1. Probabilistic
Graphical
Models
Independencies
Preliminaries

2. Independence For events , , P    if:
P(, ) = P() P()
P(|) = P()
P(|) = P()
For random variables X,Y, P X  Y if:
P(X, Y) = P(X) P(Y)
P(X|Y) = P(X)
P(Y|X) = P(Y)
forall x,y P(x,y)=P(x)P(y)
equivalently, equality of factors

3. Independence P(I,D) I D G I D G Prob. i0 d0 g1 0.126 g2 0.168 g3 d1
0.009
0.045 i1
0.252
0.0224
0.0056
0.06
0.036
0.024 I
Prob i0
0.6 i1
0.4
P(I,D) I D
Prob i0 d0
0.42 d1
0.18 i1
0.28
0.12 D
Prob d0
0.7 d1
0.3
Equality of factors

4. Conditional Independence
For (sets of) random variables X,Y,Z
P (X  Y | Z) if:
P(X, Y|Z) = P(X|Z) P(Y|Z)
P(X|Y,Z) = P(X|Z)
P(Y|X,Z) = P(X|Z)
P(X,Y,Z)  1(X,Y) 2(Y,Z)
forall x,y P(x,y|z)=P(x|z)P(y|z)

5. Conditional Independence
Coin X1 X2

6. Conditional IndependenceG S I S G
Prob. i0 s0 g1
0.126 g2
0.168 g3 s1
0.009
0.045 i1
0.252
0.0224
0.0056
0.06
0.036
0.024
P(S,G | i0) S
Prob s0
0.95 s1
0.05 S G
Prob. s0 g1
0.19 g2
0.323 g3
0.437 s1
0.01
0.017
0.023 G
Prob. g1
0.2 g2
0.34 g3
0.46

7. Conditioning can Lose Independences
Prob. i0 d0 g1
0.126 g2
0.168 g3 d1
0.009
0.045 i1
0.252
0.0224
0.0056
0.06
0.036
0.024
P(I,D | g1) I D
Prob. i0 d0
0.282 d1
0.02 i1
0.564
0.134

8. END END END