Bucket Elimination: A Unifying Framework for Probabilistic Inference

1. Bucket Elimination: A Unifying Framework for Probabilistic Inference By: Rina Dechter Presented by: Gal Lavee

2. Multiplying Tables • Often in inference we see the notation: f(A) * g(B) where A and B are sets of variables and f and g are functions (usually probability functions) given as tables of the form: • Example: a function f of two variables: A1 and A2 Where ai and ~ai are the possible states of Ai

3. Multiplying Tables • Multiply the two functions (given as tables) f and g. Where g takes one variable (B1) and f takes two variables (A1,A2)

4. Multiplying Tables • The result is a function h, which takes as 3 arguments (the union of the arguments of f and g), A1,A2,B1 • The value of the function h(A1,A2,B1) is simply the product f(A1,A2) * g(B1) • This function can be expressed as a table with 2^3 entries:

5. Summing over tables • Given a function of multiple variables it is often useful to eliminate variables by summation • Summing the function f of variables A1,A2 (given as a table) over the variable A2 yields a new function f1 of just one variable(A1). • The new function value is given by f(A1,a2)+f(A1,~a2). • This function can be specified as the table:

6. Bucket Elimination-Definition • Bucket elimination is a unifying algorithmic framework that generalizes dynamic programming to accommodate algorithms for many complex problem-solving and reasoning activities Including: • Belief Assessment • Most Probable Estimation (MPE) • Maximum A posteriori Hypothesis (MAP) • Maximum Expected Utility MEU • Uses “buckets”, an organizational device, to mimic the algebraic manipulations involved in each of these problems resulting in an easily expressible algorithmic formulation • Each bucket elimination algorithm is associated with a node ordering

7. The Bucket Elimination Algorithm • Mimics algebraic manipulation • Partition functions on the graph into “buckets” in backwards relative to the given node order • In the bucket of variable Xi we put all functions that mention Xi but do not mention any variable having a higher index • e.g. in the bucketG we will place all functions whose scope includes variable G

8. The Algorithm cntd. • Process buckets backwards relative to the node order • Processing a bucket amounts to eliminating the variable in the bucket from subsequent computation. • The computed function after elimination is placed in the bucket of the ‘highest’ variable in its scope • E.g= sum over G is computed and placed in bucket F

9. Example - Belief Updating/Assessment • Definition • Given a set of evidence compute the posterior probability of each proposition • The belief assessment task of Xk = xk is to find • where k – normalizing constant

10. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment

11. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment Initial Bucket Partition

12. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment Process buckets in reverse order (G)

13. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment Process buckets in reverse order (D)

14. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment Process buckets in reverse order (F)

15. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment Process buckets in reverse order (B)

16. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment Process buckets in reverse order (C)

17. Node Ordering: A,C,B,F,D,G D G Example - Belief Assessment Process buckets in reverse order (A)

18. Belief Assessment Summary • 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

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

20. Most Probable Explanation (MPE) • Differences from Belief Assessment • Replace Sums With Maxs • Keep track of maximizing value at each stage • “Forward Step” to determine what is the maximizing assignment tuple

21. Node Ordering: A,C,B,F,D,G D G Example - MPE

22. Node Ordering: A,C,B,F,D,G D G Example - MPE Initial Bucket Partition (same as belief assessment)

23. Node Ordering: A,C,B,F,D,G D G Example - MPE Process buckets in reverse order (G)

24. Node Ordering: A,C,B,F,D,G D G Example - MPE Process buckets in reverse order (D)

25. Node Ordering: A,C,B,F,D,G D G Example - MPE Process buckets in reverse order (F)

26. Node Ordering: A,C,B,F,D,G D G Example - MPE Process buckets in reverse order (B)

27. Node Ordering: A,C,B,F,D,G D G Example - MPE Process buckets in reverse order (C)

28. Node Ordering: A,C,B,F,D,G D G Example - MPE Process buckets in reverse order (A)

29. Node Ordering: A,C,B,F,D,G D G Example - MPE Forward Phase: determine maximizing tuple:

30. MPE Summary • 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

31. 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

32. Maximum Aposteriori Hypothesis (MAP) • Mixture of Belief Assessment and MPE • Sum over non-hypothesis vars • Max over hypothesis var • Forward phase over hypothesis vars only • Algorithm presented assumes hypothesis vars are at the beginning of the order

33. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP

34. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Initial Bucket Partition (Same as Belief Assessment, MPE)

35. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Process buckets in reverse order (G)

36. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Process buckets in reverse order (D)

37. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Process buckets in reverse order (F)

38. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Process buckets in reverse order (B)

39. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Process buckets in reverse order (C)

40. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Process buckets in reverse order (A)

41. Node Ordering: A,C,B,F,D,G Hypothesis Nodes: A,B,C D G Example - MAP Forward Phase: determine maximizing tuple:

42. MAP -Summary • In reverse node Ordering • Create bucket function by multiplying all functions (given as tables) containing the current node • If is a hypothesis node: • Perform variable elimination using the Maximization operation over the current node (recording the maximizing state function) • Else: • Perform variable elimination using Summation 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 over hypothesis nodes only

43. Generalizing Bucket Elimination • As we have seen bucket elimination algorithms can be formulated for different problems by simply altering the elimination operation. • For some problems we may also want to change the combination operation which we use to combine functions.

44. Thank You