1

Impact of Structuring on Bayesian Network Learning

and Reasoning

Mieczysław.A..Kłopotek

Institute of Computer Science,

Polish Academy of Sciences,

Warsaw, Poland,

First Warsaw International Seminar on Soft Computing Warsaw, September 8th, 2003

2

2

Agenda

Definitions Approximate Reasoning Bayesian networks

Reasoning in Bayesian networks Learning Bayesian networks from data Structured Bayesian networks (SBN) Reasoning in SBN Learning SBN from data Concluding remarks

3

3

Approximate Reasoning One possible method of expressing uncertainty: Joint

Probability Distribution Variables: causes, effects, observables Reasoning: How probable is that a variable takes a given

value if we kniow the values of some other variables Given: P(X,Y,....,Z) Find: P(X=x | T=t,...,W=w)

Difficult, if more than 40 variables have to be taken into account hard to represent, hard to reason, hard to collect data)

4

4

The method of choice for representing uncertainty in AI.

Many efficient reasoning methods and learning methods

Utilize explicit representation of structure to: provide a natural and compact

representation of large probability distributions.

allow for efficient methods for answering a wide range of queries.

Bayesian Network

5

5

Bayesian Network Efficient and effective representation of a

probability distribution Directed acyclic graph

Nodes - random variables of interests Edges - direct (causal) influence

Nodes are statistically independent of their non descendants given the state of their parents

6

6

A Bayesian network

R

Z

T

Y

X

S

Pr(r,s,x,z,y)=

Pr(z) . Pr(s|z) . Pr(y|z)

. Pr(x|y) . Pr(r|y,s)

7

7

Applications of Bayesian networks Genetic optimization algorithms with

probabilistic mutation/crossing mechanism Classification, including text classification Medical diagnosis (PathFinder, QMR), other

decision making tasks under uncertainty Hardware diagnosis (Microsoft troubleshooter,

NASA/Rockwell Vista project) Information retrieval (Ricoh helpdesk) Recommender systems other

8

8

Reasoning – the problem with a Bayesian network Fusion algorithm of Pearl elaborated for tree-like

networks only For other types of networks transformations to

trees: transformation to Markov tree (MT) is needed

(Shafer/Shenoy, Spiegelhalter/Lauritzen) – except for trees and polytrees NP hard

Cutset reasoning (Pearl) – finding cutsets difficult, the reasoning complexity grows exponentially with cutset size needed

evidence absorption reasoning by edge reversal (Shachter) – not always possible in a simple way

9

9

Towards MT – moral graph

R

Z

T

Y

X

S

Parents of a node in BN connected, edges not oriented

10

10

Towards MT – triangulated graph

R

Z

T

Y

X

S

All cycles with more than 3 nodes have at least one link between non-neighboring nodes of the cycle.

11

11

Towards MT – Hypertree

R

Z

T

Y

X

S

Hypertree = acyclic hypergraph

12

12

The Markov tree

Z,T,Y T,Y,S Y,S,R

Y,X

Hypernodes of hypertree are nodes of the Markov tree

13

13

Junction tree – alternative representation of MT

Z,T,S Z,Y,S Y,S,R

Y,X

Z,S Y,S

Y

Common BN nodes assigned to edges joining MT nodes

14

14

Efficient reasoning in Markov trees, but ....

Z,T,S Z,Y,S Y,S,R

Y,X

Z,S Y,S

Y

msg msg

msg

MT node contents projected onto common variables are passed to the neighbors

15

15

Triangulability test - Triangulation not always possible

All neighbors need to be connected

16

16

Evidence absorption reasoning

R

Z

T

Y

X

S

R

Z

T

Y

X

S

Evidence absorption

R

Z

T

Y

X

S

Edge reversal

Efficient only for good-luck selection of conditioning variables

17

17

Cutset reasoning – fixing values of some nodes creates a (poly)tree

R

Z

T

Y

X

S

Node fixed

Hence edge ignorable

18

18

How to overcome the difficulty when reasoning with BN Learn directly a triangulated graph or Markov tree

from data (Cercone N., Wong S.K.M., Xiang Y) Hard and inefficient for long dependence chains,

danger of large hypernodes Learn only tree-structured/polytree structured BN

(e.g. In Goldberg’s Bayesian Genetic Algorithms, TAN text classifiers etc.) Oversimplification, long dependence chains lost

