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CSCE 582 Computation of the Most Probable Explanation in Bayesian Networks using Bucket Elimination -Hareesh Lingareddy University of South Carolina

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Bucket Elimination -Algorithmic framework that generalizes dynamic programming to accommodate algorithms for many complex problem solving and reasoning activities. -Uses “buckets” to mimic the algebraic manipulations involved in each of these problems resulting in an easily expressible algorithmic formulation

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Bucket Elimination Algorithm -Partition functions on the graph into “buckets” in backwards relative to the given node order -In the bucket of variable X we put all functions that mention X but do not mention any variable having a higher index -Process buckets backwards relative to the node order -The computed function after elimination is placed in the bucket of the ‘highest’ variable in its scope

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Algorithms using Bucket Elimination -Belief Assessment -Most Probable Estimation(MPE) -Maximum A Posteriori Hypothesis(MAP) -Maximum Expected Utility(MEU)

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Belief Assessment Definition - Given a set of evidence compute the posterior probability of all the variables – 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 dyspnoea and has visited Asia recently ? where k – normalizing constant

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Belief Assessment Overview In reverse Node Ordering: – Create bucket function by multiplying all functions (given as tables) containing the current node – Perform variable elimination using Summation over the current node – Place the new created function table into the appropriate bucket

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Most Probable Explanation (MPE) Definition – Given evidence find the maximum probability assignment to the remaining variables – The MPE task is to find an assignment x o = (x o 1, …, x o n ) such that

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Differences from Belief Assessment – Replace Sums With Max – Keep track of maximizing value at each stage – “Forward Step” to determine what is the maximizing assignment tuple

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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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MPE Overview In reverse node Ordering – Create bucket function by multiplying all functions (given as tables) containing the current node – Perform variable elimination using the Maximization operation over the current node (recording the maximizing state function) – Place the new created function table into the appropriate bucket In forward node ordering – Calculate the maximum probability using maximizing state functions

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Maximum Aposteriori Hypothesis (MAP) Definition – Given evidence find an assignment to a subset of “hypothesis” variables that maximizes their probability – Given a set of hypothesis variables A = {A 1, …, A k },,the MAP task is to find an assignment a o = (a o 1, …, a o k ) such that

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