Learning the structure of a Bayesian network
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In [1]:
# %matplotlib inline
# from pylab import *
In [2]:
import math
import matplotlib.pyplot as plt
import pyagrum as gum
import pyagrum.lib.notebook as gnb
import pyagrum.explain as explain
import pyagrum.lib.bn_vs_bn as bnvsbn
gum.about()
gnb.configuration()
pyAgrum 3.0.0
(c) 2015-2025 Pierre-Henri Wuillemin, Christophe Gonzales
This is free software; see the source code for copying conditions.
There is ABSOLUTELY NO WARRANTY; not even for MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.
| Library | Version |
|---|---|
| OS | posix [darwin] |
| Python | 3.14.6 (main, Jun 10 2026, 10:03:53) [Clang 21.0.0 (clang-2100.0.123.102)] |
| IPython | 9.15.0 |
| Matplotlib | 3.11.0 |
| Numpy | 2.5.1 |
| pyDot | 4.0.1 |
| pyAgrum | 3.0.0 |
Thu Jul 16 18:15:09 2026 CEST
Generating the database from a BN
In [3]:
bn = gum.loadBN("res/asia.bgum")
bn
Out[3]:
In [4]:
gum.generateSample(bn, 50000, "out/sample_asia.csv", True);
out/sample_asia.csv: 100%|█████████████████████████████████|
Log2-Likelihood : -160779.95205477552
In [5]:
with open("out/sample_asia.csv", "r") as src:
for _ in range(10):
print(src.readline(), end="")
lung_cancer,bronchitis,dyspnoea,visit_to_Asia,smoking,positive_XraY,tuberculosis,tuberculos_or_cancer
1,0,1,1,0,1,1,1
1,0,0,1,0,1,1,1
1,0,0,1,1,1,1,1
0,0,0,1,0,0,1,0
1,1,1,1,1,1,1,1
1,0,0,1,0,1,1,1
1,0,0,0,0,1,1,1
1,1,1,1,0,1,1,1
1,0,0,1,1,1,1,1
In [6]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
print(learner)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : MIIC
Correction : MDL
Prior : -
In [7]:
print(f"Row of visit_to_Asia : {learner.idFromName('visit_to_Asia')}") # first row is 0
Row of visit_to_Asia : 0
In [8]:
print(f"Variable in row 4 : {learner.nameFromId(4)}")
Variable in row 4 : lung_cancer
The BNLearner is capable of recognizing missing values in databases. For this purpose, just indicate as a last argument the list of the strings that represent missing values.
In [9]:
# it is possible to add as a last argument a list of the symbols that represent missing values:
# whenever a cell of the database is equal to one of these strings, it is considered as a
# missing value
learner = gum.BNLearner("res/asia_missing.csv", bn, ["?", "N/A"])
print(f"Are there missing values in the database ? {learner.state()['Missing values'][0]}")
Are there missing values in the database ? True
type induction
When reading a csv file, BNLearner can try to find the correct type for discrete variable. Especially for numeric values.
In [10]:
%%writefile out/testTypeInduction.csv
A,B,C,D
1,2,0,hot
0,3,-2,cold
0,1,2,hot
1,2,2,warm
Overwriting out/testTypeInduction.csv
In [11]:
print("* by default, type induction is on (True) :")
learner = gum.BNLearner("out/testTypeInduction.csv")
bn3 = learner.learnBN()
for v in sorted(bn3.names()):
print(f" - {bn3.variable(v)}")
print("")
print("* but you can disable it :")
learner = gum.BNLearner("out/testTypeInduction.csv", ["?"], False)
bn3 = learner.learnBN()
for v in sorted(bn3.names()):
print(f" - {bn3.variable(v)}")
print("")
print("Note that when a Labelized variable is found, the labesl are alphabetically sorted.")
* by default, type induction is on (True) :
- A:Range([0,1])
- B:Range([1,3])
- C:Integer({-2|0|2})
- D:Labelized({cold|hot|warm})
* but you can disable it :
- A:Labelized({0|1})
- B:Labelized({1|2|3})
- C:Labelized({-2|0|2})
- D:Labelized({cold|hot|warm})
Note that when a Labelized variable is found, the labesl are alphabetically sorted.
