Markov random fields (a.k.a. Markov Networks)

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aGrUM

interactive online version

 
In [1]:

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.lib.mrf2graph as m2g

building a Markov random field

In [2]:

gum.config.reset() # back to default
gum.config['notebook','default_markovrandomfield_view']='graph'
mn=gum.fastMRF("A--B--C;C--D;B--E--F;F--D--G;H--J;E--A;J")
mn

Out[2]:
G J J E E F F E--F G G F--G D D D--F D--G H H H--J B B B--E B--F C C B--C A A A--E A--B A--C C--D

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

In [3]:

gum.config.reset() # back to default
gum.config['factorgraph','edge_length']='0.4'
gnb.showMRF(mn)
../_images/notebooks_23-Models_MarkovRandomField_6_0.svg 
In [4]:

gum.config.reset() # back to default
print("Default view for Markov random field: "+gum.config['notebook','default_markovrandomfield_view'])
gum.config['notebook','default_markovrandomfield_view']='graph'
print("modified to: "+gum.config['notebook','default_markovrandomfield_view'])
mn

Default view for Markov random field: factorgraph
modified to: graph

Out[4]:
G J J E E F F E--F G G F--G D D D--F D--G H H H--J B B B--E B--F C C B--C A A A--E A--B A--C C--D 
In [5]:

gnb.sideBySide(gnb.getMRF(mn,view="graph",size="5"),
               gnb.getMRF(mn,view="factorgraph",size="5"))
G J J E E F F E--F G G F--G D D D--F D--G H H H--J B B B--E B--F C C B--C A A A--E A--B A--C C--D
G J J E E F F D D H H B B A A C C G G f0#1#2 f0#1#2--B f0#1#2--A f0#1#2--C f2#3 f2#3--D f2#3--C f7#8 f7#8--J f7#8--H f1#4#5 f1#4#5--E f1#4#5--F f1#4#5--B f3#5#6 f3#5#6--F f3#5#6--D f3#5#6--G f0#4 f0#4--E f0#4--A f8 f8--J 
In [6]:

gnb.showMRF(mn)
print(mn)
../_images/notebooks_23-Models_MarkovRandomField_9_0.svg 
MRF{nodes: 9, edges: 12, domainSize: 512, dim: 38}

Accessors for Markov random fields

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'})}")

nodes       : {0, 1, 2, 3, 4, 5, 6, 7, 8}
node names  : {'J', 'E', 'F', 'D', 'H', 'B', 'A', 'C', 'G'}
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     || 0.3709  | 0.0939  |
1     || 0.0304  | 0.1919  |

factor(C,D) :
      ||  C                |
D     ||0        |1        |
------||---------|---------|
0     || 0.3709  | 0.0939  |
1     || 0.0304  | 0.1919  |


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 factors

In [9]:

mn.factor({'A','B','C'})

Out[9]:
A
C
B
0
1
0
0
0.44710.5408
1
0.34140.5997
1
0
0.41710.7707
1
0.70810.0170
In [10]:

mn.factor({'A','B','C'})[{'B':0}]

Out[10]:

array([[0.44714656, 0.54078863],
       [0.41709187, 0.77066469]])

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
0.34140.5997
1
0
3.00004.0000
1
0.70810.0170

Customizing graphical representation

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.MRF2UGdot(mn)),
               gnb.getGraph(m2g.MRF2UGdot(mn,nodeColor=nodemap)),
               gnb.getGraph(m2g.MRF2FactorGraphdot(mn)),
               gnb.getGraph(m2g.MRF2FactorGraphdot(mn,factorColor=fgma,nodeColor=nodemap)),
               captions=['Markov random field',
                         'MarkovRandomField with colored node w.r.t number of neighbours',
                         'MarkovRandomField as factor graph',
                         'MRF with colored factor w.r.t to the size of scope'])
G J J E E F F E--F G G F--G D D D--F D--G H H H--J B B B--E B--F C C B--C A A A--E A--B A--C C--D
Markov random field
G J J E E F F E--F G G F--G D D D--F D--G H H H--J B B B--E B--F C C B--C A A A--E A--B A--C C--D
MarkovRandomField with colored node w.r.t number of neighbours
G J J E E F F D D H H B B A A C C G G f0#1#2 f0#1#2--B f0#1#2--A f0#1#2--C f2#3 f2#3--D f2#3--C f7#8 f7#8--J f7#8--H f1#4#5 f1#4#5--E f1#4#5--F f1#4#5--B f3#5#6 f3#5#6--F f3#5#6--D f3#5#6--G f0#4 f0#4--E f0#4--A f8 f8--J
MarkovRandomField as factor graph
G J J E E F F D D H H B B A A C C G G f0#1#2 f0#1#2--B f0#1#2--A f0#1#2--C f2#3 f2#3--D f2#3--C f7#8 f7#8--J f7#8--H f1#4#5 f1#4#5--E f1#4#5--F f1#4#5--B f3#5#6 f3#5#6--F f3#5#6--D f3#5#6--G f0#4 f0#4--E f0#4--A f8 f8--J
MRF with colored factor w.r.t to the size of scope

