![Page 1: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/1.jpg)
Relationship between the directed & undirected models
Graphical ModelsGraphical Models
Siamak Ravanbakhsh Winter 2018
![Page 2: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/2.jpg)
Two directionsTwo directions
Markov network Bayes-net
⇒Markov network Bayes-net
⇐
![Page 3: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/3.jpg)
From From BayesianBayesian to to MarkovMarkov networks networks
build an I-map for the following
G1 G2 G3
![Page 4: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/4.jpg)
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
![Page 5: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/5.jpg)
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
![Page 6: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/6.jpg)
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
![Page 7: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/7.jpg)
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
![Page 8: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/8.jpg)
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
![Page 9: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/9.jpg)
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
![Page 10: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/10.jpg)
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
G1
I(G ) = I(G ) = I(H)1 2
H G2
minimal examples 1.
![Page 11: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/11.jpg)
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)
![Page 12: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/12.jpg)
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
minimal examples 3.
A
BC
D
![Page 13: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/13.jpg)
From From MarkovMarkov to to Bayesian Bayesian networksnetworks
minimal examples 3.
I(G) ⊂ I(H)
A
BC
D
B ⊥ C ∣ A
![Page 14: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/14.jpg)
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)
![Page 15: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/15.jpg)
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
![Page 16: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/16.jpg)
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
![Page 17: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/17.jpg)
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)
![Page 18: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/18.jpg)
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)
![Page 19: Relationship between the directed & undirected models ...siamakx/532R/slides/converting_directed... · Markov network Bayes-net ⇒ Markov network ... we get a perfect map directed](https://reader033.vdocuments.mx/reader033/viewer/2022060600/6053edcbde130a20361fe946/html5/thumbnails/19.jpg)
Chordal graphs = Markov Bayesian networksP-maps in both directions
Directed to undirected: moralize
Undirected to directed:the result will be chordal
∩
SummarySummary