Markov networksΒΆ

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aGrUM

interactive online version

In [1]:
%load_ext autoreload
%autoreload 2

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.lib.mn2graph as m2g

building a Markov NetworkΒΆ

In [2]:
gum.config.reset() # back to default
mn=gum.fastMN("A--B--C;C--D;B--E--F;F--D--G;H--J;E--A;J")
mn
Out[2]:
G B B J J D D G G C C A A F F E E H H f0#1#2 f0#1#2--B f0#1#2--C f0#1#2--A f2#3 f2#3--D f2#3--C f7#8 f7#8--J f7#8--H f1#4#5 f1#4#5--B f1#4#5--F f1#4#5--E f3#5#6 f3#5#6--D f3#5#6--G f3#5#6--F f0#4 f0#4--A f0#4--E f8 f8--J

Using pyAgrum.config, it is possible to adapt the graphical representations for Markov Network (see 99-Tools_configForPyAgrum.ipynb ).

In [3]:
gum.config.reset() # back to default
gum.config['factorgraph','edge_length']='0.4'
gnb.showMN(mn)
../_images/notebooks_23-Models_MarkovNetwork_6_0.svg
In [4]:
gum.config.reset() # back to default
print("Default view for Markov network: "+gum.config['notebook','default_markovnetwork_view'])
gum.config['notebook','default_markovnetwork_view']='graph'
print("modified to: "+gum.config['notebook','default_markovnetwork_view'])
mn
Default view for Markov network: factorgraph
modified to: graph
Out[4]:
G B B C C B--C F F B--F E E B--E J J D D G G D--G D--F C--D A A A--B A--C A--E F--G E--F H H H--J
In [5]:
gnb.sideBySide(gnb.getMN(mn,view="graph",size="5"),
               gnb.getMN(mn,view="factorgraph",size="5"))
G B B C C B--C F F B--F E E B--E J J D D G G D--G D--F C--D A A A--B A--C A--E F--G E--F H H H--J
G B B J J D D G G C C A A F F E E H H f0#1#2 f0#1#2--B f0#1#2--C f0#1#2--A f2#3 f2#3--D f2#3--C f7#8 f7#8--J f7#8--H f1#4#5 f1#4#5--B f1#4#5--F f1#4#5--E f3#5#6 f3#5#6--D f3#5#6--G f3#5#6--F f0#4 f0#4--A f0#4--E f8 f8--J
In [6]:
gnb.showMN(mn)
print(mn)
../_images/notebooks_23-Models_MarkovNetwork_9_0.svg
MN{nodes: 9, edges: 12, domainSize: 512, dim: 38}

Accessors for Markov NetworksΒΆ

In [7]:
print(f"nodes       : {mn.nodes()}")
print(f"node names  : {mn.names()}")
print(f"edges       : {mn.edges()}")
print(f"components  : {mn.connectedComponents()}")
print(f"factors     : {mn.factors()}")
print(f"factor(C,D) : {mn.factor({2,3})}")
print(f"factor(C,D) : {mn.factor({'C','D'})}")
print(f"factor(C,D) : {mn.factor({'D','C'})}")
nodes       : {0, 1, 2, 3, 4, 5, 6, 7, 8}
node names  : {'B', 'J', 'D', 'G', 'C', 'A', 'F', 'E', 'H'}
edges       : {(0, 1), (1, 2), (0, 4), (1, 5), (1, 4), (2, 3), (4, 5), (0, 2), (5, 6), (7, 8), (3, 6), (3, 5)}
components  : {0: {0, 1, 2, 3, 4, 5, 6}, 7: {8, 7}}
factors     : [{0, 1, 2}, {2, 3}, {8, 7}, {1, 4, 5}, {3, 5, 6}, {0, 4}, {8}]
factor(C,D) :
      ||  C                |
D     ||0        |1        |
------||---------|---------|
0     || 0.9965  | 0.9677  |
1     || 0.7258  | 0.9811  |

factor(C,D) :
      ||  C                |
D     ||0        |1        |
------||---------|---------|
0     || 0.9965  | 0.9677  |
1     || 0.7258  | 0.9811  |

factor(C,D) :
      ||  C                |
D     ||0        |1        |
------||---------|---------|
0     || 0.9965  | 0.9677  |
1     || 0.7258  | 0.9811  |

In [8]:
try:
    mn.factor({0,1})
except gum.GumException as e:
    print(e)
try:
    mn.factor({"A","B"})
except gum.GumException as e:
    print(e)
[pyAgrum] Object not found: No element with the key <{1,0}>
[pyAgrum] Object not found: No element with the key <{1,0}>