from BayesNet to MarkovRandomField

In [13]:

bn=gum.fastBN("A->B<-C->D->E->F<-B<-G;A->H->I;C->J<-K<-L")
mn=gum.MarkovRandomField.fromBN(bn)
gnb.flow.row(bn,
               gnb.getGraph(m2g.MRF2UGdot(mn)),
               captions=['a Bayesian network',
                         'the corresponding Markov random field'])
G J J E E F F E->F K K K->J D D D->E H H I I H->I B B B->F A A A->H A->B C C C->J C->D C->B L L L->K G G G->B
a Bayesian network
G J J K K J--K E E F F E--F L L K--L D D D--E H H I I H--I B B B--E B--F C C B--C G G B--G A A A--H A--B A--C A--G C--J C--K C--D C--G
the corresponding Markov random field
In [14]:

# The corresponding factor graph
m2g.MRF2FactorGraphdot(mn)

Out[14]:
G J J E E F F K K D D H H I I B B A A C C L L G G f0#1#2#6 f0#1#2#6--B f0#1#2#6--A f0#1#2#6--C f0#1#2#6--G f2#3 f2#3--D f2#3--C f3#4 f3#4--E f3#4--D f0#7 f0#7--H f0#7--A f7#8 f7#8--H f7#8--I f2#9#10 f2#9#10--J f2#9#10--K f2#9#10--C f10#11 f10#11--K f10#11--L f11 f11--L f0 f0--A f2 f2--C f1#4#5 f1#4#5--E f1#4#5--F f1#4#5--B f6 f6--G

Inference in Markov random field

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.MarkovRandomField.fromBN(bn)
iemn=gum.ShaferShenoyMRFInference(mn)
iemn.setEvidence({"A":1,"F":[0.4,0.8]})
iemn.makeInference()
iemn.posterior("B")

Out[15]:
B
0
1
0.47910.5209
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 random field mn with evidence "+str(evs)]
                  )

print("Inference for both the corresponding models in BayesNet and Markoc Random Field worlds when the MRF 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 Markoc Random Field worlds when the MRF comes from a BN
A
0
1
0.69490.3051
G
0
1
0.14000.8600
C
0
1
0.14450.8555

Inference in the Bayesian network bn with evidence {}
A
0
1
0.69490.3051
G
0
1
0.14000.8600
C
0
1
0.14450.8555

Inference in the Markov random field mn with evidence {}
C has no impact on A and G
A
0
1
0.69490.3051
G
0
1
0.14000.8600
C
0
1
0.00001.0000

Inference in the Bayesian network bn with evidence {'C': 1}
A
0
1
0.69490.3051
G
0
1
0.14000.8600
C
0
1
0.00001.0000

Inference in the Markov random field mn with evidence {'C': 1}
But if B is observed
A
0
1
0.71820.2818
G
0
1
0.19280.8072
C
0
1
0.11430.8857

Inference in the Bayesian network bn with evidence {'B': 1}
A
0
1
0.71820.2818
G
0
1
0.19280.8072
C
0
1
0.11430.8857

Inference in the Markov random field mn with evidence {'B': 1}
C has an impact on A and G
A
0
1
0.96470.0353
G
0
1
0.05100.9490
C
0
1
1.00000.0000

Inference in the Bayesian network bn with evidence {'B': 1, 'C': 0}
A
0
1
0.96470.0353
G
0
1
0.05100.9490
C
0
1
1.00000.0000

Inference in the Markov random field 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
0.69490.3051
G
0
1
0.14000.8600
C
0
1
0.14450.8555

Inference in the Bayesian network bn with evidence {}
A
0
1
0.33870.6613
G
0
1
0.74500.2550
C
0
1
0.75360.2464

