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Probablistic Inference with pyAgrum

Creative Commons LicenseaGrUMinteractive online version

In this notebook, we will show different basic features for probabilistic inference on Bayesian networks using pyagrum.

First we need some external modules:

%matplotlib inline
from pylab import *

Basic inference and display

Section titled “Basic inference and display”

Then we import pyagrum and the pyAgrum’s notebook module, that offers very usefull methods when writting a notebook.

This first example shows how you can load a BayesNet and show it as graph. Note that pyAgrum handles serveral BayesNet file format such as DSL, BIF and UAI.

import pyagrum as gum
import pyagrum.lib.notebook as gnb


bn = gum.loadBN("res/alarm.dsl")
gnb.showBN(bn, size="9")

svg

print(bn)
BN{nodes: 37, arcs: 46, domainSize: 10^16.2389, dim: 509, mem: 5Ko 896o}

From there, it is easy to get a posterior using an inference engine :

ie = gum.LazyPropagation(bn)
ie.makeInference()
print(ie.posterior(bn.idFromName("CATECHOL")))
  CATECHOL         |
NORMAL   |HIGH     |
---------|---------|
 0.0512  | 0.9488  |

But since we are in notebook, why not use pyAgrum notebook’s methods ?

gnb.showPosterior(bn, evs={}, target="CATECHOL")

svg

You may also want to see the graph with some posteriors

## due to matplotlib, format is forced to png.
gnb.showInference(bn, evs={}, targets={"VENTALV", "CATECHOL", "HR", "MINVOLSET"}, size="11")

svg

.. and then observe the impact of evidence :

gnb.showInference(
  bn,
  evs={"CO": 1, "VENTLUNG": 1},
  targets={
    "VENTALV",
    "CATECHOL",
    "HR",
    "MINVOLSET",
    "ANAPHYLAXIS",
    "STROKEVOLUME",
    "ERRLOWOUTPUT",
    "HBR",
    "PULMEMBOLUS",
    "HISTORY",
    "BP",
    "PRESS",
    "CO",
  },
  size="10",
)

svg

You can even compute all posteriors by leaving the targets parameter empty (which is its default value).

gnb.showInference(bn, evs={"CO": 1, "VENTLUNG": 1}, size="14")

svg

Showing the information graph

Section titled “Showing the information graph”

To have a global view of the knowledge brought by the inference, you can also draw the entropy of all nodes

import pyagrum.lib.explain as explain


explain.showInformation(bn, {}, size="14")
/var/folders/r1/pj4vdx_n4_d_xpsb04kzf97r0000gp/T/ipykernel_91946/3333624085.py:1: DeprecationWarning: The module 'pyagrum.lib.explain' has been deprecated since version 2.2.2. Please use the 'pyagrum.explain' module instead.
  import pyagrum.lib.explain as explain
G VENTLUNG VENTLUNG VENTALV VENTALV VENTLUNG->VENTALV MINVOL MINVOL VENTLUNG->MINVOL EXPCO2 EXPCO2 VENTLUNG->EXPCO2 HRSAT HRSAT ERRCAUTER ERRCAUTER ERRCAUTER->HRSAT HREKG HREKG ERRCAUTER->HREKG LVEDVOLUME LVEDVOLUME PCWP PCWP LVEDVOLUME->PCWP CVP CVP LVEDVOLUME->CVP PVSAT PVSAT VENTALV->PVSAT ARTCO2 ARTCO2 VENTALV->ARTCO2 ERRLOWOUTPUT ERRLOWOUTPUT HRBP HRBP ERRLOWOUTPUT->HRBP SHUNT SHUNT SAO2 SAO2 SHUNT->SAO2 KINKEDTUBE KINKEDTUBE KINKEDTUBE->VENTLUNG PRESS PRESS KINKEDTUBE->PRESS ANAPHYLAXIS ANAPHYLAXIS TPR TPR ANAPHYLAXIS->TPR MINVOLSET MINVOLSET VENTMACH VENTMACH MINVOLSET->VENTMACH PAP PAP HYPOVOLEMIA HYPOVOLEMIA HYPOVOLEMIA->LVEDVOLUME STROKEVOLUME STROKEVOLUME HYPOVOLEMIA->STROKEVOLUME CATECHOL CATECHOL SAO2->CATECHOL VENTTUBE VENTTUBE VENTTUBE->VENTLUNG VENTTUBE->PRESS FIO2 FIO2 FIO2->PVSAT HR HR HR->HRSAT HR->HRBP HR->HREKG CO CO HR->CO CATECHOL->HR HISTORY HISTORY INSUFFANESTH INSUFFANESTH INSUFFANESTH->CATECHOL INTUBATION INTUBATION INTUBATION->VENTLUNG INTUBATION->VENTALV INTUBATION->SHUNT INTUBATION->PRESS INTUBATION->MINVOL TPR->CATECHOL BP BP TPR->BP PULMEMBOLUS PULMEMBOLUS PULMEMBOLUS->SHUNT PULMEMBOLUS->PAP VENTMACH->VENTTUBE LVFAILURE LVFAILURE LVFAILURE->LVEDVOLUME LVFAILURE->HISTORY LVFAILURE->STROKEVOLUME PVSAT->SAO2 ARTCO2->CATECHOL ARTCO2->EXPCO2 DISCONNECT DISCONNECT DISCONNECT->VENTTUBE STROKEVOLUME->CO CO->BP
PyAgrum inline image