Manipulating factorsΒΆ

In [9]:
mn.factor({'A','B','C'})
Out[9]:
A
C
B
0
1
0
0
0.13550.8350
1
0.96890.2210
1
0
0.30820.5472
1
0.18840.9929
In [10]:
mn.factor({'A','B','C'})[{'B':0}]
Out[10]:
array([[0.135477  , 0.83500859],
       [0.30816705, 0.5472206 ]])
In [11]:
mn.factor({'A','B','C'})[{'B':0}]=[[1,2],[3,4]]
mn.factor({'A','B','C'})
Out[11]:
A
C
B
0
1
0
0
1.00002.0000
1
0.96890.2210
1
0
3.00004.0000
1
0.18840.9929

Customizing graphical representationΒΆ

In [12]:
gum.config.reset() # back to default
gum.config['factorgraph','edge_length']='0.5'

maxnei=max([len(mn.neighbours(n)) for n in mn.nodes()])
nodemap={n:len(mn.neighbours(mn.idFromName(n)))/maxnei for n in mn.names()}

facmax=max([len(f) for f in mn.factors()])
fgma=lambda factor: (1+len(factor)**2)/(1+facmax*facmax)

gnb.flow.row(gnb.getGraph(m2g.MN2UGdot(mn)),
               gnb.getGraph(m2g.MN2UGdot(mn,nodeColor=nodemap)),
               gnb.getGraph(m2g.MN2FactorGraphdot(mn)),
               gnb.getGraph(m2g.MN2FactorGraphdot(mn,factorColor=fgma,nodeColor=nodemap)),
               captions=['Markov network',
                         'MarkovNet with colored node w.r.t number of neighbours',
                         'Markovnet as factor graph',
                         'MN with colored factor w.r.t to the size of scope'])
G B B C C B--C F F B--F E E B--E J J D D G G D--G D--F C--D A A A--B A--C A--E F--G E--F H H H--J
Markov network
G B B C C B--C F F B--F E E B--E J J D D G G D--G D--F C--D A A A--B A--C A--E F--G E--F H H H--J
MarkovNet with colored node w.r.t number of neighbours
G B B J J D D G G C C A A F F E E H H f0#1#2 f0#1#2--B f0#1#2--C f0#1#2--A f2#3 f2#3--D f2#3--C f7#8 f7#8--J f7#8--H f1#4#5 f1#4#5--B f1#4#5--F f1#4#5--E f3#5#6 f3#5#6--D f3#5#6--G f3#5#6--F f0#4 f0#4--A f0#4--E f8 f8--J
Markovnet as factor graph
G B B J J D D G G C C A A F F E E H H f0#1#2 f0#1#2--B f0#1#2--C f0#1#2--A f2#3 f2#3--D f2#3--C f7#8 f7#8--J f7#8--H f1#4#5 f1#4#5--B f1#4#5--F f1#4#5--E f3#5#6 f3#5#6--D f3#5#6--G f3#5#6--F f0#4 f0#4--A f0#4--E f8 f8--J
MN with colored factor w.r.t to the size of scope

from BayesNet to MarkovNetΒΆ

In [13]:
bn=gum.fastBN("A->B<-C->D->E->F<-B<-G;A->H->I;C->J<-K<-L")
mn=gum.MarkovNet.fromBN(bn)
gnb.flow.row(bn,
               gnb.getGraph(m2g.MN2UGdot(mn)),
               captions=['a Bayesian network',
                         'the corresponding Markov Network'])
G B B F F B->F J J D D E E D->E G G G->B C C C->B C->J C->D A A A->B H H A->H I I L L K K L->K K->J E->F H->I
a Bayesian network
G B B G G B--G C C B--C F F B--F E E B--E J J K K J--K D D D--E C--J C--D C--G C--K A A A--B A--G A--C H H A--H I I L L K--L E--F H--I
the corresponding Markov Network
In [14]:
# The corresponding factor graph
m2g.MN2FactorGraphdot(mn)
Out[14]:
G B B J J D D G G C C A A I I L L K K F F E E H H f0#1#2#6 f0#1#2#6--B f0#1#2#6--G f0#1#2#6--C f0#1#2#6--A f2#3 f2#3--D f2#3--C f3#4 f3#4--D f3#4--E f0#7 f0#7--A f0#7--H f7#8 f7#8--I f7#8--H f2#9#10 f2#9#10--J f2#9#10--C f2#9#10--K f10#11 f10#11--L f10#11--K f11 f11--L f0 f0--A f2 f2--C f1#4#5 f1#4#5--B f1#4#5--F f1#4#5--E f6 f6--G

