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Exact Inference Algorithms for Probabilistic Reasoning;

This document explores exact inference algorithms for probabilistic reasoning, focusing on belief updating, finding most probable explanations (MPE), maximum a-posteriori hypothesis, and maximum-expected-utility (MEU) decision-making. It discusses the use of bucket elimination and the impact of ordering on complexity. The document also provides examples and explanations of induced width, moral graphs, and the impact of observations.

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Exact Inference Algorithms for Probabilistic Reasoning;

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

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

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

  4. Example of product-sum in a bucket

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

  6. B E C D D C E B A A

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

  8. 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)

  9. Different Induced graphs

  10. The impact of observations

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

  12. Use the ancestral graph only

  13. BTE in action

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