Markov Networks Electronics and Communication Engineering (ECE) Notes | EduRev

Electronics and Communication Engineering (ECE) : Markov Networks Electronics and Communication Engineering (ECE) Notes | EduRev

 Page 1


Markov Networks 
Page 2


Markov Networks 
Markov Networks 
? Undirected graphical models 
B 
D C 
A 
1
() ()
c
c
PX X
Z
= F
?
3.7   if A and B
( , ) 2.1   if A and B 
0.7  otherwise
2.3   if B and C and D
(, , )
5.1  otherwise
A B
BC D
?
?
F=
?
?
?
?
F=
?
?
()
c
Xc
ZX = F
??
? Potential functions defined over cliques 
Page 3


Markov Networks 
Markov Networks 
? Undirected graphical models 
B 
D C 
A 
1
() ()
c
c
PX X
Z
= F
?
3.7   if A and B
( , ) 2.1   if A and B 
0.7  otherwise
2.3   if B and C and D
(, , )
5.1  otherwise
A B
BC D
?
?
F=
?
?
?
?
F=
?
?
()
c
Xc
ZX = F
??
? Potential functions defined over cliques 
Markov Networks 
? Undirected graphical models 
B 
D C 
A 
? Potential functions defined over cliques 
Weight of Feature i Feature i 
exp ( )
i i
Xi
Z wf X
??
=
??
??
??
1
() exp ()
i i
i
P X wf X
Z
??
=
??
??
?
1   if A and B 
(, )
0  otherwise
1   if B and C and D
(, , )
0  
f A B
f BC D
?
=
?
?
?
=
?
?
Page 4


Markov Networks 
Markov Networks 
? Undirected graphical models 
B 
D C 
A 
1
() ()
c
c
PX X
Z
= F
?
3.7   if A and B
( , ) 2.1   if A and B 
0.7  otherwise
2.3   if B and C and D
(, , )
5.1  otherwise
A B
BC D
?
?
F=
?
?
?
?
F=
?
?
()
c
Xc
ZX = F
??
? Potential functions defined over cliques 
Markov Networks 
? Undirected graphical models 
B 
D C 
A 
? Potential functions defined over cliques 
Weight of Feature i Feature i 
exp ( )
i i
Xi
Z wf X
??
=
??
??
??
1
() exp ()
i i
i
P X wf X
Z
??
=
??
??
?
1   if A and B 
(, )
0  otherwise
1   if B and C and D
(, , )
0  
f A B
f BC D
?
=
?
?
?
=
?
?
Hammersley-Clifford Theorem 
If Distribution is strictly positive (P(x) > 0) 
And Graph encodes conditional independences 
Then Distribution is product of potentials over       
  cliques of graph 
 
Inverse is also true. 
(“Markov network = Gibbs distribution”) 
Page 5


Markov Networks 
Markov Networks 
? Undirected graphical models 
B 
D C 
A 
1
() ()
c
c
PX X
Z
= F
?
3.7   if A and B
( , ) 2.1   if A and B 
0.7  otherwise
2.3   if B and C and D
(, , )
5.1  otherwise
A B
BC D
?
?
F=
?
?
?
?
F=
?
?
()
c
Xc
ZX = F
??
? Potential functions defined over cliques 
Markov Networks 
? Undirected graphical models 
B 
D C 
A 
? Potential functions defined over cliques 
Weight of Feature i Feature i 
exp ( )
i i
Xi
Z wf X
??
=
??
??
??
1
() exp ()
i i
i
P X wf X
Z
??
=
??
??
?
1   if A and B 
(, )
0  otherwise
1   if B and C and D
(, , )
0  
f A B
f BC D
?
=
?
?
?
=
?
?
Hammersley-Clifford Theorem 
If Distribution is strictly positive (P(x) > 0) 
And Graph encodes conditional independences 
Then Distribution is product of potentials over       
  cliques of graph 
 
Inverse is also true. 
(“Markov network = Gibbs distribution”) 
Markov Nets vs. Bayes Nets 
Property Markov Nets Bayes Nets 
Form Prod. potentials Prod. potentials 
Potentials Arbitrary Cond. probabilities 
Cycles Allowed Forbidden 
Partition func. Z = ? Z = 1 
Indep. check Graph separation D-separation 
Indep. props. Some Some 
Inference MCMC, BP, etc. Convert to Markov 
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