Parameters learning from the database
We give the \(bn\) as a parameter for the learner in order to have the variables and the order of the labels for each variables. Please try to remove the argument \(bn\) in the first line below to see the difference …
In [12]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables and labels
bn2 = learner.learnParameters(bn.dag())
gnb.showBN(bn2)
In [13]:
gnb.sideBySide(
"<H3>Original BN</H3>",
"<H3>Learned NB</H3>",
bn.cpt("visit_to_Asia"),
bn2.cpt("visit_to_Asia"),
bn.cpt("tuberculosis"),
bn2.cpt("tuberculosis"),
ncols=2,
)
Structural learning a BN from the database
Note that, currently, the BNLearner is not yet able to learn in the presence of missing values. This is the reason why, when it discovers that there exist such values, it raises a gum.MissingValueInDatabase exception.
In [14]:
with open("res/asia_missing.csv", "r") as asiafile:
for _ in range(10):
print(asiafile.readline(), end="")
try:
learner = gum.BNLearner("res/asia_missing.csv", bn, ["?", "N/A"])
bn2 = learner.learnBN()
except gum.MissingValueInDatabase:
print("exception raised: there are missing values in the database")
smoking,lung_cancer,bronchitis,visit_to_Asia,tuberculosis,tuberculos_or_cancer,dyspnoea,positive_XraY
0,0,0,1,1,0,0,0
1,1,0,1,1,1,0,1
1,1,1,1,1,1,1,1
1,1,0,1,1,1,0,N/A
0,1,0,1,1,1,1,1
1,1,1,1,1,1,1,1
1,1,1,1,1,1,0,1
1,1,0,1,1,1,0,1
1,1,1,1,1,1,1,1
exception raised: there are missing values in the database
Different learning algorithms
For now, there are three scored-based algorithms that are wrapped in pyAgrum : LocalSearchWithTabuList, GreedyHillClimbing and K2
In [15]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useLocalSearchWithTabuList()
print(learner)
bn2 = learner.learnBN()
print("Learned in {0}ms".format(1000 * learner.currentTime()))
gnb.flow.row(bn, bn2, explain.getInformation(bn2), captions=["Original BN", "Learned BN", "information"])
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : Local Search with Tabu List
Tabu list size : 2
Score : BDeu
Prior : -
Learned in 74.803375ms
Out[15]:
To apprehend the distance between the original and the learned BN, we have several tools :
Compute the KL divergence (and other distance) between original and learned joint distribution
In [16]:
kl = gum.ExactBNdistance(bn, bn2)
kl.compute()
Out[16]:
{'klPQ': 0.0003524370027529867,
'errorPQ': 0,
'klQP': 0.000317085802666716,
'errorQP': 128,
'hellinger': 0.011248490414486263,
'bhattacharya': 6.32606273479283e-05,
'jensen-shannon': 8.892241380241719e-05}
Compute some scores on the BNs (as binary classifiers) abd show the graphical diff between the two graphs
In [17]:
gnb.flow.row(bn, bn2, captions=["bn", "bn2"])
gnb.flow.row(
bnvsbn.graphDiff(bn, bn2),
bnvsbn.graphDiff(bn2, bn),
bnvsbn.graphDiffLegend(),
captions=["bn versus bn2", "bn2 versus bn", ""],
)
gcmp = bnvsbn.GraphicalBNComparator(bn, bn2)
gnb.flow.add_html(
"<br/>".join([f"{k} : {v:.2f}" for k, v in gcmp.skeletonScores().items() if k != "count"]), "Skeleton scores"
)
gnb.flow.add_html("<br/>".join([f"{k} : {v:.2f}" for k, v in gcmp.scores().items() if k != "count"]), "Scores")
gnb.flow.display()
recall : 0.88
precision : 0.70
fscore : 0.78
dist2opt : 0.33
Skeleton scores
precision : 0.70
fscore : 0.78
dist2opt : 0.33
recall : 0.88
precision : 0.70
fscore : 0.78
dist2opt : 0.33
sid : 18.00
Scores
precision : 0.70
fscore : 0.78
dist2opt : 0.33
sid : 18.00
A greedy Hill Climbing algorithm (with insert, remove and change arc as atomic operations).