… and then observe the impact of an evidence on the whole bayes network :

explain.showInformation(bn, {"CO": 0}, size="9")
G VENTLUNG VENTLUNG VENTALV VENTALV VENTLUNG->VENTALV MINVOL MINVOL VENTLUNG->MINVOL EXPCO2 EXPCO2 VENTLUNG->EXPCO2 HRSAT HRSAT ERRCAUTER ERRCAUTER ERRCAUTER->HRSAT HREKG HREKG ERRCAUTER->HREKG LVEDVOLUME LVEDVOLUME PCWP PCWP LVEDVOLUME->PCWP CVP CVP LVEDVOLUME->CVP PVSAT PVSAT VENTALV->PVSAT ARTCO2 ARTCO2 VENTALV->ARTCO2 ERRLOWOUTPUT ERRLOWOUTPUT HRBP HRBP ERRLOWOUTPUT->HRBP SHUNT SHUNT SAO2 SAO2 SHUNT->SAO2 KINKEDTUBE KINKEDTUBE KINKEDTUBE->VENTLUNG PRESS PRESS KINKEDTUBE->PRESS ANAPHYLAXIS ANAPHYLAXIS TPR TPR ANAPHYLAXIS->TPR MINVOLSET MINVOLSET VENTMACH VENTMACH MINVOLSET->VENTMACH PAP PAP HYPOVOLEMIA HYPOVOLEMIA HYPOVOLEMIA->LVEDVOLUME STROKEVOLUME STROKEVOLUME HYPOVOLEMIA->STROKEVOLUME CATECHOL CATECHOL SAO2->CATECHOL VENTTUBE VENTTUBE VENTTUBE->VENTLUNG VENTTUBE->PRESS FIO2 FIO2 FIO2->PVSAT HR HR HR->HRSAT HR->HRBP HR->HREKG CO CO HR->CO CATECHOL->HR HISTORY HISTORY INSUFFANESTH INSUFFANESTH INSUFFANESTH->CATECHOL INTUBATION INTUBATION INTUBATION->VENTLUNG INTUBATION->VENTALV INTUBATION->SHUNT INTUBATION->PRESS INTUBATION->MINVOL TPR->CATECHOL BP BP TPR->BP PULMEMBOLUS PULMEMBOLUS PULMEMBOLUS->SHUNT PULMEMBOLUS->PAP VENTMACH->VENTTUBE LVFAILURE LVFAILURE LVFAILURE->LVEDVOLUME LVFAILURE->HISTORY LVFAILURE->STROKEVOLUME PVSAT->SAO2 ARTCO2->CATECHOL ARTCO2->EXPCO2 DISCONNECT DISCONNECT DISCONNECT->VENTTUBE STROKEVOLUME->CO CO->BP
PyAgrum inline image

Exploring the junction tree

Section titled “Exploring the junction tree”

Lazy Propagation, like several other inference algorithms, uses a junction tree to propagate information.

