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# 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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