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Bayesian Networks Bucket Elimination AlgorithmPowerPoint Presentation

Bayesian Networks Bucket Elimination Algorithm

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Content

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

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.

Examples:

Satisfiable

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

Satisfiable?

BucketB

BucketC

BucketD

Direct ResolutionExample:

Given a set of clauses

and an order d=ABCD

Set initial buckets as follows:

BucketB

BucketC

BucketD

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

How to get a truth assignment?

Queries on Bayesian Networks

- 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.

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,

Eliminate parameterX fromh

Defined overU = S– {X}.

Preliminary – Elimination Functions

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

Preliminary – Elimination Functions

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

Defined over

Preliminary – Elimination Functions

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

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?

C

B

F

D

G

Consider the ordering AFDCBG.

Determination of the ArityBucketG

BucketB

1

G

4

BucketC

B

1

,3

C

BucketD

0

,2

D

BucketF

,1

0

F

BucketA

0

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.

Determination of the ArityThe 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

Definition of Tree-Width

Goal: Finding an ordering with smallest induced width.

Greedy heuristic and Approximation methods

Are available.

NP-Hard

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.

C

B

F

D

G

Exercises- 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?

MPE

Goal:

xi

NotationsMPE

Let

Xn

MPESome terms involve xn,

some terms not.

Xn is conditioned by its parents.

Xnconditions its children.

Exercise

Consider ordering ACBFDG

MAP

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

C

B

F

g = 1

D

G

Consider orderingCBAFDG

ExerciseBucketG

BucketD

BucketF

BucketA

Give the detail

BucketB

BucketC

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