You can show the junction tree used by Lazy Propagation with pyAgrum:

jt = ie.junctionTree()
gnb.showJunctionTree(bn, size="12")

svg

## another representation of the junction, more convenient for investigating the flow of data in the jt
## the size/width of cliques and separators are proportionnal to the number of nodes in the factor.
jt.map()
0 0~16 0--0~16 1 1~32 1--1~32 2 2~33 2--2~33 3 3~4 3--3~4 4 4~22 4--4~22 5 5~22 5--5~22 6 6~23 6--6~23 7 7~26 7--7~26 8 8~17 8--8~17 10 10~14 10--10~14 11 11~16 11--11~16 12 12~13 12--12~13 13 13~30 13--13~30 14 14~26 14--14~26 16 16~17 16--16~17 17 17~24 17--17~24 19 19~27 19--19~27 20 20~33 20--20~33 22 22~33 22--22~33 23 23~27 23--23~27 23~31 23--23~31 24 24~26 24--24~26 26 26~27 26--26~27 27 30 30~31 30--30~31 31 31~32 31--31~32 32 32~33 32--32~33 33 19~27--27 12~13--13 2~33--33 23~27--27 22~33--33 11~16--16 24~26--26 31~32--32 10~14--14 26~27--27 13~30--30 5~22--22 7~26--26 20~33--33 16~17--17 32~33--33 23~31--31 8~17--17 1~32--32 3~4--4 4~22--22 17~24--24 14~26--26 30~31--31 6~23--23 0~16--16

Introspection in junction trees

Section titled “Introspection in junction trees”

One can easily walk through the junction tree.

for n in jt.nodes():
  print([bn.variable(n).name() for n in jt.clique(n)])
['CVP', 'LVEDVOLUME']
['FIO2', 'VENTALV', 'PVSAT']
['ARTCO2', 'EXPCO2', 'VENTLUNG']
['VENTMACH', 'MINVOLSET']
['VENTMACH', 'DISCONNECT', 'VENTTUBE']
['PRESS', 'KINKEDTUBE', 'INTUBATION', 'VENTTUBE']
['ANAPHYLAXIS', 'TPR']
['HRBP', 'ERRLOWOUTPUT', 'HR']
['LVFAILURE', 'HISTORY']
['HREKG', 'HR', 'ERRCAUTER']
['PCWP', 'LVEDVOLUME']
['PAP', 'PULMEMBOLUS']
['SHUNT', 'INTUBATION', 'PULMEMBOLUS']
['HRSAT', 'HR', 'ERRCAUTER']
['LVFAILURE', 'HYPOVOLEMIA', 'LVEDVOLUME']
['HYPOVOLEMIA', 'STROKEVOLUME', 'LVFAILURE']
['CO', 'BP', 'TPR']
['INTUBATION', 'VENTLUNG', 'MINVOL']
['KINKEDTUBE', 'INTUBATION', 'VENTTUBE', 'VENTLUNG']
['INSUFFANESTH', 'TPR', 'ARTCO2', 'SAO2', 'CATECHOL']
['CO', 'STROKEVOLUME', 'HR']
['CO', 'CATECHOL', 'HR']
['CO', 'TPR', 'CATECHOL']
['INTUBATION', 'SHUNT', 'PVSAT', 'SAO2']
['INTUBATION', 'ARTCO2', 'PVSAT', 'SAO2']
['ARTCO2', 'VENTALV', 'INTUBATION', 'PVSAT']
['VENTALV', 'INTUBATION', 'ARTCO2', 'VENTLUNG']
for e in jt.edges():
  print(f"Separator for {e} : {jt.clique(e[0]).intersection(jt.clique(e[1]))}")
Separator for (13, 30) : {18, 2}
Separator for (2, 33) : {26, 22}
Separator for (3, 4) : {16}
Separator for (26, 27) : {34, 30}
Separator for (7, 26) : {31}
Separator for (12, 13) : {4}
Separator for (31, 32) : {27, 26, 2}
Separator for (23, 31) : {26, 28}
Separator for (5, 22) : {0, 2, 20}
Separator for (17, 24) : {13}
Separator for (19, 27) : {34, 14}
Separator for (24, 26) : {34, 31}
Separator for (32, 33) : {25, 2, 26}
Separator for (6, 23) : {14}
Separator for (23, 27) : {14, 30}
Separator for (11, 16) : {15}
Separator for (10, 14) : {7, 31}
Separator for (8, 17) : {9}
Separator for (0, 16) : {15}
Separator for (1, 32) : {25, 27}
Separator for (20, 33) : {2, 22}
Separator for (4, 22) : {20}
Separator for (14, 26) : {31}
Separator for (22, 33) : {2, 22}
Separator for (30, 31) : {2, 27, 28}
Separator for (16, 17) : {1, 9}
jt.hasRunningIntersection()
True

