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