Bucket Elimination: A Unifying Framework for Probabilistic Inference

Bucket Elimination: A Unifying Framework for Bucket Elimination: A Unifying Framework for Probabilistic InferenceProbabilistic Inference

By: Rina Dechter

Presented by:Gal Lavee

Multiplying TablesMultiplying 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

a1~a1

a2f(a1,a2)f(~a1,a2)

~a2f(a1,~a2)f(~a1,~a2)


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

a1~a1

a2f(a1,a2)f(~a1,a2)

~a2f(a1,~a2)f(~a1,~a2)

b1~b1

g(b1)g(~b1)


b1a1~a1

a2f(a1,a2)*g(b1)f(~a1,a2)*g(b1)

~a2f(a1,~a2)*g(b1)f(~a1,~a2)*g(b1)

~b1a1~a1

a2f(a1,a2)*g(~b1)f(~a1,a2)*g(~b1)

~a2f(a1,~a2)*g(~b1)f(~a1,~a2)*g(~b1)

• 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:

Summing over tablesSumming 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:

a1~a1

a2f(a1,a2)f(~a1,a2)

~a2f(a1,~a2)f(~a1,~a2)

a1~a1

f(a1,a2)+ f(a1,~a2)

f(~a1,a2)+ f(~a1,~a2)

Bucket Elimination-DefinitionBucket 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

The Bucket Elimination AlgorithmThe 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

The AlgorithmThe Algorithm cntd.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

Example - Belief Updating/AssessmentExample - 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

}{ 1

)),(|()|()(kxX

n

iiikkk expaxPkexXPxbel

where k – normalizing constant

Example - Belief AssessmentExample - Belief Assessment

A

F

BC

D

G

Node Ordering: A,C,B,F,D,G

Example - Belief AssessmentExample - Belief AssessmentA

F

B C

D GNode Ordering: A,C,B,F,D,G

( | ), 1

( | , )

( | , )

( | )

( | )

( )

G

D

F

B

C

A

bucket P g f g

bucket P d b a

bucket P f b c

bucket P b a

bucket P c a

bucket P a

Initial Bucket Partition


F

B C


1

Eliminate ( ) ( | )

( | , )

( | , ) ( )

( | )

( | )

( )

G Gg

D

F G

B

C

A

bucket f P g f

bucket P d b a

bucket P f b c f

bucket P b a

bucket P c a

bucket P a

Process buckets in reverse order (G)


F

B C


Eliminate ( , ) ( | , )

( | , ) ( )

( | ) ( , )

( | )

( )

D Dd

F G

B D

C

A

bucket b a P d b a

bucket P f b c f

bucket P b a b a

bucket P c a

bucket P a

Process buckets in reverse order (D)


F

B C


Eliminate ( , ) ( | , ) ( )

( | ) ( , ) ( , )

( | )

( )

F F Gf

B D F

C

A

bucket c b P f b c f

bucket P b a b a c b

bucket P c a

bucket P a

Process buckets in reverse order (F)


F

B C


Eliminate ( , ) ( | ) ( , ) ( , )

( | ) ( , )

( )

B B D Fb

C B

A

bucket c a P b a b a c b

bucket P c a c a

bucket P a

Process buckets in reverse order (B)


F

B C


Eliminate ( ) ( | ) ( , )

( ) ( )

C C Bc

A C

bucket a P c a c a

bucket P a a

Process buckets in reverse order (C)


F

B C


1Eliminate Bel(a)= ( ) ( )A Cbucket P a a

Z

Process buckets in reverse order (A)

Belief Assessment SummaryBelief 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

Most Probable Explanation (MPE)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

n

iii

XexpaxPxP o

1

)),(|(max)(

Most Probable Explanation (MPE)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

Example - MPEExample - MPE

A

F

BC

D

G


Example - MPEExample - MPEA

F

B C


( | ), 1

( | , )

( | , )

( | )

( | )

( )

G

D

F

B

C

A

bucket P g f g

bucket P d b a

bucket P f b c

bucket P b a

bucket P c a

bucket P a

Initial Bucket Partition (same as belief assessment)


F

B C


1

1

Eliminate ( ) max ( | ) ,

( ) arg max ( | )

( )

( | , )

( | , ) ( )

( | )

( | )

( )

G Gg

g

G

D

F G

B

C

A

bucket h f P g f

G f P g f

bucket G f

bucket P d b a

bucket P f b c h f

bucket P b a

bucket P c a

bucket P a



F

B C


Eliminate ( , ) max ( | , ) ,

( , ) arg max ( | , )

( )

( , )

( | , ) ( )

( | ) ( , )

( | )

( )

D Dd

d

G

D

F G

B D

C

A

bucket h b a P d b a

D b a P d b a

bucket G f

bucket D b a

bucket P f b c h f

bucket P b a h b a

bucket P c a

bucket P a



F

B C


Eliminate ( , ) max ( | , ) ( ) ,

( , ) arg max ( | , ) ( )