Introspecting junction trees and friends

Section titled “Introspecting junction trees and friends”

The junction tree created by a LazyPropagation is optimized for the query (see RelevanceReasonning notebook). But you can also introspect a junction tree directly from a BN or a graph using the JunctionTreeGenerator’s class.

bn = gum.fastBN("0->1->2<-3->4->5->6<-2->7")
jtg = gum.JunctionTreeGenerator()
gnb.sideBySide(
  bn,
  jtg.junctionTree(bn),
  jtg.eliminationOrder(bn),
  jtg.binaryJoinTree(bn),
  captions=[
    "A Bayesien network",
    "a junction tree for this BN",
    "its elimination order",
    "an (optimized) binary join tree",
  ],
)
G 3 3 2 2 3->2 4 4 3->4 0 0 1 1 0->1 1->2 7 7 2->7 6 6 2->6 5 5 5->6 4->5
A Bayesien network
(0) 2-5-6 2-5-6 (0) 2-5-6^(4) 2-3-5 2-5 (0) 2-5-6--(0) 2-5-6^(4) 2-3-5 (1) 2-7 2-7 (1) 2-7^(4) 2-3-5 2 (1) 2-7--(1) 2-7^(4) 2-3-5 (2) 0-1 0-1 (2) 0-1^(3) 1-2-3 1 (2) 0-1--(2) 0-1^(3) 1-2-3 (3) 1-2-3 1-2-3 (3) 1-2-3^(4) 2-3-5 2-3 (3) 1-2-3--(3) 1-2-3^(4) 2-3-5 (4) 2-3-5 2-3-5 (4) 2-3-5^(5) 3-4-5 3-5 (4) 2-3-5--(4) 2-3-5^(5) 3-4-5 (5) 3-4-5 3-4-5 (2) 0-1^(3) 1-2-3--(3) 1-2-3 (4) 2-3-5^(5) 3-4-5--(5) 3-4-5 (0) 2-5-6^(4) 2-3-5--(4) 2-3-5 (1) 2-7^(4) 2-3-5--(4) 2-3-5 (3) 1-2-3^(4) 2-3-5--(4) 2-3-5
a junction tree for this BN
[6, 7, 0, 1, 2, 3, 4, 5]
its elimination order
(0) 2-5-6 2-5-6 (0) 2-5-6^(6) 2-5 2-5 (0) 2-5-6--(0) 2-5-6^(6) 2-5 (1) 2-7 2-7 (1) 2-7^(6) 2-5 2 (1) 2-7--(1) 2-7^(6) 2-5 (2) 0-1 0-1 (2) 0-1^(3) 1-2-3 1 (2) 0-1--(2) 0-1^(3) 1-2-3 (3) 1-2-3 1-2-3 (3) 1-2-3^(4) 2-3-5 2-3 (3) 1-2-3--(3) 1-2-3^(4) 2-3-5 (4) 2-3-5 2-3-5 (4) 2-3-5^(5) 3-4-5 3-5 (4) 2-3-5--(4) 2-3-5^(5) 3-4-5 (4) 2-3-5^(6) 2-5 2-5 (4) 2-3-5--(4) 2-3-5^(6) 2-5 (5) 3-4-5 3-4-5 (6) 2-5 2-5 (2) 0-1^(3) 1-2-3--(3) 1-2-3 (4) 2-3-5^(5) 3-4-5--(5) 3-4-5 (3) 1-2-3^(4) 2-3-5--(4) 2-3-5 (1) 2-7^(6) 2-5--(6) 2-5 (0) 2-5-6^(6) 2-5--(6) 2-5 (4) 2-3-5^(6) 2-5--(6) 2-5
an (optimized) binary join tree

junction tree from graphs (using uniform domainSize)

