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