Bayesian Networks Bucket Elimination Algorithm

Bayesian NetworksBucket Elimination Algorithm

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大同大學資工所智慧型多媒體研究室

Content

Basic ConceptBelief UpdatingMost Probable Explanation (MPE)Maximum A Posteriori (MAP)


Basic Concept


Satisfiability

Given a statement of clauses (in disjunction normal form), the satisfiability problem is to determine whether there exists a truth assignment to make the statement true.

1. ( ) ( ) ( ) ( )A B C C D A B D A D Examples:

A=True, B=True, C=False, D=False Satisfiable2. ( ) ( ) ( ) ( ) ( )A B C B C D B C A C D

Satisfiable?

Resolution

( ) ( )q q can be true if and only if

can be true.

q q ?TURE

unsatisfiable

Direct Resolution

A B C

B C D

B C

A C

D

Example:

Given a set of clauses

( ) ( ) ( ) ( ) ( )A B C B C D B C A C D

and an order d=ABCD

BucketA BucketB BucketC BucketD

B C D DA C

B C A B C

Set initial buckets as follows:

Direct Resolution

( ) ( ) ( ) ( ) ( )A B C B C D B C A C D


B C D DA C

B C A B C

B C

A B B

A B

Because no empty clause () is resulted, the statement is satisfiable.

How to get a truth assignment?

Direct Resolution

( ) ( ) ( ) ( ) ( )A B C B C D B C A C D


B C D DA C

B C A B C

B C

A B B

A B

True or FalseA TrueB

True if True

True/False if False

AC

A

FalseD

Direct Resolution

Queries on Bayesian Networks

Belief updating

Finding the most probable explanation (mpe)– Given evidence, finding a maximum probability assignment to t

he rest of variables. Maximizing a posteriori hypothesis (map)

– Given evidence, finding an assignment to a subset of hypothesis variables that maximize their probability.

Maximizing the expected utility of the problem (meu)– Given evidence and utility function, finding a subset of decision

variables that maximize the expected utility.

( | ) ?P X E

Bucket Elimination

The algorithm will be used as a framework for various probabilistic inferences on Bayesian Networks.

Preliminary – Elimination Functions

Given a function h defined over subset of variables S, where X S,

min

max

mean

X

X

X

X

h

h

h

h

Eliminate parameter X from h

Defined over U = S – {X}.


Given a function h defined over subset of variables S, where X S,

min

max

mean

X

X

X

X

h

h

h

h

min min( ) ( , )X Xh h xu u

max max( ) ( , )X Xh h xu u

1

|( ) ( ,n )

|mea X XX

h xh u u

( ) ( , )X Xh h x u u


Given function h1,…, hn defined over subset of variables S1,…, Sn, respectively,

j

j

j

j

h

h

Defined overjj

U S


Given function h1,…, hn defined over subset of variables S1,…, Sn, respectively,

j

j

j

j

h

h

) (( )jj jj jh h Su u

) (( )jj jj jh h Su u


Belief Updating


Goal

( | ) ?X xP E e

( , )

( )

X xP

P

E e

E eNormalization

Factor

Basic Concept of Variable Elimination

Example:

A

B

D

C

F

G

( | ) ?1a gP

( , , , , , ) ( | ) ( | , ) ( | , ) ( | ) ( | ) ( )P g f d c b a P g f P f b c P d a b P b a P c a P a

( , ) ?1a gP


Example:

( | ) ?1a gP

, , , , 1

( , ) ( , , , )1 , ,b c d f g

P P a b c d f gga

( , ) ?1a gP

, , , , 1

( ) ( | ) ( | , ) ( | , ) ( | ) ( | )b c d f g

P a P g f P f b c P d a b P b a P c a

1

( ) ( | ) ( | ) ( | , ) ( | , ) ( | )c b f d g

P a P c a P b a P f b c P d a b P g f

( , , , , , ) ( | ) ( | , ) ( | , ) ( | ) ( | ) ( )P g f d c b a P g f P f b c P d a b P b a P c a P a


