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Bucket Elimination: A unifying framework for Probabilistic inference Rina Dechter

Bucket Elimination: A unifying framework for Probabilistic inference Rina Dechter. presented by Anton Bezuglov, Hrishikesh Goradia CSCE 582 Fall02 Instructor: Dr. Marco Valtorta. Contributions. For a Bayesian network, the paper presents algorithms for Belief Assessment

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Bucket Elimination: A unifying framework for Probabilistic inference Rina Dechter

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  1. Bucket Elimination: A unifying framework for Probabilistic inferenceRina Dechter presented by Anton Bezuglov, Hrishikesh Goradia CSCE 582 Fall02 Instructor: Dr. Marco Valtorta

  2. Contributions • For a Bayesian network, the paper presents algorithms for • Belief Assessment • Most Probable Explanation (MPE) • Maximum Aposteriori Hypothesis (MAP) • All of the above are bucket elimination algorithms.

  3. Belief Assessment • Definition • The belief assessment task of Xk = xk is to find • In the Visit to Asia example, the belief assessment problem answers questions like • What is the probability that a person has tuberculosis, given that he/she has dyspnea and has visited Asia recently ? • where k – normalizing constant

  4. Most Probable Explanation (MPE) • Definition • The MPE task is to find an assignment xo = (xo1, …, xon) such that • In the Visit to Asia example, the MPE problem answers questions like • What are the most probable values for allvariables such that a person doesn’t catch dyspnea ?

  5. Maximum Aposteriori Hypothesis (MAP) • Definition • Given a set of hypothesized variables A = {A1, …, Ak}, , the MAP taskis to find an assignment ao = (ao1, …, aok) such that • Inthe Visit to Asia example, the MAP problem answers questions like • What are the most probable values for a person having both lung cancerand bronchitis, given that he/she has dyspnea and that his/her X-ray is positive?

  6.        Ordering the Variables Method 1 (Minimum deficiency) Begin elimination with the node which adds the fewest number of edges 1. , ,  (nothing added) 2.  (nothing added) 3. , , ,  (one edge added) Method 2 (Minimum degree) Begin elimination with the nodewhich has the lowest degree 1. ,  (degree = 1) 2. , ,  (degree = 2) 3. , ,  (degree = 2)

  7. Elimination Algorithm for Belief Assessment P(| =“yes”, =“yes”) = X\ {} (P(|)* P(|)* P(|,)* P(|,)* P()*P(|)*P(|)*P()) Bucket : P(|)*P(), =“yes” Hn(u)=xnПji=1Ci(xn,usi) Bucket : P(|) Bucket : P(|,), =“yes” Bucket : P(|,) H(,) H() Bucket : P(|) H(,,) Bucket : P(|)*P() H(,,) Bucket : H(,) k-normalizing constant Bucket : H() H() *k P(| =“yes”, =“yes”)

  8. Elimination Algorithm for Most Probable Explanation Finding MPE = max ,,,,,,, P(,,,,,,,) MPE= MAX{,,,,,,,} (P(|)* P(|)* P(|,)* P(|,)* P()*P(|)*P(|)*P()) Bucket : P(|)*P() Hn(u)=maxxn (ПxnFnC(xn|xpa)) Bucket : P(|) Bucket : P(|,), =“no” Bucket : P(|,) H(,) H() Bucket : P(|) H(,,) Bucket : P(|)*P() H(,,) Bucket : H(,) Bucket : H() H() MPE probability

  9. Elimination Algorithm for Most Probable Explanation Forward part ’ = arg maxP(’|)*P() Bucket : P(|)*P() Bucket : P(|) ’ = arg maxP(|’) Bucket : P(|,), =“no” ’ = “no” Bucket : P(|,) H(,) H() ’ = arg maxP(|’,’)*H(,’)*H() Bucket : P(|) H(,,) ’ = arg maxP(|’)*H(’,,’) Bucket : P(|)*P() H(,,) ’ = arg maxP(’|)*P()* H(’,’,) Bucket : H(,) ’ = arg maxH(’,) Bucket : H() H() ’ = arg maxH()* H() Return: (’, ’, ’, ’, ’, ’, ’, ’)

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