
---------
*data* : a nested numpy array
The data from which the Bayesian network
structure will be learned.
*metric* : a string
Which score metric to use.
Options:
- AIC
- BIC / MDL
- LL (log-likelihood)
*max_iter* : an integer
The maximum number of iterations of the
hill-climbing algorithm to run. Note that
the algorithm will terminate on its own if no
improvement is made in a given iteration.
*debug* : boolean
Whether to print(the scores/moves of the)
algorithm as its happening.
*restriction* : a list of 2-tuples
For MMHC algorithm, the list of allowable edge additions.
Returns
-------
*bn* : a BayesNet object
"""
nrow = data.shape[0]
ncol = data.shape[1]
names = range(ncol)
# INITIALIZE NETWORK W/ NO EDGES
# maintain children and parents dict for fast lookups
c_dict = dict([(n,[]) for n in names])
p_dict = dict([(n,[]) for n in names])
# COMPUTE INITIAL LIKELIHOOD SCORE
value_dict = dict([(n, np.unique(data[:,i])) for i,n in enumerate(names)])
bn = BayesNet(c_dict)
mle_estimator(bn, data)
max_score = info_score(bn, nrow, metric)
tabu_list = [None]*k
_iter = 0
improvement = True
while improvement:
improvement = False
max_delta = 0
if debug:
print('ITERATION: ' , _iter)
### TEST ARC ADDITIONS ###
for u in bn.nodes():
for v in bn.nodes():
# CHECK TABU LIST - can't delete an addition on the tabu list
if (u,v,'Deletion') not in tabu_list:
# CHECK EDGE EXISTENCE AND CYCLICITY
if v not in c_dict[u] and u!=v and not would_cause_cycle(c_dict, u, v):
# FOR MMHC ALGORITHM -> Edge Restrictions
if restriction is None or (u,v) in restriction:
# SCORE FOR 'V' -> gaining a parent
old_cols = (v,) + tuple(p_dict[v]) # without 'u' as parent
mi_old = mutual_information(data[:,old_cols])
new_cols = old_cols + (u,) # with'u' as parent
mi_new = mutual_information(data[:,new_cols])
delta_score = nrow * (mi_old - mi_new)
if delta_score > max_delta:
if debug:
print('Improved Arc Addition: ' , (u,v))
print('Delta Score: ' , delta_score)
max_delta = delta_score
max_operation = 'Addition'
max_arc = (u,v)
### TEST ARC DELETIONS ###
for u in bn.nodes():
for v in bn.nodes():
# CHECK TABU LIST - can't add back a deletion on the tabu list
if (u,v,'Addition') not in tabu_list:
if v in c_dict[u]:
# SCORE FOR 'V' -> losing a parent
old_cols = (v,) + tuple(p_dict[v]) # with 'u' as parent
mi_old = mutual_information(data[:,old_cols])
new_cols = tuple([i for i in old_cols if i != u]) # without 'u' as parent
mi_new = mutual_information(data[:,new_cols])
delta_score = nrow * (mi_old - mi_new)
if delta_score > max_delta:
if debug:
print('Improved Arc Deletion: ' , (u,v))
print('Delta Score: ' , delta_score)
max_delta = delta_score
max_operation = 'Deletion'
max_arc = (u,v)
### TEST ARC REVERSALS ###
for u in bn.nodes():
for v in bn.nodes():
# CHECK TABU LIST - can't reverse back a reversal on the tabu list
if (u,v,'Reversal') not in tabu_list:
if v in c_dict[u] and not would_cause_cycle(c_dict,v,u, reverse=True):
# SCORE FOR 'U' -> gaining 'v' as parent
old_cols = (u,) + tuple(p_dict[v]) # without 'v' as parent
mi_old = mutual_information(data[:,old_cols])
new_cols = old_cols + (v,) # with 'v' as parent
mi_new = mutual_information(data[:,new_cols])
delta1 = nrow * (mi_old - mi_new)
# SCORE FOR 'V' -> losing 'u' as parent
old_cols = (v,) + tuple(p_dict[v]) # with 'u' as parent