( | ) ?1a gP

1( , )P a g

( , ) ?1a gP

1

( ) ( | ) ( | ) ( | , ) ( | , ) ( | )c b f d g


G(f)

( ) ( | ) ( | ) ( | , () ( | , ))c b f

Gd

P a P c a P b a P f b c P d a bf D(a, b)

( ) ( | ) ( | ) ( , ) ( )( | , )D Gc b f

P a P c a P b a P f ba cb f F(b, c)

( , ) ( , )( ) ( | ) ( | )c b

D FP a P c a P b a a b b c B(a, c)

( ) ( | ) ( , )c

BP a P c ca a C(a)

)( ) (C aP a


1

( ) ( | )Gg

P g ff

( )( ,, |)Dd

P db aa b

( | ,( , ) ( ))F Gf

P f b cb c f

( , ) ( , )) ( ,( )|b

B D Fa c a b bP a cb

( ) ( , )( | ) Bc

C P c aa a c ( ) ), ((1) CP P a aga

BucketG

BucketD

BucketF

BucketB

BucketC

BucketA

1( , )P a g

1

( ) ( | ) ( | ) ( | , ) ( | , ) ( | )c b f d g



( | )P g f

( | , )P d a b

( | , )P f b c

( | )P b a

( | )P c a

( )P a

BucketG

BucketD

BucketF

BucketB

BucketC

BucketA

1( , )P a g

1

( ) ( | ) ( | ) ( | , ) ( | , ) ( | )c b f d g


1g

d

f

b

c

( )G f

( , )D a b

( , )F b c

( , )B a c

( )C a

( , 1)P a g

( )G f

( , )D a b ( , )F b c

( , )B a c

( )C a


1

( ) ( | )Gg

P g ff

f G(f )

+ 0.1

0.7


f G(f )

+ 0.1

0.7

( )( ,, |)Dd

P db aa b a b D(a,

b)

0 0 1

0 1 1

1 0 1

1 1 1


f G(f )

+ 0.1

0.7

a b D(a, b)

0 0 1

0 1 1

1 0 1

1 1 1

( | ,( , ) ( ))F Gf

P f b cb c f

0.70.1

0.70.10.70.10.70.1

b c F(b, c)

0 0 0.701

0 1 0.610

1 0 0.400

1 1 0.340


f G(f )

+ 0.1

0.7

a b D(a, b)

0 0 1

0 1 1

1 0 1

1 1 1

b c F(b, c)

0 0 0.701

0 1 0.610

1 0 0.400

1 1 0.340

( , ) ( , )) ( ,( )|b

B D Fa c a b bP a cb a c B(a, c)

0 0 0.90.701+0.1 0.400=0.6709

0 1 0.90.610+0.1 0.340=0.5830

1 0 0.60.701+0.4 0.400=0.5806

1 1 0.60.610+0.4 0.340=0.5020


f G(f )

+ 0.1

0.7

a b D(a, b)

0 0 1

0 1 1

1 0 1

1 1 1

b c F(b, c)

0 0 0.701

0 1 0.610

1 0 0.400

1 1 0.340

a c B(a, c)

0 0 0.6709

0 1 0.5830

1 0 0.5806

1 1 0.5020

( ) ( , )( | ) Bc

C P c aa a c

a C(a )

1 0.67 0.5806+0.33 0.5020=0.554662

0 0.75 0.6709+0.25 0.5830=0.648925


f G(f )

+ 0.1

0.7

a b D(a, b)

0 0 1

0 1 1

1 0 1

1 1 1

b c F(b, c)

0 0 0.701

0 1 0.610

1 0 0.400

1 1 0.340

a c B(a, c)

0 0 0.6709

0 1 0.5830

1 0 0.5806

1 1 0.5020

a C(a )

1 0.554662

0 0.648925

( ) ), ((1) CP P a aga

a P(a, g=1)

1 0.30.554662=0.1663986

0 0.70.648925=0.4542475

( , )( | )

(

11

1)

aPP

P ga

gg

( 1) 0.1663986 0.4542475

0.6206461

P g

a P(a | g=1)1 0.1663986/0.6206461=0.268110 0.4542475/0.6206461=0.73189

Bucket Elimination Algorithm

Complexity The BuckElim Algorithm can be applied to any ordering. The arity of the function recorded in a bucket

– the numbers of variables appearing in the processed bucked, excluding the bucket’s variable.