In [18]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useGreedyHillClimbing()
print(learner)
bn2 = learner.learnBN()
print("Learned in {0}ms".format(1000 * learner.currentTime()))
gnb.sideBySide(
bn,
bn2,
gnb.getBNDiff(bn, bn2),
explain.getInformation(bn2),
captions=["Original BN", "Learned BN", "Graphical diff", "information"],
)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : Greedy Hill Climbing
Score : BDeu
Prior : -
Learned in 54.989ms
And a K2 for those who likes it :)
In [19]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useK2([0, 1, 2, 3, 4, 5, 6, 7])
print(learner)
bn2 = learner.learnBN()
print("Learned in {0}ms".format(1000 * learner.currentTime()))
gnb.sideBySide(
bn,
bn2,
gnb.getBNDiff(bn, bn2),
explain.getInformation(bn2),
captions=["Original BN", "Learned BN", "Graphical diff", "information"],
)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : K2
K2 order : visit_to_Asia, tuberculosis, tuberculos_or_cancer, positive_XraY, lung_cancer, smoking, bronchitis, dyspnoea
Score : BDeu
Prior : -
Learned in 25.979375ms
K2 can be very good if the order is the good one (a topological order of nodes in the reference)
In [20]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useK2([7, 6, 5, 4, 3, 2, 1, 0])
print(learner)
bn2 = learner.learnBN()
print("Learned in {0}s".format(learner.currentTime()))
gnb.sideBySide(
bn,
bn2,
gnb.getBNDiff(bn, bn2),
explain.getInformation(bn2),
captions=["Original BN", "Learned BN", "Graphical diff", "information"],
)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : K2
K2 order : dyspnoea, bronchitis, smoking, lung_cancer, positive_XraY, tuberculos_or_cancer, tuberculosis, visit_to_Asia
Score : BDeu
Prior : -
Learned in 0.018740041000000002s
Following the learning curve
In [21]:
import numpy as np
%matplotlib inline
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useLocalSearchWithTabuList()
# we could prefere a log2likelihood score
# learner.useScoreLog2Likelihood()
learner.setMaxTime(10)
# representation of the error as a pseudo log (negative values really represents negative epsilon
@np.vectorize
def pseudolog(x):
res = np.log(x) # np.log(y)
return res if x > 0 else -res
# in order to control the complexity, we limit the number of parents
learner.setMaxIndegree(7) # no more than 3 parent by node
learner.setEpsilon(1e-10)
gnb.animApproximationScheme(learner, scale=pseudolog) # scale by default is np.log10
bn2 = learner.learnBN()
Customizing the learning algorithms
1. Learn a tree ?
In [22]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useGreedyHillClimbing()
learner.setMaxIndegree(1) # no more than 1 parent by node
print(learner)
bntree = learner.learnBN()
gnb.sideBySide(
bn,
bntree,
gnb.getBNDiff(bn, bntree),
explain.getInformation(bntree),
captions=["Original BN", "Learned BN", "Graphical diff", "information"],
)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : Greedy Hill Climbing
Score : BDeu
Prior : -
Constraint Max InDegree : 1
2. with prior structural knowledge
In [23]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useGreedyHillClimbing()
# I know that smoking causes cancer
learner.addMandatoryArc("smoking", "lung_cancer") # smoking->lung_cancer
# I know that visit to Asia may change the risk of tuberculosis
learner.addMandatoryArc("visit_to_Asia", "tuberculosis") # visit_to_Asia->tuberculosis
print(learner)
bn2 = learner.learnBN()
gnb.sideBySide(
bn,
bn2,
gnb.getBNDiff(bn, bn2),
explain.getInformation(bn2),
captions=["Original BN", "Learned BN", "Graphical diff", "information"],
)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : Greedy Hill Climbing
Score : BDeu
Prior : -
Constraint Mandatory Arcs : {visit_to_Asia->tuberculosis, smoking->lung_cancer}
3. changing the scores
By default, a BDEU score is used. But it can be changed.