mi_old = mutual_information(data[:,old_cols])
new_cols = tuple([u for i in old_cols if i != u]) # without 'u' as parent
mi_new = mutual_information(data[:,new_cols])
delta2 = nrow * (mi_old - mi_new)
# COMBINED DELTA-SCORES
delta_score = delta1 + delta2
if delta_score > max_delta:
if debug:
print('Improved Arc Reversal: ' , (u,v))
print('Delta Score: ' , delta_score)
max_delta = delta_score
max_operation = 'Reversal'
max_arc = (u,v)
### DETERMINE IF/WHERE IMPROVEMENT WAS MADE ###
if max_delta != 0:
improvement = True
u,v = max_arc
if max_operation == 'Addition':
if debug:
print('ADDING: ' , max_arc , '\n')
c_dict[u].append(v)
p_dict[v].append(u)
tabu_list[_iter % 5] = (u,v,'Addition')
elif max_operation == 'Deletion':
if debug:
print('DELETING: ' , max_arc , '\n')
c_dict[u].remove(v)
p_dict[v].remove(u)
tabu_list[_iter % 5] = (u,v,'Deletion')
elif max_operation == 'Reversal':
if debug:
print('REVERSING: ' , max_arc, '\n')
c_dict[u].remove(v)
p_dict[v].remove(u)
c_dict[v].append(u)
p_dict[u].append(v)
tabu_list[_iter % 5] = (u,v,'Reversal')
else:
if debug:
print('No Improvement on Iter: ' , _iter)
### TEST FOR MAX ITERATION ###
_iter += 1
if _iter > max_iter:
if debug:
print('Max Iteration Reached')
break
bn = BayesNet(c_dict)
return bn
def hc(data,
metric='LL',
max_iter=100,
debug=False,
init_nodes=None,
restriction=None,
init_edges=None,
remove_geo_edges=True,
black_list=None):
"""
Greedy Hill Climbing search proceeds by choosing the move
which maximizes the increase in fitness of the
network at the current step. It continues until
it reaches a point where there does not exist any
feasible single move that increases the network fitness.
It is called "greedy" because it simply does what is
best at the current iteration only, and thus does not
look ahead to what may be better later on in the search.
For computational saving, a Priority Queue (python's heapq)
can be used to maintain the best operators and reduce the
complexity of picking the best operator from O(n^2) to O(nlogn).
This works by maintaining the heapq of operators sorted by their
delta score, and each time a move is made, we only have to recompute
the O(n) delta-scores which were affected by the move. The rest of
the operator delta-scores are not affected.
For additional computational efficiency, we can cache the
sufficient statistics for various families of distributions -
therefore, computing the mutual information for a given family
only needs to happen once.
The possible moves are the following:
- add edge
- delete edge
- invert edge
Arguments
---------
*data* : a nested numpy array
The data from which the Bayesian network
structure will be learned.
*metric* : a string
Which score metric to use.
Options:
- AIC
- BIC / MDL
- LL (log-likelihood)
*max_iter* : an integer
The maximum number of iterations of the
hill-climbing algorithm to run. Note that
the algorithm will terminate on its own if no
improvement is made in a given iteration.
*debug* : boolean
Whether to print(the scores/moves of the)
algorithm as its happening.
*init_nodes* : a list of initialize nodes (number of nodes according to the dataset)
*restriction* : a list of 2-tuples
For MMHC algorithm, the list of allowable edge additions.
