1. 1 Exact Inference Algorithms for Probabilistic Reasoning; COMPSCI 276 Fall 2007

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5. 5 Probabilistic Inference Tasks  Belief updating:  Finding most probable explanation (MPE)  Finding maximum a-posteriory hypothesis  Finding maximum-expected-utility (MEU) decision

6. 6 Example with a chain ABC D P(D)=?P(D|A=a)=? P(A|D=d)=? O(4k^2) instead of O(k^4), k is the domain size

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10. 10 Example of product-sum in a bucket

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12. 12 Bucket elimination Algorithm elim-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

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14. 14 E D C B A B C D E A

15. 15 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

16. 16 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)

17. 17 Different Induced graphs

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19. 19 The impact of observations

20. 20 “Moral” graph A D E C B Theorem: elim-bel is exponential in the adjusted induced-width w*(e,d)

21. 21 Use the ancestral graph only

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26. 26 BTE in action