In [24]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useGreedyHillClimbing()
# I know that smoking causes cancer
learner.addMandatoryArc(0, 1)
# we prefere a log2likelihood score
learner.useScoreLog2Likelihood()
# in order to control the complexity, we limit the number of parents
learner.setMaxIndegree(1) # no more than 1 parent by node
print(learner)
bn2 = learner.learnBN()
kl = gum.ExactBNdistance(bn, bn2)
gnb.sideBySide(
bn,
bn2,
gnb.getBNDiff(bn, bn2),
"<br/>".join(["<b>" + k + "</b> :" + str(v) for k, v in kl.compute().items()]),
captions=["original", "learned BN", "diff", "distances"],
)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : Greedy Hill Climbing
Score : Log2Likelihood
Prior : -
Constraint Max InDegree : 1
Constraint Mandatory Arcs : {visit_to_Asia->tuberculosis}
4. comparing BNs
There are multiple ways to compare Bayes net…
In [25]:
help(gnb.getBNDiff)
Help on function getBNDiff in module pyagrum.lib.notebook:
getBNDiff(
bn1: pyagrum.BayesNet,
bn2: pyagrum.BayesNet,
size: float | str | None = None,
noStyle: bool = False
) -> str
get a HTML string representation of a graphical diff between the arcs of _bn1 (reference) with those of _bn2.
if `noStyle` is False use 4 styles (fixed in pyagrum.config) :
- the arc is common for both
- the arc is common but inverted in `bn2`
- the arc is added in `bn2`
- the arc is removed in `bn2`
Parameters
----------
bn1: pyagrum.BayesNet
the reference
bn2: pyagrum.BayesNet
the compared one
size: float|str
size (for graphviz) of the rendered graph
noStyle: bool
with style or not.
Returns
-------
str
the HTML representation of the comparison
In [26]:
gnb.showBNDiff(bn, bn2)
In [27]:
import pyagrum.lib.bn_vs_bn as gbnbn
help(gbnbn.graphDiff)
Help on function graphDiff in module pyagrum.lib.bn_vs_bn:
graphDiff(bnref, bncmp, noStyle: bool = False) -> dot.Dot
Return a pydot graph that compares the arcs of bnref to bncmp.
graphDiff allows bncmp to have less nodes than bnref. (this is not the case in GraphicalBNComparator.dotDiff())
if noStyle is False use 4 styles (fixed in pyagrum.config) :
- the arc is common for both
- the arc is common but inverted in _bn2
- the arc is added in _bn2
- the arc is removed in _bn2
See graphDiffLegend() to add a legend to the graph.
Warning
-------
if pydot is not installed, this function just returns None
Returns
-------
pydot.Dot
the result dot graph or None if pydot can not be imported
In [28]:
gbnbn.GraphicalBNComparator?
In [29]:
gcmp = gbnbn.GraphicalBNComparator(bn, bn2)
gnb.sideBySide(
bn, bn2, gcmp.dotDiff(), gbnbn.graphDiffLegend(), bn2, bn, gbnbn.graphDiff(bn2, bn), gbnbn.graphDiffLegend(), ncols=4
)
In [30]:
print("But also gives access to different scores :")
print(gcmp.scores())
print(gcmp.skeletonScores())
print(gcmp.hamming())
But also gives access to different scores :
{'count': {'tp': 3, 'tn': 22, 'fp': 4, 'fn': 2}, 'recall': 0.6, 'precision': 0.42857142857142855, 'fscore': 0.5, 'dist2opt': 0.6975174637562116, 'sid': 16}
{'count': {'tp': 6, 'tn': 19, 'fp': 1, 'fn': 2}, 'recall': 0.75, 'precision': 0.8571428571428571, 'fscore': 0.7999999999999999, 'dist2opt': 0.2879377767249482}
{'hamming': 3, 'structural hamming': 6}
In [31]:
print("KL divergence can be computed")
kl = gum.ExactBNdistance(bn, bn2)
kl.compute()
KL divergence can be computed
Out[31]:
{'klPQ': 0.11601936110927323,
'errorPQ': 0,
'klQP': 0.012770989477639258,
'errorQP': 64,
'hellinger': 0.19683305875484047,
'bhattacharya': 0.019561709604055653,
'jensen-shannon': 0.02197812746019102}
5. Mixing algorithms
First we learn a structure with HillClimbing (faster ?)