Returns
-------
*bn* : a BayesNet object
"""
nrow = data.shape[0]
ncol = data.shape[1]
names = range(ncol)
# INITIALIZE NETWORK W/ NO EDGES
# maintain children and parents dict for fast lookups
c_dict = dict([(n, []) for n in names])
p_dict = dict([(n, []) for n in names])
if init_edges:
for edge in init_edges:
c_dict[edge[0]].append(edge[1])
p_dict[edge[1]].append(edge[0])
score_list = []
# COMPUTE INITIAL LIKELIHOOD SCORE
value_dict = dict([(n, np.unique(data[:, i]))
for i, n in enumerate(names)])
bn = BayesNet(c_dict)
mle_estimator(bn, data)
max_score = info_score(bn, data, metric)
# CREATE EMPIRICAL DISTRIBUTION OBJECT FOR CACHING
#ED = EmpiricalDistribution(data,names)
_iter = 0
improvement = True
while improvement:
improvement = False
max_delta = 0
if debug:
print('ITERATION: ', _iter)
### TEST ARC ADDITIONS ###
for u in bn.nodes():
for v in bn.nodes():
if v not in c_dict[u] and u != v and not would_cause_cycle(
c_dict, u, v) and len(p_dict[v]) != 3:
# FOR MMHC ALGORITHM -> Edge Restrictions
if (init_nodes is None or not (v in init_nodes)) and (
restriction is None or
(u, v) in restriction) and (black_list is None or not (
(u, v) in black_list)):
# SCORE FOR 'V' -> gaining a parent
old_cols = (v, ) + tuple(
p_dict[v]) # without 'u' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = old_cols + (u, ) # with'u' as parent
mi_new = mutual_information(data[:, new_cols])
delta_score = nrow * (mi_old - mi_new)
if delta_score > max_delta:
if debug:
print('Improved Arc Addition: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Addition'
max_arc = (u, v)
# ### TEST ARC DELETIONS ###
for u in bn.nodes():
for v in bn.nodes():
if v in c_dict[u]:
# SCORE FOR 'V' -> losing a parent
old_cols = (v, ) + tuple(p_dict[v]) # with 'u' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = tuple([i for i in old_cols
if i != u]) # without 'u' as parent
mi_new = mutual_information(data[:, new_cols])
delta_score = nrow * (mi_old - mi_new)
if (delta_score > max_delta):
if init_edges == None:
if debug:
print('Improved Arc Deletion: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Deletion'
max_arc = (u, v)
else:
if (u, v) in init_edges:
if remove_geo_edges:
if debug:
print('Improved Arc Deletion: ',
(u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Deletion'
max_arc = (u, v)
else:
if debug:
print('Improved Arc Deletion: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Deletion'
max_arc = (u, v)
# ### TEST ARC REVERSALS ###
for u in bn.nodes():
for v in bn.nodes():
if v in c_dict[u] and not would_cause_cycle(
c_dict, v, u,
reverse=True) and len(p_dict[u]) != 3 and (
init_nodes is None or not (u in init_nodes)) and (
restriction is None or
(v, u) in restriction) and (
black_list is None
or not ((v, u) in black_list)):
# SCORE FOR 'U' -> gaining 'v' as parent
old_cols = (u, ) + tuple(
p_dict[v]) # without 'v' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = old_cols + (v, ) # with 'v' as parent
mi_new = mutual_information(data[:, new_cols])
delta1 = -1 * nrow * (mi_old - mi_new)
# SCORE FOR 'V' -> losing 'u' as parent
old_cols = (v, ) + tuple(p_dict[v]) # with 'u' as parent
mi_old = mutual_information(data[:, old_cols])
new_cols = tuple([u for i in old_cols
if i != u]) # without 'u' as parent
mi_new = mutual_information(data[:, new_cols])
delta2 = nrow * (mi_old - mi_new)
# COMBINED DELTA-SCORES
delta_score = delta1 + delta2
if (delta_score > max_delta):
if init_edges == None:
if debug:
print('Improved Arc Reversal: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Reversal'
max_arc = (u, v)
else:
if (u, v) in init_edges:
if remove_geo_edges:
if debug:
print('Improved Arc Reversal: ',
(u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Reversal'
max_arc = (u, v)
else:
if debug:
print('Improved Arc Reversal: ', (u, v))
print('Delta Score: ', delta_score)
max_delta = delta_score
max_operation = 'Reversal'
max_arc = (u, v)
if max_delta != 0:
improvement = True
u, v = max_arc
if max_operation == 'Addition':
if debug:
print('ADDING: ', max_arc, '\n')
c_dict[u].append(v)
p_dict[v].append(u)
elif max_operation == 'Deletion':
if debug:
print('DELETING: ', max_arc, '\n')
c_dict[u].remove(v)
p_dict[v].remove(u)
elif max_operation == 'Reversal':
if debug:
print('REVERSING: ', max_arc, '\n')
c_dict[u].remove(v)
p_dict[v].remove(u)
c_dict[v].append(u)
p_dict[u].append(v)
else:
if debug:
print('No Improvement on Iter: ', _iter)
### TEST FOR MAX ITERATION ###
_iter += 1
if _iter > max_iter:
if debug:
print('Max Iteration Reached')
break
bn = BayesNet(c_dict)
return bn