Time and Space complexity is exponentially grow with a function of arity r.

The arity is dependent on the ordering. How many possible orderings for BN’s variables?

Determination of the Arity

( | )P g f

( | , )P d a b ( | , )P f b c( | )P b a

( | )P c a

( )P a

BucketG

BucketB

BucketC

BucketD

BucketF

BucketA

1g

b

c

d

f

( )G f

( , , , )B a c d f

( , )D a f

( )F a

( , 1)P a g

( )G f

( , , )C a d f

A

B

D

C

F

G

Consider the ordering AFDCBG.

G

B

C

D

F

A( )F a

( , , , )B a c d f ( , , )C a d f

( , )D a f

1

4

1

0

0

0

,3

,2

,1

Determination of the Arity

Given the ordering, e.g., AFDCBG.

1

4

0

3

2

1

G

B

C

D

F

A

G

B

C

D

F

A

1

4

1

0

0

0

A

B

D

C

F

G

InitialGraph

Width of node

InducedGraph

Width of node

d

w(d): width of initial graph for ordering d.w*(d): width of induced graph for ordering d.

w(d): width of initial graph for ordering d.w*(d): width of induced graph for ordering d.

The width of a graph is the maximum width of its nodes.

The width of a graph is the maximum width of its nodes. w(d) = 4 w*(d) = 4

Definition of Tree-Width

Goal: Finding an ordering with smallest induced width.

NP-Hard Greedy heuristic and Approximation methodsAre available.

Summary

The complexity of BuckElim algorithm is dominated by the time and space needed to process a bucket.

It is time and space is exponential in number of bucket variables.

Induced width bounds the arity of bucket functions.

Exercises

A

B

D

C

F

G

Use BuckElim to evaluate P(a|b=1) with the following two ordering:

1. d1=ACBFDG

2. d2=AFDCBG

Give the details and make some conclusion.

How to improve the algorithm?


Most Probable Explanation (MPE)


MPE

Goal:

arg max ( | )* ?P x

X x E exevidence

1( , , )nx xx

arg max ( , ) ?* P x

exx

MPE

Goal:

arg max ( | )* ?P x

X x E ex

Notationsarg max (* , )P

xx ex

i

iF

xi

i

arg max (* , )Px

x ex

MPE1

( ) ( , ) ( , | )i

n

i iP P xP

X e ex x

( ) max ( )* max ( )n

nP P P X Xx x

x x x

Let 1( , , )n nx xx

1max ( , )|

n

n

ii ixP

x

e

1 , 1m x ( , )|a

n nx i i

n

ixP

x

e

arg max (* , )Px

x ex

MPE*( )P x

1 , 1m x ( , )|a

n nx i i

n

ixP

x

e

Some terms involve xn,some terms not.

Xn

n

n

Xn is conditioned by its parents.

Xn conditions its children.

( , | )n nxP e

|( , ), ,k n k nP x x x e

nF

arg max (* , )Px

x ex

MPE*( )P x

1 , 1m x ( , )|a

n nx i i

n

ixP

x

e

1

max ( , ) max ( , ) ( , )| | |n i nn i nX F i i n

xiXn ix xP P P x

Xxe e e

Xn

n

n

Not conditioned by x n Conditioned by x nItself

x n appears in these CPT’s

arg max (* , )Px

x ex

MPE*( )P x

1 , 1m x ( , )|a

n nx i i

n

ixP

x

e

1

max ( , ) max ( , ) ( , )| | |n i nn i nX F i i n

xiXn ix xP P P x

Xxe e e

( )nnh xU

1

max ( , ) (| )i n n

ni iF nX

P xhx

X U

xe

Eliminate variable xn at Bucketn.