Section titled “junction tree from graphs (using uniform domainSize)”
## creating a dag slightly different
dag = bn.dag()
dag.addArc(0, 3)
dag.addArc(0, 7)
gnb.sideBySide(
  dag,
  dag.moralGraph(),
  jtg.junctionTree(dag),
  jtg.eliminationOrder(dag),
  jtg.binaryJoinTree(dag),
  captions=[
    "A DAG",
    "its moral graph",
    "a junction tree for this dag (with partial order)",
    "its elimination order (with partial order)",
    "an (optipmized) binary jointree (with partial order)",
  ],
)
0 0 1 1 0->1 3 3 0->3 7 7 0->7 2 2 1->2 6 6 2->6 2->7 3->2 4 4 3->4 5 5 4->5 5->6
A DAG
no_name 0 0 1 1 0->1 2 2 0->2 3 3 0->3 7 7 0->7 1->2 1->3 2->3 5 5 2->5 6 6 2->6 2->7 4 4 3->4 4->5 5->6
its moral graph
(0) 2-5-6 2-5-6 (0) 2-5-6^(4) 2-3-5 2-5 (0) 2-5-6--(0) 2-5-6^(4) 2-3-5 (1) 0-1-2-3 0-1-2-3 (1) 0-1-2-3^(4) 2-3-5 2-3 (1) 0-1-2-3--(1) 0-1-2-3^(4) 2-3-5 (1) 0-1-2-3^(2) 0-2-7 0-2 (1) 0-1-2-3--(1) 0-1-2-3^(2) 0-2-7 (2) 0-2-7 0-2-7 (4) 2-3-5 2-3-5 (4) 2-3-5^(5) 3-4-5 3-5 (4) 2-3-5--(4) 2-3-5^(5) 3-4-5 (5) 3-4-5 3-4-5 (4) 2-3-5^(5) 3-4-5--(5) 3-4-5 (0) 2-5-6^(4) 2-3-5--(4) 2-3-5 (1) 0-1-2-3^(4) 2-3-5--(4) 2-3-5 (1) 0-1-2-3^(2) 0-2-7--(2) 0-2-7
a junction tree for this dag (with partial order)
[6, 1, 7, 0, 2, 3, 4, 5]
its elimination order (with partial order)
(0) 2-5-6 2-5-6 (0) 2-5-6^(4) 2-3-5 2-5 (0) 2-5-6--(0) 2-5-6^(4) 2-3-5 (1) 0-1-2-3 0-1-2-3 (1) 0-1-2-3^(4) 2-3-5 2-3 (1) 0-1-2-3--(1) 0-1-2-3^(4) 2-3-5 (1) 0-1-2-3^(2) 0-2-7 0-2 (1) 0-1-2-3--(1) 0-1-2-3^(2) 0-2-7 (2) 0-2-7 0-2-7 (4) 2-3-5 2-3-5 (4) 2-3-5^(5) 3-4-5 3-5 (4) 2-3-5--(4) 2-3-5^(5) 3-4-5 (5) 3-4-5 3-4-5 (4) 2-3-5^(5) 3-4-5--(5) 3-4-5 (0) 2-5-6^(4) 2-3-5--(4) 2-3-5 (1) 0-1-2-3^(4) 2-3-5--(4) 2-3-5 (1) 0-1-2-3^(2) 0-2-7--(2) 0-2-7
an (optipmized) binary jointree (with partial order)
## creating an undigraph slightly different
ug = bn.dag().moralGraph()
ug.addEdge(0, 7)
gnb.sideBySide(
  ug,
  jtg.junctionTree(ug),
  jtg.eliminationOrder(ug),
  jtg.binaryJoinTree(ug),
  captions=[
    "A undigraph",
    "a junction tree for this undigraph",
    "its elimination order",
    "an (optipmized) binary jointree",
  ],
)
no_name 0 0 1 1 0->1 7 7 0->7 2 2 1->2 3 3 1->3 2->3 5 5 2->5 6 6 2->6 2->7 4 4 3->4 4->5 5->6
A undigraph
(0) 2-5-6 2-5-6 (0) 2-5-6^(2) 2-3-5 2-5 (0) 2-5-6--(0) 2-5-6^(2) 2-3-5 (1) 3-4-5 3-4-5 (1) 3-4-5^(2) 2-3-5 3-5 (1) 3-4-5--(1) 3-4-5^(2) 2-3-5 (2) 2-3-5 2-3-5 (2) 2-3-5^(3) 1-2-3 2-3 (2) 2-3-5--(2) 2-3-5^(3) 1-2-3 (3) 1-2-3 1-2-3 (3) 1-2-3^(4) 1-2-7 1-2 (3) 1-2-3--(3) 1-2-3^(4) 1-2-7 (4) 1-2-7 1-2-7 (4) 1-2-7^(5) 0-1-7 1-7 (4) 1-2-7--(4) 1-2-7^(5) 0-1-7 (5) 0-1-7 0-1-7 (2) 2-3-5^(3) 1-2-3--(3) 1-2-3 (4) 1-2-7^(5) 0-1-7--(5) 0-1-7 (0) 2-5-6^(2) 2-3-5--(2) 2-3-5 (1) 3-4-5^(2) 2-3-5--(2) 2-3-5 (3) 1-2-3^(4) 1-2-7--(4) 1-2-7
a junction tree for this undigraph
[6, 4, 5, 3, 2, 1, 7, 0]
its elimination order
(0) 2-5-6 2-5-6 (0) 2-5-6^(2) 2-3-5 2-5 (0) 2-5-6--(0) 2-5-6^(2) 2-3-5 (1) 3-4-5 3-4-5 (1) 3-4-5^(2) 2-3-5 3-5 (1) 3-4-5--(1) 3-4-5^(2) 2-3-5 (2) 2-3-5 2-3-5 (2) 2-3-5^(3) 1-2-3 2-3 (2) 2-3-5--(2) 2-3-5^(3) 1-2-3 (3) 1-2-3 1-2-3 (3) 1-2-3^(4) 1-2-7 1-2 (3) 1-2-3--(3) 1-2-3^(4) 1-2-7 (4) 1-2-7 1-2-7 (4) 1-2-7^(5) 0-1-7 1-7 (4) 1-2-7--(4) 1-2-7^(5) 0-1-7 (5) 0-1-7 0-1-7 (2) 2-3-5^(3) 1-2-3--(3) 1-2-3 (4) 1-2-7^(5) 0-1-7--(5) 0-1-7 (0) 2-5-6^(2) 2-3-5--(2) 2-3-5 (1) 3-4-5^(2) 2-3-5--(2) 2-3-5 (3) 1-2-3^(4) 1-2-7--(4) 1-2-7
an (optipmized) binary jointree

