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 –Most Probable Explanation (MPE) –Maximum Aposteriori Hypothesis (MAP) All of the above are bucket elimination algorithms.

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

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 ?

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?

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)

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

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 ))

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: (  ’,  ’,  ’, ’,  ’,  ’,  ’,  ’)