Relationship between the directed & undirected models...
Transcript of Relationship between the directed & undirected models...
Relationship between the directed & undirected models
Graphical ModelsGraphical Models
Siamak Ravanbakhsh Winter 2018
Two directionsTwo directions
Markov network Bayes-net
⇒Markov network Bayes-net
⇐
From From BayesianBayesian to to MarkovMarkov networks networks
build an I-map for the following
G1 G2 G3
From From BayesianBayesian to to MarkovMarkov 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
From From BayesianBayesian to to MarkovMarkov 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
From From BayesianBayesian to to MarkovMarkov 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
From From BayesianBayesian to to MarkovMarkov 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
G
From From BayesianBayesian to to MarkovMarkov networks networks
connect each node to its Markov blanket
children +parents +
parents of children
in both directed and undirected modelsX ⊥ every other var. ∣MB(X )i i
G M[G]
From From BayesianBayesian to to MarkovMarkov 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
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
G1
I(G ) = I(G ) = I(H)1 2
H G2
minimal examples 1.
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
G1
I(G ) = I(G ) = I(H)1 2
H G2
minimal examples 1.
minimal examples 2.
H G
I(G) = I(H)
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
minimal examples 3.
A
BC
D
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
minimal examples 3.
I(G) ⊂ I(H)
A
BC
D
B ⊥ C ∣ A
H
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
minimal examples 3.
examples 4.I(G) ⊂ I(H)
A
BC
D
B ⊥ C ∣ A
GI(G) ⊂ I(H)
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
H
build a minimal Imap from CIs in :
pick an ordering e.g., A,B,C,...,F
select a minimal parent set
Hexamples 4.
GI(G) ⊂ I(H)
have to triangulate the loopsGtherefore, is chordal
loops of size >3 have chords
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
cannot have any immoralitiesI(G) ⊆ I(H) ⇒ G
any non-triangulated loop of size 4 (or more) will have immoralities
Gtherefore, is chordal
loops of size >3 have chords
?
alternatively
Chordal = Chordal = Markov Markov Bayesian Bayesian networksnetworks
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
GH I(G) = I(H)
directeddirectedparameter-estimation is easy
can represent causal relations
better for encoding expert
domain knowledge
undirectedundirectedsimpler CI semantics
less interpretable form for local factors
less restrictive in structural form (loops)
Chordal graphs = Markov Bayesian networksP-maps in both directions
Directed to undirected: moralize
Undirected to directed:the result will be chordal
∩
SummarySummary