Using partial order to specify the elimination order

Section titled “Using partial order to specify the elimination order”
## adding a partial order for the elimination order
po = [[1, 2, 3], [0, 4, 7], [5, 6]]
gnb.sideBySide(
  bn,
  jtg.junctionTree(bn, po),
  jtg.eliminationOrder(bn, po),
  jtg.binaryJoinTree(bn),
  captions=[
    "A Bayesien network",
    "a junction tree for this BN using partial order",
    "its elimination order following partial order",
    "an (optimized) binary join tree",
  ],
)
G 3 3 2 2 3->2 4 4 3->4 0 0 1 1 0->1 1->2 7 7 2->7 6 6 2->6 5 5 5->6 4->5
A Bayesien network
(0) 1-2-3-4 1-2-3-4 (0) 1-2-3-4^(1) 0-1-2-4 1-2-4 (0) 1-2-3-4--(0) 1-2-3-4^(1) 0-1-2-4 (1) 0-1-2-4 0-1-2-4 (1) 0-1-2-4^(2) 0-2-4-5-6-7 0-2-4 (1) 0-1-2-4--(1) 0-1-2-4^(2) 0-2-4-5-6-7 (2) 0-2-4-5-6-7 0-2-4-5-6-7 (0) 1-2-3-4^(1) 0-1-2-4--(1) 0-1-2-4 (1) 0-1-2-4^(2) 0-2-4-5-6-7--(2) 0-2-4-5-6-7
a junction tree for this BN using partial order
[3, 1, 2, 7, 0, 4, 6, 5]
its elimination order following partial order
(0) 2-5-6 2-5-6 (0) 2-5-6^(6) 2-5 2-5 (0) 2-5-6--(0) 2-5-6^(6) 2-5 (1) 2-7 2-7 (1) 2-7^(6) 2-5 2 (1) 2-7--(1) 2-7^(6) 2-5 (2) 0-1 0-1 (2) 0-1^(3) 1-2-3 1 (2) 0-1--(2) 0-1^(3) 1-2-3 (3) 1-2-3 1-2-3 (3) 1-2-3^(4) 2-3-5 2-3 (3) 1-2-3--(3) 1-2-3^(4) 2-3-5 (4) 2-3-5 2-3-5 (4) 2-3-5^(5) 3-4-5 3-5 (4) 2-3-5--(4) 2-3-5^(5) 3-4-5 (4) 2-3-5^(6) 2-5 2-5 (4) 2-3-5--(4) 2-3-5^(6) 2-5 (5) 3-4-5 3-4-5 (6) 2-5 2-5 (2) 0-1^(3) 1-2-3--(3) 1-2-3 (4) 2-3-5^(5) 3-4-5--(5) 3-4-5 (3) 1-2-3^(4) 2-3-5--(4) 2-3-5 (1) 2-7^(6) 2-5--(6) 2-5 (0) 2-5-6^(6) 2-5--(6) 2-5 (4) 2-3-5^(6) 2-5--(6) 2-5
an (optimized) binary join tree
## adding a partial order for the elimination order also for the graphs
po = [[0, 4, 7], [1, 3], [5, 6, 2]]