Process the next bucket recursively.

Example

A

B

D

C

F

G

, , , ,, ,max ( 1) ?, , ,

a b c d fa b c gP d f

( ) ( | ) ( | , ) ( | , ) (, | ) ( | ), 1 1, , , ( )P P f P f b c Pg g d a b P b a P c aa b c f ad P

ExampleA

B

D

C

F

G

, , , ,, ,max ( 1) ?, , ,

a b c d fa b c gP d f

( | )P g f

( | , )P d a b

( | , )P f b c

( | )P b a

( | )P c a

( )P a

BucketG

BucketD

BucketF

BucketB

BucketC

BucketA

1max

g

maxd

maxf

maxb

maxc

( )Gh f

( , )Dh a b

( , )Fh b c

( , )Bh a c

( )C a

, , , ,max ( , , , , , )1

a b c d fa b c d f gP

( )Gh f

( , )Dh a b ( , )Fh b c

( , )B a c

( )Ch amaxa

Consider ordering ACBFDG


Exercise, , , ,

, ,max ( 1) ?, , ,a b c d f

a b c gP d f

Consider ordering ACBFDG


Maximum A Posteriori (MAP)


MAP

arg max ( , ) ?* P a

eaa

arg max ( | )* ?P a

A a E ea

Given a belief network, a subset of hypothesized variables A=(A1, …, Ak), and evidence E=e, the goal is to determine

ExampleA

B

D

C

F

G

Hypothesis (Decision) Variables

,arg max ( | )*, * 1( ?) ,

b cc P c gbb

g = 1

MAP

Ordering

1 1, ,, ,kk nd XX X X

1 1, , ,, kk nX XA A

1, nk k XA

Some of them may be observed

MAP1, n

kkd XA

* arg max ( | )k

k kPa

a ea

( , )arg max

( )k

kP

P

a

ea

e

arg max ( , )k

kPa

a e

MAP1, n

kkd XA

* arg max ( , )k

k kPa

a ea

11( , ) ( , , )n

kk k

nkP P

x

x eaea

1 1( , | )n

k

n

i iiP x

x

e

1

*

1( ) max ( , | )

knk

n

i iikP P x

xaa e

MAP1, n

kkd XA

* arg max ( , )k

k kPa

a ea

1

*

1( ) max ( , | )

knk

n

i iikP P x

xaa e

Bucket Elimination for belief updating

Bucket Elimination for belief updating

Bucket Elimination for MPE

Bucket Elimination for MPE


ExampleA

B

D

C

F

G g = 1

Consider ordering CBAFDG

,arg max ( | )*, * 1( ?) ,

b cc P c gbb

( | )P g f

( | , )P d a b

( | , )P f b c

( | )P b a( | )P c a ( )P a

BucketG

BucketD

BucketF

BucketA

BucketB

BucketC

1g

d

f

a

maxb

( )G f

( , )D a b

( , )F b c

( , )A b c

( )b c

,max ( |, )1

b cb c gP

( )G f

( , )D a b

( , )F b c ( , )A b c

maxc

( )b c

ExerciseA

B

D

C

F

G g = 1

Consider ordering CBAFDG

,arg max ( | )*, * 1( ?) ,

b cc P c gbb

( | )P g f

( | , )P d a b

( | , )P f b c

( | )P b a( | )P c a ( )P a

BucketG

BucketD

BucketF

BucketA

BucketB

BucketC

1g

d

f

a

maxb

( )G f

( , )D a b

( , )F b c

( , )A b c

( )b c

,max ( |, )1

b cb c gP

( )G f

( , )D a b

( , )F b c ( , )A b c

maxc

( )b c

Give the detailGive the detail

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Bayesian Networks Bucket Elimination Algorithm