Our approach: Propose a more general class of Bayesian networks that is still efficient for reasoning

19

19

What is a structured Bayesian network An analogon of well-structured

programs Graphical structure: nested sequences

and alternatives By collapsing sequences and

alternatives to single nodes, one single node obtainable

Efficient reasoning possible

20

20

Structured Bayesian Network (SBN), an example

For comparison: a tree-structured BN

21

21

SBN collapsing

22

22

SBN construction steps

means 0,1 or 2 arrows

23

23

Reasoning in SBN

Either directly in the structure Or easily transformable to Markov tree Direct reasoning consisting of

Forward step (leave node/root node valuation calculation)

Backward step (intermediate node valuation calculation

24

24

Reasoning in SBN forward step

means 0,1 or 2 arrows

A

B

A

B

CEP(B|A)

P(B|C,E)

25

25

Reasoning in SBN backward step: local context

AC

B

D

.....

.....A

C

B

D

.....

.....A

C

B

.....

.....A

C

B

.....

.....

(a) (b) (c) (d)

Joint distribu-tion of A,B known, joint C,D or C sought

26

26

Reasoning in SBN – backward step: local reasoning

A,B,............ A,B,C,DA,B

Msg2(A,B)

Msg1(A,B)

P(A)*P(B|A,D)

Not needed

27

27

SBN –towards a MT

28

28

SBN –towards a MT

29

29

SBN –towards a MT

30

30

B

C D

E

A

J

I

S

R

H

F

G

P

K

L

M N

O

Towards a Markobv tree – an example

31

31

B

C D

E

A

J

I

S

R

H

F

G

P

K

L

M N

O

Towards a Markobv tree – an example

32

32

A,B,I

B,C,D,I

C,D,E,I

F,G,I

G,H,I

I,H,E,R D,E,I

E,H,R,J H,R,J

K,L,R

L,M,N,R

M,N,O,R

N,O,R

O,P,R

R,J,P

P,J,S

Markov tree from SBN

33

33

B

C D

E

A

J

I

S

R

H

F

G

P

K

L

M N

O

Structured Bayesian network – a Hierarchical (Object-Oriented) Bayesian network

34

34

Learning SBN from Data

Define the DEP() measure as follows:

DEP(Y,X)=P(x|y)-P(x|y).

Define DEP[](Y,X)= (DEP(Y,X) )2

Construct a tree according to Chow/Liu algorithm using DEP[](Y,X) with Y

belonging to the tree and X not.

35

35

Continued ....

Let us call all the edges obtained by the previous algorithm “free edges”.

During the construction process the following type of edges may additionally appear “node X loop unoriented edge”, “node X loop oriented edge”, “node X loop transient edge”.

Do in a loop (till termination condition below is satisfied):

For each two properly connected non-neighboring nodes identify the unique connecting path between them.

36

36

Continued ....

Two nodes are properly connected if the path between them consists either of edges having the status of free edges or of oriented, unoriented (but not suspended) edges of the same loop, with no pair of oriented or transient oriented edges pointing in different directions and no transient edge pointing to one of the two connected points.

Note that in this sense there is at most one path properly connecting two nodes.

37

37

Continued ....

Connect that a pair of non-neighboring nodes X,Y by an edge, that maximizes DEP[](X,Y), the minimum of

unconditional DEP and conditional DEP given a direct successor of X on the path to Y.

Identify the loop that has emerged from this operation.

38

38

Continued ....

We can have one of the following cases: (1)it consists entirely of free edges (2)it contains some unoriented loop edges, but

no oriented edge. (3)It contains at least one oriented edge. Depending on this, give a proper status to

edges contained in a loop: “node X loop unoriented edge”, “node X loop oriented edge”, “node X loop transient edge”. (details in written presentation).

39

39

Places of edge insertion

X

C

D

Y

B

Y

C

D

E

X

Y

G

D

E

X

C

Y

C

D

X

B

H

X

C

D

E

Y

X

G

D

E

Y

C

40

40

Concluding Remarks

new class of Bayesian networks defined completely new method of reasoning in Bayesian

networks outlined Local computation – at most 4 nodes involved applicable to a more general class of networks then

known reasoning methods new class Bayesian networks easily transfornmed to

Markov trees new class Bayesian networks – a kind of hierarchical or

object-oriented Bayesian networks Can be learned from data

41

41

THANK YOU