## creating a dag slightly different
dag = bn.dag()
dag.addArc(0, 3)
dag.addArc(0, 7)


gnb.sideBySide(
  dag,
  dag.moralGraph(),
  jtg.junctionTree(dag, po),
  jtg.eliminationOrder(dag, po),
  jtg.binaryJoinTree(dag, po),
  captions=[
    "A DAG",
    "its moral graph",
    "a junction tree for this dag (with partial order)",
    "its elimination order (with partial order)",
    "an (optimized) binary jointree (with partial order)",
  ],
)
0 0 1 1 0->1 3 3 0->3 7 7 0->7 2 2 1->2 6 6 2->6 2->7 3->2 4 4 3->4 5 5 4->5 5->6
A DAG
no_name 0 0 1 1 0->1 2 2 0->2 3 3 0->3 7 7 0->7 1->2 1->3 2->3 5 5 2->5 6 6 2->6 2->7 4 4 3->4 4->5 5->6
its moral graph
(0) 0-2-7 0-2-7 (0) 0-2-7^(1) 0-1-2-3 0-2 (0) 0-2-7--(0) 0-2-7^(1) 0-1-2-3 (1) 0-1-2-3 0-1-2-3 (1) 0-1-2-3^(4) 2-3-5 2-3 (1) 0-1-2-3--(1) 0-1-2-3^(4) 2-3-5 (2) 3-4-5 3-4-5 (2) 3-4-5^(4) 2-3-5 3-5 (2) 3-4-5--(2) 3-4-5^(4) 2-3-5 (4) 2-3-5 2-3-5 (4) 2-3-5^(5) 2-5-6 2-5 (4) 2-3-5--(4) 2-3-5^(5) 2-5-6 (5) 2-5-6 2-5-6 (0) 0-2-7^(1) 0-1-2-3--(1) 0-1-2-3 (4) 2-3-5^(5) 2-5-6--(5) 2-5-6 (2) 3-4-5^(4) 2-3-5--(4) 2-3-5 (1) 0-1-2-3^(4) 2-3-5--(4) 2-3-5
a junction tree for this dag (with partial order)
[7, 0, 4, 1, 3, 2, 6, 5]
its elimination order (with partial order)
(0) 0-2-7 0-2-7 (0) 0-2-7^(1) 0-1-2-3 0-2 (0) 0-2-7--(0) 0-2-7^(1) 0-1-2-3 (1) 0-1-2-3 0-1-2-3 (1) 0-1-2-3^(4) 2-3-5 2-3 (1) 0-1-2-3--(1) 0-1-2-3^(4) 2-3-5 (2) 3-4-5 3-4-5 (2) 3-4-5^(4) 2-3-5 3-5 (2) 3-4-5--(2) 3-4-5^(4) 2-3-5 (4) 2-3-5 2-3-5 (4) 2-3-5^(5) 2-5-6 2-5 (4) 2-3-5--(4) 2-3-5^(5) 2-5-6 (5) 2-5-6 2-5-6 (0) 0-2-7^(1) 0-1-2-3--(1) 0-1-2-3 (4) 2-3-5^(5) 2-5-6--(5) 2-5-6 (2) 3-4-5^(4) 2-3-5--(4) 2-3-5 (1) 0-1-2-3^(4) 2-3-5--(4) 2-3-5
an (optimized) binary jointree (with partial order)