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Bayesian Networks Bucket Elimination Algorithm

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Bayesian NetworksBucket Elimination Algorithm

主講人：虞台文

大同大學資工所

智慧型多媒體研究室

- Basic Concept
- Belief Updating
- Most Probable Explanation (MPE)
- Maximum A Posteriori (MAP)

Bayesian NetworksBucket Elimination Algorithm

Basic Concept

大同大學資工所

智慧型多媒體研究室

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.

Examples:

Satisfiable

A=True, B=True, C=False, D=False

Satisfiable?

can be true if and only if

can be true.

unsatisfiable

BucketA

BucketB

BucketC

BucketD

Example:

Given a set of clauses

and an order d=ABCD

Set initial buckets as follows:

BucketA

BucketB

BucketC

BucketD

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

How to get a truth assignment?

BucketA

BucketB

BucketC

BucketD

- Belief updating
- Finding the most probable explanation (mpe)
- Given evidence, finding a maximum probability assignment to the 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.

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

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

Eliminate parameterX fromh

Defined overU = S– {X}.

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

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

Defined over

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

Bayesian NetworksBucket Elimination Algorithm

Belief Updating

大同大學資工所

智慧型多媒體研究室

Normalization

Factor

A

C

B

F

D

G

Example:

Example:

G(f)

D(a, b)

F(b, c)

B(a, c)

C(a)

BucketG

BucketD

BucketF

BucketB

BucketC

BucketA

BucketG

BucketD

BucketF

BucketB

BucketC

BucketA

0.7

0.1

0.7

0.1

0.7

0.1

0.7

0.1

- 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?

A

C

B

F

D

G

Consider the ordering AFDCBG.

BucketG

BucketB

1

G

4

BucketC

B

1

,3

C

BucketD

0

,2

D

BucketF

,1

0

F

BucketA

0

A

A

C

B

1

1

F

G

D

4

4

B

G

3

1

C

2

0

D

1

0

F

0

0

A

d

Given the ordering, e.g., AFDCBG.

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

w(d) = 4

w*(d) = 4

w(d): width of initial graph

for ordering d.

w*(d): width of induced graph

for ordering d.

Width of node

Width of node

G

B

C

Induced

Graph

D

Initial

Graph

F

A

Goal: Finding an ordering with smallest induced width.

Greedy heuristic and Approximation methods

Are available.

NP-Hard

- 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.

A

C

B

F

D

G

- Use BuckElim to evaluate P(a|b=1) with the following two ordering:
- d1=ACBFDG
- d2=AFDCBG

Give the details and make some conclusion.

How to improve the algorithm?

Bayesian NetworksBucket Elimination Algorithm

Most Probable Explanation (MPE)

大同大學資工所

智慧型多媒體研究室

Goal:

evidence

Goal:

xi

Let

Xn

Some terms involve xn,

some terms not.

Xn is conditioned by its parents.

Xnconditions its children.

Xn

xnappears in these CPT’s

Not conditioned by xn

Conditioned by xn

Itself

Process the next bucket recursively.

Eliminate variable xnatBucketn.

A

C

B

F

D

G

A

C

B

F

D

G

Consider ordering ACBFDG

BucketG

BucketD

BucketF

BucketB

BucketC

BucketA

Consider ordering ACBFDG

Bayesian NetworksBucket Elimination Algorithm

Maximum

A Posteriori (MAP)

大同大學資工所

智慧型多媒體研究室

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

A

C

B

F

D

G

Hypothesis (Decision)

Variables

g = 1

Ordering

Some of them may be observed

Bucket Elimination for MPE

Bucket Elimination for belief updating

A

C

B

F

g = 1

D

G

Consider orderingCBAFDG

BucketG

BucketD

BucketF

BucketA

BucketB

BucketC

A

C

B

F

g = 1

D

G

Consider orderingCBAFDG

BucketG

BucketD

BucketF

BucketA

Give the detail

BucketB

BucketC