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CS b553 : A lgorithms for Optimization and Learning

CS b553 : A lgorithms for Optimization and Learning. Variable Elimination. Last Time. Variable elimination on polytrees Top down inference Linear in size of network Variable elimination in general No guarantees… NP hard in worst case… but when?. Variable Elimination in General Networks.

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CS b553 : A lgorithms for Optimization and Learning

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  1. CS b553: Algorithms for Optimization and Learning Variable Elimination

  2. Last Time • Variable elimination on polytrees • Top down inference • Linear in size of network • Variable elimination in general • No guarantees… • NP hard in worst case… but when?

  3. Variable Elimination in General Networks Coherence Difficulty Intelligence Grade SAT Letter Job Happy

  4. Variable Elimination in General Networks Coherence Difficulty Intelligence Grade SAT Letter Job Happy

  5. Joint distribution • P(X) = P(C)P(D|C)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L

  6. Going through VE • P(X) = P(C)P(D|C)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L • 1(D)=SCP(C)P(D|C)

  7. Going through VE • SCP(X) = 1(D)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L • 1(D)=SCP(C)P(D|C)

  8. Going through VE • SCP(X) = 1(D)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L • 2(G,I)=SD1(D)P(G|I,D)

  9. Going through VE • SC,DP(X) = 2(G,I)P(I)P(S|I)P(L|G) P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L • 2(G,I)=SD1(D)P(G|I,D)

  10. Going through VE • SC,DP(X) = 2(G,I)P(I)P(S|I)P(L|G) P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L • 3(G,S)=SI2(G,I)P(I)P(S|I)

  11. Going through VE • SC,D,IP(X) = 3(G,S)P(L|G)P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L • 3(G,S)=SI2(G,I)P(I)P(S|I)

  12. Going through VE • SC,D,IP(X) = 3(G,S)P(L|G)P(J|L,S)P(H|G,J) • Apply elimination ordering C,D,I,H,G,S,L • 4(G,J)=SHP(H|G,J) What values does this factor store?

  13. Going through VE • SC,D,I,HP(X) = 3(G,S)P(L|G)P(J|L,S)4(G,J) • Apply elimination ordering C,D,I,H,G,S,L • 4(G,J)=SHP(H|G,J)

  14. Going through VE • SC,D,I,HP(X) = 3(G,S)P(L|G)P(J|L,S)4(G,J) • Apply elimination ordering C,D,I,H,G,S,L • 5(S,L,J)=SG3(G,S)P(L|G)4(G,J)

  15. Going through VE • SC,D,I,H,GP(X) = 5(S,L,J)P(J|L,S) • Apply elimination ordering C,D,I,H,G,S,L • 5(S,L,J)=SG3(G,S)P(L|G)4(G,J)

  16. Going through VE • SC,D,I,H,GP(X) = 5(S,L,J)P(J|L,S) • Apply elimination ordering C,D,I,H,G,S,L • 6(L,J)=SS 5(S,L,J)P(J|L,S)

  17. Going through VE • SC,D,I,H,G,SP(X) = 6(L,J) • Apply elimination ordering C,D,I,H,G,S,L • 6(L,J)=SS 5(S,L,J)

  18. Going through VE • SC,D,I,H,G,SP(X) = 6(L,J) • Apply elimination ordering C,D,I,H,G,S,L • 7(J)=SL 6(S,L)

  19. Going through VE • SC,D,I,H,G,S,LP(X) = 7(J) • Apply elimination ordering C,D,I,H,G,S,L • 7(J)=SL 6(L,J)

  20. Comparing Orderings • Consider G,I,S,L,H,C,D

  21. Understanding VE: From BNs to Undirected Graphs • Consider each factor as a variable i • Draw an edge between any variables appearing in the same factor

  22. Building the Undirected Graph P(C) Coherence P(I) P(D|C) Difficulty Intelligence P(S|I) P(G|I,D) Grade SAT P(L|G) Letter P(J|S,L) Job Happy P(H|G,J)

  23. Building the Undirected Graph P(C) Coherence P(I) P(D|C) Difficulty Intelligence P(S|I) P(G|I,D) Grade SAT P(L|G) Letter P(J|S,L) Job Happy P(H|G,J)

  24. Building the Undirected Graph Coherence Difficulty Intelligence Grade SAT Letter Job Happy

  25. Variable Elimination Coherence Difficulty Intelligence Grade SAT Letter Job Happy

  26. Variable Elimination Difficulty Intelligence Grade SAT Letter Job Happy

  27. Variable Elimination Difficulty Intelligence Grade SAT Letter Job Happy

  28. Variable Elimination Intelligence Grade SAT Letter Job Happy

  29. Variable Elimination Intelligence Grade SAT Letter Job Happy

  30. Variable Elimination New fill edge Grade SAT Letter Job Happy

  31. Variable Elimination Grade SAT Letter Job Happy

  32. Variable Elimination Grade SAT Letter Job

  33. Variable Elimination Grade SAT Letter Job

  34. Variable Elimination SAT Letter Job

  35. Induced Graph from a VE ordering Coherence Difficulty Intelligence Grade SAT Letter Job Happy

  36. Induced Graph from a VE ordering Coherence Difficulty Intelligence Grade SAT • Theorem: • The scope of every intermediate factor in VE is a clique in the induced graph • Every maximal clique in the induced graph is the scope of an intermediate factor Letter Job Happy

  37. Determining Optimal orderings • Again, NP hard! • Good heuristics in practice: • Min-neighbors, min-fill, etc • Search among elimination orderings while counting size of introduced factors • Greedy search often works well

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