Inference in Markov networkΒΆ

In [15]:
bn=gum.fastBN("A->B<-C->D->E->F<-B<-G;A->H->I;C->J<-K<-L")
iebn=gum.LazyPropagation(bn)

mn=gum.MarkovNet.fromBN(bn)
iemn=gum.ShaferShenoyMNInference(mn)
iemn.setEvidence({"A":1,"F":[0.4,0.8]})
iemn.makeInference()
iemn.posterior("B")
Out[15]:
B
0
1
0.61520.3848
In [16]:
def affAGC(evs):
    gnb.sideBySide(gnb.getSideBySide(gum.getPosterior(bn,target="A",evs=evs),
                                     gum.getPosterior(bn,target="G",evs=evs),
                                     gum.getPosterior(bn,target="C",evs=evs)),
                   gnb.getSideBySide(gum.getPosterior(mn,target="A",evs=evs),
                                     gum.getPosterior(mn,target="G",evs=evs),
                                     gum.getPosterior(mn,target="C",evs=evs)),
                   captions=["Inference in the Bayesian network bn with evidence "+str(evs),
                             "Inference in the markov network mn with evidence "+str(evs)]
                  )

print("Inference for both the corresponding models in BayesNet and MarkovNet worlds when the MN comes from a BN")
affAGC({})
print("C has no impact on A and G")
affAGC({'C':1})

print("But if B is observed")
affAGC({'B':1})
print("C has an impact on A and G")
affAGC({'B':1,'C':0})
Inference for both the corresponding models in BayesNet and MarkovNet worlds when the MN comes from a BN
A
0
1
0.65170.3483
G
0
1
0.26060.7394
C
0
1
0.54980.4502

Inference in the Bayesian network bn with evidence {}
A
0
1
0.65170.3483
G
0
1
0.26060.7394
C
0
1
0.54980.4502

Inference in the markov network mn with evidence {}
C has no impact on A and G
A
0
1
0.65170.3483
G
0
1
0.26060.7394
C
0
1
0.00001.0000

Inference in the Bayesian network bn with evidence {'C': 1}
A
0
1
0.65170.3483
G
0
1
0.26060.7394
C
0
1
0.00001.0000

Inference in the markov network mn with evidence {'C': 1}
But if B is observed
A
0
1
0.66510.3349
G
0
1
0.25360.7464
C
0
1
0.52620.4738

Inference in the Bayesian network bn with evidence {'B': 1}
A
0
1
0.66510.3349
G
0
1
0.25360.7464
C
0
1
0.52620.4738

Inference in the markov network mn with evidence {'B': 1}
C has an impact on A and G
A
0
1
0.64720.3528
G
0
1
0.17900.8210
C
0
1
1.00000.0000

Inference in the Bayesian network bn with evidence {'B': 1, 'C': 0}
A
0
1
0.64720.3528
G
0
1
0.17900.8210
C
0
1
1.00000.0000

Inference in the markov network mn with evidence {'B': 1, 'C': 0}
In [17]:
mn.generateFactors()
print("But with more general factors")
affAGC({})
print("C has impact on A and G even without knowing B")
affAGC({'C':1})

But with more general factors
A
0
1
0.65170.3483
G
0
1
0.26060.7394
C
0
1
0.54980.4502

Inference in the Bayesian network bn with evidence {}
A
0
1
0.88280.1172
G
0
1
0.39920.6008
C
0
1
0.15570.8443

Inference in the markov network mn with evidence {}
C has impact on A and G even without knowing B
A
0
1
0.65170.3483
G
0
1
0.26060.7394
C
0
1
0.00001.0000

Inference in the Bayesian network bn with evidence {'C': 1}
A
0
1
0.88160.1184
G
0
1
0.45330.5467
C
0
1
0.00001.0000

Inference in the markov network mn with evidence {'C': 1}

Graphical inference in markov networkΒΆ

In [18]:
bn=gum.fastBN("A->B<-C->D->E->F<-B<-G;A->H->I;C->J<-K<-L")
mn=gum.MarkovNet.fromBN(bn)

