Learning the structure of a Bayesian network

# %matplotlib inline
## from pylab import *

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 2.3.0.9
(c) 2015-2024 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.  For details, see 'pyagrum.warranty'.

Library	Version
OS	posix [darwin]
Python	3.14.0 (main, Oct 7 2025, 09:34:52) [Clang 17.0.0 (clang-1700.3.19.1)]
IPython	9.6.0
Matplotlib	3.10.7
Numpy	2.3.4
pyDot	4.0.1
pyAgrum	2.3.0.9

Wed Oct 29 13:58:45 2025 CET

Generating the database from a BN

bn = gum.loadBN("res/asia.bif")
bn

gum.generateSample(bn, 50000, "out/sample_asia.csv", True);

out/sample_asia.csv: 0%| |

out/sample_asia.csv: 100%|█████████████████████████████████|

Log2-Likelihood : -161108.69919616362

with open("out/sample_asia.csv", "r") as src:
  for _ in range(10):
    print(src.readline(), end="")

bronchitis,tuberculos_or_cancer,tuberculosis,dyspnoea,positive_XraY,visit_to_Asia,smoking,lung_cancer
1,1,1,1,1,1,0,1
1,1,1,1,1,1,1,1
1,1,1,1,1,1,0,1
1,1,1,1,1,1,1,1
0,1,1,0,1,1,1,1
1,1,1,1,1,1,1,1
1,1,1,1,1,1,1,1
1,1,1,1,1,1,1,1
0,1,1,0,1,1,1,1

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          : -

print(f"Row of visit_to_Asia : {learner.idFromName('visit_to_Asia')}")  # first row is 0

Row of visit_to_Asia : 0

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.

## 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.

%%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

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 …

learner = gum.BNLearner("out/sample_asia.csv", bn)  # using bn as template for variables and labels
bn2 = learner.learnParameters(bn.dag())
gnb.showBN(bn2)

svg

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,
)

Original BN

Learned NB

visit_to_Asia
0	1
0.0100	0.9900

visit_to_Asia
0	1
0.0100	0.9900

	tuberculosis
visit_to_Asia	0	1
0	0.0500	0.9500
1	0.0100	0.9900

	tuberculosis
visit_to_Asia	0	1
0	0.0579	0.9421
1	0.0090	0.9910

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.

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

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 15.564ms

Original BN

Learned BN

information

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

kl = gum.ExactBNdistance(bn, bn2)
kl.compute()

{'klPQ': 0.0002454972790806624,
 'errorPQ': 0,
 'klQP': 0.0002170115306462794,
 'errorQP': 128,
 'hellinger': 0.009492813820926288,
 'bhattacharya': 4.5052130085963663e-05,
 'jensen-shannon': 6.276601520482401e-05}

Compute some scores on the BNs (as binary classifiers) abd show the graphical diff between the two graphs

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()

bn2

bn versus bn2

bn2 versus bn

recall : 1.00
precision : 0.73
fscore : 0.84
dist2opt : 0.27
Skeleton scores

recall : 0.38
precision : 0.27
fscore : 0.32
dist2opt : 0.96
Scores

A greedy Hill Climbing algorithm (with insert, remove and change arc as atomic operations).

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 13.482ms

Original BN

Learned BN

Graphical diff

information

And a K2 for those who likes it :)

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 6.264ms

Original BN

Learned BN

Graphical diff

information

K2 can be very good if the order is the good one (a topological order of nodes in the reference)

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 0.007757s

Original BN

Learned BN

Graphical diff

information

Following the learning curve

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()

svg

Customizing the learning algorithms

1. Learn a tree ?

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

Original BN

Learned BN

Graphical diff

information

2. with prior structural knowledge

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}

Original BN

Learned BN

Graphical diff

information

3. changing the scores

By default, a BDEU score is used. But it can be changed.

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}

original

learned BN

diff

klPQ :0.122659583144537
errorPQ :0
klQP :0.035362312624309146
errorQP :64
hellinger :0.2008652891479617
bhattacharya :0.020379688845562502
jensen-shannon :0.0233148500701501
distances

4. comparing BNs

There are multiple ways to compare Bayes net…

help(gnb.getBNDiff)

Help on function getBNDiff in module pyagrum.lib.notebook:

getBNDiff(bn1, bn2, size=None, noStyle=False)
    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

gnb.showBNDiff(bn, bn2)

svg

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=False)
    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

gbnbn.GraphicalBNComparator?

gcmp = gbnbn.GraphicalBNComparator(bn, bn2)
gnb.sideBySide(
  bn, bn2, gcmp.dotDiff(), gbnbn.graphDiffLegend(), bn2, bn, gbnbn.graphDiff(bn2, bn), gbnbn.graphDiffLegend(), ncols=4
)

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': 44, 'fp': 4, 'fn': 5}, 'recall': 0.375, 'precision': 0.42857142857142855, 'fscore': 0.39999999999999997, 'dist2opt': 0.8468504072413839}
{'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': 7}

print("KL divergence can be computed")
kl = gum.ExactBNdistance(bn, bn2)
kl.compute()

KL divergence can be computed





{'klPQ': 0.122659583144537,
 'errorPQ': 0,
 'klQP': 0.035362312624309146,
 'errorQP': 64,
 'hellinger': 0.2008652891479617,
 'bhattacharya': 0.020379688845562502,
 'jensen-shannon': 0.0233148500701501}

5. Mixing algorithms

First we learn a structure with HillClimbing (faster ?)

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"],
)

original

learned BN

diff

klPQ :0.00024105451496356508
errorPQ :0
klQP :0.00021087593292772806
errorQP :128
hellinger :0.009400672333381922
bhattacharya :4.4181654289395046e-05
jensen-shannon :6.150358231389019e-05
distances

And then we refine with tabuList

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;
     1;
     2;
     3;
     4;
     5;
     6;
     7;

     0 -> 1;
     6 -> 7;
     2 -> 3;
     6 -> 5;
     2 -> 7;
     4 -> 5;
     4 -> 2;
     1 -> 2;
     6 -> 4;
}

)

original

learned BN

diff

Impact of the size of the database for the learning

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)

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)

size=400

size=500

size=700

size=1000

size=2000

size=5000

size=10000

size=50000

size=75000

size=100000

size=150000

size=175000

size=200000

size=300000

size=500000

final value computed : -8.33048744688638

svg

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)

size=400

size=500

size=700

size=1000

size=2000

size=5000

size=10000

size=50000

size=75000

size=100000

size=150000

size=175000

size=200000

size=300000

size=500000

-8.42789925936379

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