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Lec 7: April 18th, 2006 EE512 - Graphical Models - J. Bilmes Page 1
University of WashingtonDepartment of Electrical Engineering
EE512 Spring, 2006 Graphical Models
Jeff A. Bilmes <[email protected]>Jeff A. Bilmes <[email protected]>
Lecture 7 Slides
April 18th, 2006
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• If you see a typo, please tell me during lecture– everyone will then benefit.– note, corrected slides will go on web.
• READING: – Chapter 3 & 17 in Jordan’s book– Lauritzen chapters 1-3 (on reserve in library)– Möbius Inversion Lemma handout (to be on web site)
• Reminder: TA discussions and office hours:– Office hours: Thursdays 3:30-4:30, Sieg Ground Floor
Tutorial Center– Discussion Sections: Fridays 9:30-10:30, Sieg Ground Floor
Tutorial Center Lecture Room
• Reminder: take-home Midterm: May 5th-8th, you must work alone on this.
Announcements
Lec 7: April 18th, 2006 EE512 - Graphical Models - J. Bilmes Page 3
• L1: Tues, 3/28: Overview, GMs, Intro BNs.• L2: Thur, 3/30: semantics of BNs + UGMs• L3: Tues, 4/4: elimination, probs, chordal I• L4: Thur, 4/6: chrdal, sep, decomp, elim• L5: Tue, 4/11: chdl/elim, mcs, triang, ci props.• L6: Thur, 4/13: MST,CI axioms, Markov prps.• L7: Tues, 4/18: Mobius, HC-thm, (F)=(G)• L8: Thur, 4/20• L9: Tue, 4/25• L10: Thur, 4/27
• L11: Tues, 5/2• L12: Thur, 5/4• L13: Tues, 5/9• L14: Thur, 5/11• L15: Tue, 5/16• L16: Thur, 5/18• L17: Tues, 5/23• L18: Thur, 5/25• L19: Tue, 5/30• L20: Thur, 6/1: final presentations
Class Road Map
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• L1: Tues, 3/28: • L2: Thur, 3/30:• L3: Tues, 4/4: • L4: Thur, 4/6:• L5: Tue, 4/11:• L6: Thur, 4/13:• L7: Tues, 4/18: Today• L8: Thur, 4/20: Team Lists, short abstracts I• L9: Tue, 4/25:• L10: Thur, 4/27: short abstracts II
• L11: Tues, 5/2• L12: Thur, 5/4: abstract II + progress• L13: Tues, 5/9• L14: Thur, 5/11: 1 page progress report• L15: Tue, 5/16• L16: Thur, 5/18: 1 page progress report• L17: Tues, 5/23• L18: Thur, 5/25: 1 page progress report• L19: Tue, 5/30• L20: Thur, 6/1: final presentations
• L21: Tue, 6/6 4-page papers due (like a conference paper).
Final Project Milestone Due Dates
• Team lists, abstracts, and progress reports must be turned in, in class and using paper (dead tree versions only).
• Final reports must be turned in electronically in PDF (no other formats accepted).
• Progress reports must report who did what so far!!
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• when are trees of maxcliques JTs?• max/min spanning trees• conditional independence relations• logical axioms of conditional independence relations• axioms and positivity• independence and knowledge• independence and separation• completeness conjecture• Markov properties on MRFs, (G),(L),(P)
Summary of Last Time
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• Factorization property on MRF, (F)• When (F) = (G) = (L) = (P)
• inclusion-exclusion
• Möbius Inversion lemma
• Hammersley/Clifford theorem, when (G) => (F)
• Factorization and decomposability• Factorization and junction tree• Directed factorization (DF), and (G)• Markov blanket• Bayesian networks and moralization
Outline of Today’s Lecture
Lec 7: April 18th, 2006 EE512 - Graphical Models - J. Bilmes Page 7
Books and Sources for Today
• M. Jordan: Chapters 17.• S. Lauritzen, 1996. Chapters 1-3.• J. Pearl, Probabilistic Reasoning in Intelligent Systems:
Networks of Plausible Inference, 1988.• Any good graph theory text.
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Properties of Markov Properties
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The alphabetical theorem: (F) (G) (L) (P)
Lec 7: April 18th, 2006 EE512 - Graphical Models - J. Bilmes Page 13
The alphabetical theorem: (F) (G) (L) (P)
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The equivalence theorem: (F) (G) (L) (P)
Lec 7: April 18th, 2006 EE512 - Graphical Models - J. Bilmes Page 23
Hammersley/Clifford
by pairwise Markov property
since we have unity
ratios
pairwise Markov
property and chain rule
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Factorization and decomposability
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Factorization and decomposability
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(G), factorization, and decomposability
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Recursive application + positivity
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Recursive application + positivity
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Recall from Lecture 3: Ancestral Sets
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Preservation of (DF) in ancestral sets