- Hareesh Lingareddy University of South Carolina

1. CSCE 582Computation of the Most Probable Explanation in Bayesian Networks using Bucket Elimination -HareeshLingareddy University of South Carolina

2. 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

3. 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

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

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

6. 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

7. Most Probable Explanation (MPE) • Definition • Given evidence find the maximum probability assignment to the remaining variables • The MPE task is to find an assignment xo = (xo1, …, xon) such that

8. 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

9. 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

10. 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: (’, ’, ’, ’, ’, ’, ’, ’)

11. 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

12. 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 = {A1, …, Ak}, ,the MAP taskis to find an assignment ao = (ao1, …, aok) such that