gnb.sideBySide(gnb.getJunctionTree(bn),gnb.getJunctionTree(mn),captions=["Junction tree for the BN","Junction tree for the induced MN"])
gnb.sideBySide(gnb.getJunctionTreeMap(bn,size="3!"),gnb.getJunctionTreeMap(mn,size="3!"),captions=["Map of the junction tree for the BN","Map of the junction tree for the induced MN"])
G (0) 8-7 I H (0) 8-7^(4) 7-0 H (0) 8-7--(0) 8-7^(4) 7-0 (1) 11-10 L K (1) 11-10^(3) 9-2-10 K (1) 11-10--(1) 11-10^(3) 9-2-10 (2) 6-1-0-2 G B A C (2) 6-1-0-2^(8) 1-4-2 B C (2) 6-1-0-2--(2) 6-1-0-2^(8) 1-4-2 (2) 6-1-0-2^(4) 7-0 A (2) 6-1-0-2--(2) 6-1-0-2^(4) 7-0 (3) 9-2-10 J C K (3) 9-2-10^(8) 1-4-2 C (3) 9-2-10--(3) 9-2-10^(8) 1-4-2 (4) 7-0 H A (6) 1-5-4 B F E (6) 1-5-4^(8) 1-4-2 B E (6) 1-5-4--(6) 1-5-4^(8) 1-4-2 (8) 1-4-2 B E C (8) 1-4-2^(9) 3-4-2 E C (8) 1-4-2--(8) 1-4-2^(9) 3-4-2 (9) 3-4-2 D E C (8) 1-4-2^(9) 3-4-2--(9) 3-4-2 (1) 11-10^(3) 9-2-10--(3) 9-2-10 (0) 8-7^(4) 7-0--(4) 7-0 (6) 1-5-4^(8) 1-4-2--(8) 1-4-2 (2) 6-1-0-2^(8) 1-4-2--(8) 1-4-2 (2) 6-1-0-2^(4) 7-0--(4) 7-0 (3) 9-2-10^(8) 1-4-2--(8) 1-4-2
Junction tree for the BN
G (0) 8-7 I H (0) 8-7^(4) 7-0 H (0) 8-7--(0) 8-7^(4) 7-0 (1) 11-10 L K (1) 11-10^(3) 9-2-10 K (1) 11-10--(1) 11-10^(3) 9-2-10 (2) 6-1-0-2 G B A C (2) 6-1-0-2^(8) 1-4-2 B C (2) 6-1-0-2--(2) 6-1-0-2^(8) 1-4-2 (2) 6-1-0-2^(4) 7-0 A (2) 6-1-0-2--(2) 6-1-0-2^(4) 7-0 (3) 9-2-10 J C K (3) 9-2-10^(8) 1-4-2 C (3) 9-2-10--(3) 9-2-10^(8) 1-4-2 (4) 7-0 H A (6) 1-5-4 B F E (6) 1-5-4^(8) 1-4-2 B E (6) 1-5-4--(6) 1-5-4^(8) 1-4-2 (8) 1-4-2 B E C (8) 1-4-2^(9) 3-4-2 E C (8) 1-4-2--(8) 1-4-2^(9) 3-4-2 (9) 3-4-2 D E C (8) 1-4-2^(9) 3-4-2--(9) 3-4-2 (1) 11-10^(3) 9-2-10--(3) 9-2-10 (0) 8-7^(4) 7-0--(4) 7-0 (6) 1-5-4^(8) 1-4-2--(8) 1-4-2 (2) 6-1-0-2^(8) 1-4-2--(8) 1-4-2 (2) 6-1-0-2^(4) 7-0--(4) 7-0 (3) 9-2-10^(8) 1-4-2--(8) 1-4-2
Junction tree for the induced MN
G 0 0~4 0--0~4 1 1~3 1--1~3 2 2~8 2--2~8 2~4 2--2~4 3 3~8 3--3~8 4 6 6~8 6--6~8 8 8~9 8--8~9 9 8~9--9 1~3--3 0~4--4 6~8--8 2~8--8 2~4--4 3~8--8
Map of the junction tree for the BN
G 0 0~4 0--0~4 1 1~3 1--1~3 2 2~8 2--2~8 2~4 2--2~4 3 3~8 3--3~8 4 6 6~8 6--6~8 8 8~9 8--8~9 9 8~9--9 1~3--3 0~4--4 6~8--8 2~8--8 2~4--4 3~8--8
Map of the junction tree for the induced MN
In [19]:
gnb.showInference(bn,evs={"D":1,"H":0})
../_images/notebooks_23-Models_MarkovNetwork_28_0.svg
In [20]:
gum.config.reset()
gnb.showInference(mn,size="8",evs={"D":1,"H":0})
../_images/notebooks_23-Models_MarkovNetwork_29_0.svg
In [21]:
gum.config['factorgraph','edge_length_inference']='1.1'
gnb.showInference(mn,size="11",evs={"D":1,"H":0})
../_images/notebooks_23-Models_MarkovNetwork_30_0.svg
In [22]:
gum.config['notebook','default_markovnetwork_view']='graph'
gnb.showInference(mn,size="8",evs={"D":1,"H":0})
../_images/notebooks_23-Models_MarkovNetwork_31_0.svg
In [ ]: