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Lecture 7 Slides April 18 th , 2006

University of Washington Department of Electrical Engineering EE512 Spring, 2006 Graphical Models Jeff A. Bilmes <bilmes@ee.washington.edu>. Lecture 7 Slides April 18 th , 2006. Announcements. If you see a typo, please tell me during lecture everyone will then benefit.

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Lecture 7 Slides April 18 th , 2006

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  1. University of WashingtonDepartment of Electrical Engineering EE512 Spring, 2006 Graphical ModelsJeff A. Bilmes <bilmes@ee.washington.edu> Lecture 7 Slides April 18th, 2006 EE512 - Graphical Models - J. Bilmes

  2. Announcements • 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. EE512 - Graphical Models - J. Bilmes

  3. Class Road Map • 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 EE512 - Graphical Models - J. Bilmes

  4. Final Project Milestone Due Dates • 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). • 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!! EE512 - Graphical Models - J. Bilmes

  5. Summary of Last Time • 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) EE512 - Graphical Models - J. Bilmes

  6. Outline of Today’s Lecture • 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 EE512 - Graphical Models - J. Bilmes

  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. EE512 - Graphical Models - J. Bilmes

  8. Properties of Markov Properties EE512 - Graphical Models - J. Bilmes

  9. Markov Properties of Graphs EE512 - Graphical Models - J. Bilmes

  10. Properties of Markov Properties EE512 - Graphical Models - J. Bilmes

  11. (F) Factorization Property EE512 - Graphical Models - J. Bilmes

  12. The alphabetical theorem: (F)  (G)  (L)  (P) EE512 - Graphical Models - J. Bilmes

  13. The alphabetical theorem: (F)  (G)  (L)  (P) EE512 - Graphical Models - J. Bilmes

  14. The equivalence theorem: (F)  (G)  (L)  (P) EE512 - Graphical Models - J. Bilmes

  15. Inclusion-Exclusion EE512 - Graphical Models - J. Bilmes

  16. Möbius Inversion Lemma EE512 - Graphical Models - J. Bilmes

  17. Möbius Inversion Lemma EE512 - Graphical Models - J. Bilmes

  18. Hammersley/Clifford EE512 - Graphical Models - J. Bilmes

  19. Hammersley/Clifford EE512 - Graphical Models - J. Bilmes

  20. Hammersley/Clifford EE512 - Graphical Models - J. Bilmes

  21. Hammersley/Clifford EE512 - Graphical Models - J. Bilmes

  22. Hammersley/Clifford EE512 - Graphical Models - J. Bilmes

  23. Hammersley/Clifford by pairwise Markov property since we have unity ratios pairwise Markov property and chain rule EE512 - Graphical Models - J. Bilmes

  24. Hammersley/Clifford EE512 - Graphical Models - J. Bilmes

  25. Factorization and decomposability EE512 - Graphical Models - J. Bilmes

  26. Factorization and decomposability EE512 - Graphical Models - J. Bilmes

  27. (G), factorization, and decomposability EE512 - Graphical Models - J. Bilmes

  28. Recursive application + positivity EE512 - Graphical Models - J. Bilmes

  29. Recursive application + positivity EE512 - Graphical Models - J. Bilmes

  30. (DF) EE512 - Graphical Models - J. Bilmes

  31. (DF) and (G) EE512 - Graphical Models - J. Bilmes

  32. Markov Blanket EE512 - Graphical Models - J. Bilmes

  33. Recall from Lecture 3: Ancestral Sets EE512 - Graphical Models - J. Bilmes

  34. Preservation of (DF) in ancestral sets EE512 - Graphical Models - J. Bilmes

  35. Example (DF) – (G) EE512 - Graphical Models - J. Bilmes

  36. Example (DF) – (G) EE512 - Graphical Models - J. Bilmes

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