In [32]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useGreedyHillClimbing()
learner.addMandatoryArc(0, 1)
bn2 = learner.learnBN()
kl = gum.ExactBNdistance(bn, bn2)
gnb.sideBySide(
bn,
bn2,
gnb.getBNDiff(bn, bn2),
"<br/>".join(["<b>" + k + "</b> :" + str(v) for k, v in kl.compute().items()]),
captions=["original", "learned BN", "diff", "distances"],
)
And then we refine with tabuList
In [33]:
learner = gum.BNLearner("out/sample_asia.csv", bn) # using bn as template for variables
learner.useLocalSearchWithTabuList()
learner.setInitialDAG(bn2.dag())
print(learner)
bn3 = learner.learnBN()
kl = gum.ExactBNdistance(bn, bn3)
gnb.sideBySide(
bn,
bn2,
gnb.getBNDiff(bn, bn2),
"<br/>".join(["<b>" + k + "</b> :" + str(v) for k, v in kl.compute().items()]),
captions=["original", "learned BN", "diff", "distances"],
)
Filename : out/sample_asia.csv
Size : (50000,8)
Variables : visit_to_Asia[2], tuberculosis[2], tuberculos_or_cancer[2], positive_XraY[2], lung_cancer[2], smoking[2], bronchitis[2], dyspnoea[2]
Induced types : False
Missing values : False
Algorithm : Local Search with Tabu List
Tabu list size : 2
Score : BDeu
Prior : -
Initial DAG : True (digraph {
0 [label="(0) visit_to_Asia"];
1 [label="(1) tuberculosis"];
2 [label="(2) tuberculos_or_cancer"];
3 [label="(3) positive_XraY"];
4 [label="(4) lung_cancer"];
5 [label="(5) smoking"];
6 [label="(6) bronchitis"];
7 [label="(7) dyspnoea"];
0 -> 1;
2 -> 3;
6 -> 5;
1 -> 7;
6 -> 7;
4 -> 7;
4 -> 2;
1 -> 2;
5 -> 4;
}
)
Impact of the size of the database for the learning
In [34]:
rows = 3
sizes = [400, 500, 700, 1000, 2000, 5000, 10000, 50000, 75000, 100000, 150000, 175000, 200000, 300000, 500000]
def extract_asia(n):
"""
extract n line from asia.csv to extract.csv
"""
with open("out/sample_asia.csv", "r") as src:
with open("out/extract_asia.csv", "w") as dst:
for _ in range(n + 1):
print(src.readline(), end="", file=dst)
In [35]:
gnb.flow.clear()
nbr = 0
l = []
for i in sizes:
extract_asia(i)
learner = gum.BNLearner("out/extract_asia.csv", bn) # using bn as template for variables
learner.useGreedyHillClimbing()
print(learner.state()["Size"][0])
bn2 = learner.learnBN()
kl = gum.ExactBNdistance(bn, bn2)
r = kl.compute()
l.append(math.log(r["klPQ"]))
gnb.flow.add(gnb.getBNDiff(bn, bn2, size="3!"), f"size={i}")
gnb.flow.display()
plt.plot(sizes, l)
print(f"final value computed : {l[-1]}")
(400,8)
(500,8)
(700,8)
(1000,8)
(2000,8)
(5000,8)
(10000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
final value computed : -7.900098734769177
In [36]:
gnb.flow.clear()
nbr = 0
l = []
for i in sizes:
extract_asia(i)
learner = gum.BNLearner("out/extract_asia.csv", bn) # using bn as template for variables
learner.useMIIC()
print(learner.state()["Size"][0])
bn2 = learner.learnBN()
kl = gum.ExactBNdistance(bn, bn2)
r = kl.compute()
l.append(math.log(r["klPQ"]))
gnb.flow.add(gnb.getBNDiff(bn, bn2, size="3!"), f"size={i}")
gnb.flow.display()
plt.plot(sizes, l)
print(l[-1])
(400,8)
(500,8)
(700,8)
(1000,8)
(2000,8)
(5000,8)
(10000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
(50000,8)
-8.123694059066613
Algorithms constraint-based
In [37]:
learner = gum.BNLearner("out/extract_asia.csv", bn)
learner.usePC()
learner.setPCStable(True)
learner.setPCAlpha(0.05)
learner.learnDAG()
Out[37]:
In [38]:
gum.generateSample(bn, 50000, "out/sample_asia.csv", True)
learner = gum.BNLearner("out/extract_asia.csv", bn)
learner.useFCI()
learner.learnPAG() # only for FCI
out/sample_asia.csv: 100%|█████████████████████████████████|
Log2-Likelihood : -161513.27130078577
Out[38]:
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