( )

( , )

( , )

( | ) ( , ) ( , )

( | )

( )

F F Gf

Gf

G

D

F

B D F

C

A

bucket h b c P f b c h f

F b c P f b c h f

bucket G f

bucket D b a

bucket F b c

bucket P b a h b a h b c

bucket P c a

bucket P a



F

B C


Eliminate ( , ) max ( | ) ( , ) ( , ) ,

( , ) arg max ( | ) ( , ) ( , )

( )

( , )

( , )

( , )

( | ) ( , )

( )

B B D fb

D fb

G

D

F

B

C B

A

bucket h c a P b a h b a h b c

B c a P b a h b a h b c

bucket G f

bucket D b a

bucket F b c

bucket B c a

bucket P c a h c a

bucket P a



F

B C


Eliminate ( ) max ( | ) ( , ) ,

( ) arg max ( | ) ( , )

( )

( , )

( , )

( , )

( )

( ) ( )

C C Bc

Bc

G

D

F

B

C

A C

bucket h a P c a h c a

C a P c a h c a

bucket G f

bucket D b a

bucket F b c

bucket B c a

bucket C a

bucket P a h a



F

B C


Eliminate M max ( ) ( ) ,

arg max ( ) ( )

( )

( , )

( , )

( , )

( )

A Ca

Ca

G

D

F

B

C

A

bucket P a h a

A P a h a

bucket G f

bucket D b a

bucket F b c

bucket B c a

bucket C a

bucket A



F

B C


'

' ')

' ', ')

' ', ')

' ', ')

' ')

a A

c C a

b B c a

f F b c

d D b a

g G f

Forward Phase: determine maximizing tuple:

MPE SummaryMPE 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

Maximum Aposteriori Hypothesis Maximum Aposteriori Hypothesis (MAP)(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 task is to find an assignment

ao = (ao1, …, ao

k) such that

XA

AX

n

iii

A

o expaxPaP1

)),(|(max)(

Maximum Aposteriori Hypothesis Maximum Aposteriori Hypothesis (MAP)(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

Example - MAPExample - MAP

A

F

BC

D

G


Hypothesis Nodes: A,B,C


( | ), 1

( | , )

( | , )

( | )

( | )

( )

G

D

F

B

C

A

bucket P g f g

bucket P d b a

bucket P f b c

bucket P b a

bucket P c a

bucket P a

Initial Bucket Partition (Same as Belief Assessment, MPE)

A

F

B C

DG




1

Eliminate ( ) ( | )

( | , )

( | , ) ( )

( | )

( | )

( )

G Gg

D

F G

B

C

A

bucket f P g f

bucket P d b a

bucket P f b c f

bucket P b a

bucket P c a

bucket P a


A

F

B C

DG




Eliminate ( , ) ( | , )

( | , ) ( )

( | ) ( , )

( | )

( )

D Dd

F G

B D

C

A

bucket b a P d b a

bucket P f b c f

bucket P b a b a

bucket P c a

bucket P a


A

F

B C

DG




Eliminate ( , ) ( | , ) ( )

( | ) ( , ) ( , )

( | )

( )

F F Gf

B D F

C

A

bucket c b P f b c f

bucket P b a b a c b

bucket P c a

bucket P a


A

F

B C

DG




Eliminate ( , ) max ( | ) ( , ) ( , ) ,

( , ) arg max ( | ) ( , ) ( , )

( , )

( | ) ( , )

( )

B B D fb

D fb

B

C B

A

bucket h c a P b a b a b c

B c a P b a b a b c

bucket B c a

bucket P c a h c a

bucket P a


A

F

B C

DG




Eliminate ( ) max ( | ) ( , ) ,

( ) arg max ( | ) ( , )

( , )

( )

( ) ( )

C C Bc

Bc

B

C

A C

bucket h a P c a h c a

C a P c a h c a

bucket B c a

bucket C a

bucket P a h a


A

F

B C

DG




Eliminate M max ( ) ( ) ,

arg max ( ) ( )

( , )

( )

A Ca

Ca

B

C

A

bucket P a h a

A P a h a

bucket B c a

bucket C a

bucket A


A

F

B C

DG




'

' ')

' ', ')

a A

c C a

b B c a

Forward Phase: determine maximizing tuple:

A

F

B C

DG



MAP -SummaryMAP -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

Generalizing Bucket EliminationGeneralizing 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.

ProblemCombination OperatorElimination Operator

Belief AssessmentMultiplicationSummation

MPEMultiplicationMaximization

CSPJoinMultiplication

Max-CSPSummationMinimization

Thank YouThank You

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Bucket Elimination: A Unifying Framework for Probabilistic Inference