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

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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.

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Belief Assessment Definition –The belief assessment task of X k = x k 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

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Most Probable Explanation (MPE) Definition –The MPE task is to find an assignment x o = (x o 1, …, x o n ) such that In the Visit to Asia example, the MPE problem answers questions like –What are the most probable values for all variables such that a person doesn’t catch dyspnea ?

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Maximum Aposteriori Hypothesis (MAP) Definition –Given a set of hypothesized variables A = {A 1, …, A k },, the MAP task is to find an assignment a o = (a o 1, …, a o k ) such that In the Visit to Asia example, the MAP problem answers questions like –What are the most probable values for a person having both lung cancer and bronchitis, given that he/she has dyspnea and that his/her X-ray is positive?

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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 node which has the lowest degree 1. , (degree = 1) 2. , , (degree = 2) 3., , (degree = 2)

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Elimination Algorithm for Belief Assessment Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : P( | ) P( | )*P( ), =“yes” P( | , ) P( | , ), =“yes” P( | =“yes”, =“yes”) = X\ { } (P( | )* P( | )* P( | , )* P( | , )* P( )*P( | )*P( | )*P( )) P( | ) P( | )*P( ) H()H() H()H() H(,)H(,) H ( ,, ) H ( , , ) H()H() H(,)H(,) P( | =“yes”, =“yes”) H n (u)= xn П j i=1 C i (x n,u si ) *k k-normalizing constant

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Elimination Algorithm for Most Probable Explanation Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : P( | ) P( | )*P( ) P( | , ) P( | , ), =“no” MPE= MAX { , , , ,, , , } (P( | )* P( | )* P( | , )* P( | , )* P( )*P( | )*P( | )*P( )) P( | ) P( | )*P( ) H()H() H()H() H(,)H(,) H ( ,, ) H ( , , ) H()H() H(,)H(,) MPE probability Finding MPE = max , , , ,, , , P( , , , ,, , , ) H n (u)=max xn ( П xn Fn C(x n |x pa ))

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Elimination Algorithm for Most Probable Explanation Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : Bucket : P( | ) P( | )*P( ) P( | , ) P( | , ), =“no” P( | ) P( | )*P( ) H()H() H()H() H(,)H(,) H ( ,, ) H ( , , ) H()H() H(,)H(,) Forward part ’ = arg max H ( )* H ( ) ’ = arg max H ( ’, ) ’ = arg max P( ’| )*P( )* H ( ’, ’, ) ’ = arg max P( | ’)*H ( ’,, ’) ’ = arg max P( | ’, ’)*H ( , ’)*H ( ) ’ = “no” ’ = arg max P( | ’) ’ = arg max P( ’| )*P( ) Return: ( ’, ’, ’, ’, ’, ’, ’, ’)

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