Inference in the Markov random field mn with evidence {}
C has impact on A and G even without knowing B
A
0
1
0.69490.3051
G
0
1
0.14000.8600
C
0
1
0.00001.0000

Inference in the Bayesian network bn with evidence {'C': 1}
A
0
1
0.47720.5228
G
0
1
0.57420.4258
C
0
1
0.00001.0000

Inference in the Markov random field mn with evidence {'C': 1}

Graphical inference in Markov random field

In [18]:

bn=gum.fastBN("A->B<-C->D->E->F<-B<-G;A->H->I;C->J<-K<-L")
mn=gum.MarkovRandomField.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) 7-8 H I (0) 7-8^(4) 0-7 H (0) 7-8--(0) 7-8^(4) 0-7 (1) 10-11 K L (1) 10-11^(3) 2-9-10 K (1) 10-11--(1) 10-11^(3) 2-9-10 (2) 0-1-2-6 A B C G (2) 0-1-2-6^(8) 1-2-4 B C (2) 0-1-2-6--(2) 0-1-2-6^(8) 1-2-4 (2) 0-1-2-6^(4) 0-7 A (2) 0-1-2-6--(2) 0-1-2-6^(4) 0-7 (3) 2-9-10 C J K (3) 2-9-10^(8) 1-2-4 C (3) 2-9-10--(3) 2-9-10^(8) 1-2-4 (4) 0-7 A H (6) 1-4-5 B E F (6) 1-4-5^(8) 1-2-4 B E (6) 1-4-5--(6) 1-4-5^(8) 1-2-4 (8) 1-2-4 B C E (8) 1-2-4^(9) 2-3-4 C E (8) 1-2-4--(8) 1-2-4^(9) 2-3-4 (9) 2-3-4 C D E (8) 1-2-4^(9) 2-3-4--(9) 2-3-4 (1) 10-11^(3) 2-9-10--(3) 2-9-10 (0) 7-8^(4) 0-7--(4) 0-7 (6) 1-4-5^(8) 1-2-4--(8) 1-2-4 (2) 0-1-2-6^(8) 1-2-4--(8) 1-2-4 (2) 0-1-2-6^(4) 0-7--(4) 0-7 (3) 2-9-10^(8) 1-2-4--(8) 1-2-4
Junction tree for the BN
G (0) 7-8 H I (0) 7-8^(4) 0-7 H (0) 7-8--(0) 7-8^(4) 0-7 (1) 10-11 K L (1) 10-11^(3) 2-9-10 K (1) 10-11--(1) 10-11^(3) 2-9-10 (2) 0-1-2-6 A B C G (2) 0-1-2-6^(8) 1-2-4 B C (2) 0-1-2-6--(2) 0-1-2-6^(8) 1-2-4 (2) 0-1-2-6^(4) 0-7 A (2) 0-1-2-6--(2) 0-1-2-6^(4) 0-7 (3) 2-9-10 C J K (3) 2-9-10^(8) 1-2-4 C (3) 2-9-10--(3) 2-9-10^(8) 1-2-4 (4) 0-7 A H (6) 1-4-5 B E F (6) 1-4-5^(8) 1-2-4 B E (6) 1-4-5--(6) 1-4-5^(8) 1-2-4 (8) 1-2-4 B C E (8) 1-2-4^(9) 2-3-4 C E (8) 1-2-4--(8) 1-2-4^(9) 2-3-4 (9) 2-3-4 C D E (8) 1-2-4^(9) 2-3-4--(9) 2-3-4 (1) 10-11^(3) 2-9-10--(3) 2-9-10 (0) 7-8^(4) 0-7--(4) 0-7 (6) 1-4-5^(8) 1-2-4--(8) 1-2-4 (2) 0-1-2-6^(8) 1-2-4--(8) 1-2-4 (2) 0-1-2-6^(4) 0-7--(4) 0-7 (3) 2-9-10^(8) 1-2-4--(8) 1-2-4
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_MarkovRandomField_28_0.svg 
In [20]:

gum.config.reset()
gnb.showInference(mn,size="8",evs={"D":1,"H":0})
../_images/notebooks_23-Models_MarkovRandomField_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_MarkovRandomField_30_0.svg 
In [22]:

gum.config['notebook','default_markovrandomfield_view']='graph'
gnb.showInference(mn,size="8",evs={"D":1,"H":0})
../_images/notebooks_23-Models_MarkovRandomField_31_0.svg