1. 1 Exact Inference Algorithms Bucket-elimination and more COMPSCI 179, Spring 2010 Set 8: Rina Dechter (Reading: chapter 14, Russell and Norvig

3. Counting 1234 4321 55 555 How many people? SUM operator CHAINstructure

4. Maximization What is the maximum? 15 23 10 32 10 100 65 47 50 77 100 15 23 77 100 47 100 77 100 23 77 10 100 32 MAX operator TREEstructure

5. 12”14”15” S IIIIII P 60G80G H 6C9C B Min-Cost Assignment What is minimum cost configuration? 6C9C I3050 II4055 III∞60 IIIIII 12”45∞∞ 14”506070 15”∞6580 60G80G 12”3050 14”4045 15”50∞ IIIIII 12”75∞∞ 14”80100130 15”∞105140 12”14”15” 105120155 40 II 30 I 60 III + MIN-SUMoperators CHAINstructure 60 40 30 105 80 75 80 14” 75 12” 105 15” 50 40 30 40 14” 30 12” 50 15” +=

6. Belief Updating B uzz sound M echanical problem H igh temperature F aulty head R ead delays HP(H) 0.9 1.1 FP(F) 0.99 1.01 HFMP(M|H,F) 000.9 001.1 010 011.9 100.8 101.2 110.01 111.99 FRP(R|F) 00.8 01.2 10.3 01.7 P(F | B=1) = ? Mh 1 (M) 0.05 1.8 HFMBel(M,H,F) 000.0405 001.072 010.0045 011.648 100.004 101.008 110.00005 111.0792 Hh 2 (H) 0.9 1.1 Fh 3 (F) 0.1245 1.73175 Fh 4 (F) 01 11 HFMP(M|H,F) 000.9 001.1 010 011.9 100.8 101.2 110.01 111.99 ** = MBP(B|M) 00.95 01.05 10.2 11.8 ** = FP(F,B=1) 0.123255 1.073175 P(B=1) =.19643 Probability of evidence P(F=1|B=1) =.3725 Updated belief SUM-PRODoperators POLY-TREEstructure P(h,f,r,m,b) = P(h) P(f) P(m|h,f) P(r|f) P(b|m)

7. X YZ TRLM Belief updating (sum-prod)

8. X YZ TRLM MPE (max-prod)

9. CSP – consistency (projection-join) X YZ TRLM

10. X YZ TRLM #CSP (sum-prod)

11. X YZ TRLM Tree-solving Belief updating (sum-prod) MPE (max-prod) CSP – consistency (projection- join) #CSP (sum-prod)

12. Belief Propagation Instances of tree message passing algorithm Exact for trees Linear in the input size Importance: – One of the first algorithms for inference in Bayesian networks – Gives a cognitive dimension to its computations – Basis for conditioning algorithms for arbitrary Bayesian network – Basis for Loopy Belief Propagation (approximate algorithms) [Pearl, 1988]

21. 21 Exact Inference Algorithms Bucket-elimination COMPSCI 179, Spring 2010 Set 8: Rina Dechter (Reading: chapter 14, Russell and Norvig

26. 26 Belief Updating lung Cancer Smoking X-ray Bronchitis Dyspnoea P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?

27. 27 Belief updating: P(X|evidence)=? “Moral” graph A D E C B P(a|e=0) P(a,e=0)= P(a)P(b|a)P(c|a)P(d|b,a)P(e|b,c)= P(a) P(b|a)P(d|b,a)P(e|b,c) BC ED Variable Elimination P(c|a)

28. 28 Bucket elimination Algorithm BE-bel (Dechter 1996) Elimination operator P(a|e=0) W*=4 ”induced width” (max clique size) bucket B: P(a) P(c|a) P(b|a) P(d|b,a) P(e|b,c) bucket C: bucket D: bucket E: bucket A: e=0 B C D E A

29. 29 “Moral” graph A D E C B

30. 30 BE-BEL

35. IntelligenceDifficulty Grade Letter SAT Job Apply Student Network example P(J)?

36. 36 E D C B A B C D E A

37. Fall 2003 ICS 275A - Constraint Networks 37 The induced-width width: is the max number of parents in the ordered graph Induced-width: width of induced graph: recursively connecting parents going from last node to first. Induced-width w*(d) = the max induced-width over all nodes Induced-width of a graph: max w*(d) over all d

38. 38 Complexity of elimination The effect of the ordering: “Moral” graph A D E C B B C D E A E D C B A

39. 39 More accurately: O(r exp(w*(d)) where r is the number of cpts. For Bayesian networks r=n. For Markov networks? BE-BEL

40. 40

41. 41 The impact of observations Moral graph Induced Moral graph Adjusted Graph for evidence in B Induced- adjusted.

42. 42 Probabilistic Inference Tasks  Belief updating:  Finding most probable explanation (MPE)

43. 43 Elimination operator MPE W*=4 ”induced width” (max clique size) bucket B: P(a) P(c|a) P(b|a) P(d|b,a) P(e|b,c) bucket C: bucket D: bucket E: bucket A: e=0 B C D E A Algorithm elim-mpe (Dechter 1996)

44. 44 Generating the MPE-tuple C: E: P(b|a) P(d|b,a) P(e|b,c)B: D: A: P(a) P(c|a) e=0

45. 12”14”15” S IIIIII P 60G80G H 6C9C B Min-Cost Assignment What is minimum cost configuration? 6C9C I3050 II4055 III∞60 IIIIII 12”45∞∞ 14”506070 15”∞6580 60G80G 12”3050 14”4045 15”50∞ IIIIII 12”75∞∞ 14”80100130 15”∞105140 12”14”15” 105120155 40 II 30 I 60 III + MIN-SUMoperators CHAINstructure 60 40 30 105 80 75 80 14” 75 12” 105 15” 50 40 30 40 14” 30 12” 50 15” +=

46. 46

47. 47 BE-MPE

48. 48 Finding small induced-width NP-complete A tree has induced-width of ? Greedy algorithms: – Min width – Min induced-width – Max-cardinality – Fill-in (thought as the best) – See anytime min-width (Gogate and Dechter)