SLIDE 1
Relationship between the directed & undirected models
Graphical Models Graphical Models
Siamak Ravanbakhsh Winter 2018
Two directions Two directions
Markov network Bayes-net
⇒
Markov network Bayes-net
Graphical Models Graphical Models Relationship between the directed - - PowerPoint PPT Presentation
Graphical Models Graphical Models Relationship between the directed - - PowerPoint PPT Presentation
Graphical Models Graphical Models Relationship between the directed & undirected models Siamak Ravanbakhsh Winter 2018 Two directions Two directions Markov network Bayes-net Markov network Bayes-net
SLIDE 2
SLIDE 3
From From Bayesian Bayesian to to Markov Markov networks networks
build an I-map for the following
G1 G2 G3
SLIDE 4
From From Bayesian Bayesian to to Markov Markov networks networks
build an I-map for the following
I(M[G ]) ⊆ I(G )
3 3
G1
I(M[G ]) = I(G )
1 1
G2 G3
moralized
SLIDE 5
From From Bayesian Bayesian to to Markov Markov networks networks
build an I-map for the following
I(M[G ]) ⊆ I(G )
3 3
G1
I(M[G ]) = I(G )
1 1
G2 G3
I(M[G ]) = I(G )
3 3
G4
moralized
SLIDE 6
From From Bayesian Bayesian to to Markov Markov networks networks
build an I-map for the following
I(M[G ]) ⊆ I(G )
3 3
G1
I(M[G ]) = I(G )
1 1
G2 G3
I(M[G ]) = I(G )
3 3
G4
moralize & keep the skeleton
moralized
SLIDE 7
From From Bayesian Bayesian to to Markov Markov networks networks
for moral , we get a perfect map directed and undirected CI tests are equivalent G
M[G]
G
I(M[G]) = I(G)
moralize & keep the skeleton
SLIDE 8
G
From From Bayesian Bayesian to to Markov Markov networks networks
connect each node to its Markov blanket
children + parents + parents of children
in both directed and undirected models X ⊥ every other var. ∣ MB(X )
i i
SLIDE 9
G
M[G]
From From Bayesian Bayesian to to Markov Markov networks networks
connect each node to its Markov blanket
children + parents + parents of children
in both directed and undirected models gives the same moralized graph X ⊥ every other var. ∣ MB(X )
i i
SLIDE 10
From From Markov Markov to to Bayesian Bayesian networks networks
G1
I(G ) = I(G ) = I(H)
1 2
H G2
minimal examples 1.
SLIDE 11
From From Markov Markov to to Bayesian Bayesian networks networks
G1
I(G ) = I(G ) = I(H)
1 2
H G2
minimal examples 1. minimal examples 2.
H G
I(G) = I(H)
SLIDE 12
From From Markov Markov to to Bayesian Bayesian networks networks
minimal examples 3.
A B C D
SLIDE 13
From From Markov Markov to to Bayesian Bayesian networks networks
minimal examples 3.
I(G) ⊂ I(H)
A B C D
B ⊥ C ∣ A
SLIDE 14
H
From From Markov Markov to to Bayesian Bayesian networks networks
minimal examples 3. examples 4.
I(G) ⊂ I(H)
A B C D
B ⊥ C ∣ A
G
I(G) ⊂ I(H)
SLIDE 15
From From Markov Markov to to Bayesian Bayesian networks networks
H
build a minimal Imap from CIs in : pick an ordering e.g., A,B,C,...,F select a minimal parent set
H
examples 4.
G
I(G) ⊂ I(H)
have to triangulate the loops
G
therefore, is chordal loops of size >3 have chords
SLIDE 16
From From Markov Markov to to Bayesian Bayesian networks networks
cannot have any immoralities I(G) ⊆ I(H) ⇒ G any non-triangulated loop of size 4 (or more) will have immoralities
G
therefore, is chordal loops of size >3 have chords ? alternatively
SLIDE 17
Chordal = Chordal = Markov Markov Bayesian Bayesian networks networks
is not chordal, then for every no perfect MAP in the form of Bayes-net I(G) ≠ I(H)
∩
is chordal, then for some has a Bayes-net perfect map H G G H I(G) = I(H)
SLIDE 18
directed directed
parameter-estimation is easy can represent causal relations better for encoding expert domain knowledge
undirected undirected
simpler CI semantics less interpretable form for local factors less restrictive in structural form (loops)
SLIDE 19
Chordal graphs = Markov Bayesian networks P-maps in both directions Directed to undirected: moralize Undirected to directed: the result will be chordal