presentace karny

7/31/2019 Presentace Karny

1/27

Some applications of Bayesian networks

Ji r Vomlel

Institute of Information Theory and AutomationAcademy of Sciences of the Czech Republic

This presentation is available athttp://www.utia.cas.cz/vomlel/

1


2/27

Contents Brief introduction to Bayesian networks

Typical tasks that can be solved using Bayesian networks

1: Medical diagnosis (a very simple example)

2: Decision making maximizing expected utility (another simpleexample)

3: Adaptive testing (a case study)

4: Decision-theoretic troubleshooting (a commercial product)

2


3/27

Bayesian network a directed acyclic graph G = ( V , E)

each node i V corresponds to a random variable Xi with anite set X i of mutually exclusive states

pa(i) denotes the set of parents of node i in graph G

to each node i V corresponds a conditional probability tableP(Xi | (X j) j pa(i) )

the DAG implies conditional independence relations between(Xi) i V

d-separation (Pearl, 1986) can be used to read the CI relationsfrom the DAG

3


4/27

Using the chain rule we have that:

P(( Xi) i V ) = i V

P(Xi | Xi 1, . . . , X1)

Assume an ordering of Xi, i V such that if j pa(i) then j < i.From the DAG we can read conditional independence relations

Xi Xk | (X j) j pa(i) for i V and k < i and k pa(i)

Using the conditional independence relations from the DAG we get

P(( Xi) i V ) = i V

P(Xi | (X j) j pa(i) ) .

It is the joint probability distribution represented by the Bayesiannetwork .

4


5/27

Example:X1 X2P(X1) P(X2)

P(X3 | X1)P(X4 | X2)

P(X6 | X3 , X4)

P(X9 | X6)P(X8 | X7 , X6)

P(X5 | X1)

P(X7 | X5)

X5

X7

X3

X8

X6

X9

X4

P(X1, . . . , X9) == P(X9 |X8, . . . , X1) P(X8 |X7, . . . , X1) . . . P(X2 |X1) P(X1)

= P(X9 |X6) P(X8 |X7, X6) P(X7 |X5) P(X6 |X4, X3)P(X5 |X1) P(X4 |X2) P(X3 |X1) P(X2) P(X1)

5


6/27


7/27

Simplied diagnostic exampleWe have a patient.Possible diagnoses: tuberculosis, lung cancer, bronchitis.

7


8/27

We dont know anything about the pa-tient

Patient is a smoker.

8


9/27

Patient is a smoker. ... and he complains about dyspnoea

9


10/27

Patient is a smoker and complainsabout dyspnoea

... and his X-ray is positive

10


11/27


12/27

Application 2 :Decision making

The goal: maximize expected utility

Hugin example: mildew4.net

12


13/27

Fixed and Adaptive Test Strategies

wrong

correct wrong

wrong correct

wrong

wrong

correct

correctwrong wrongcorrectcorrectcorrect

Q5

Q4

Q3

Q2

Q1

Q2

Q1

Q3

Q4

Q5

Q6

Q7

Q8

Q9

Q10

Q6 Q8

Q7

Q8

Q4 Q7 Q10

Q6

Q7

Q9

13


14/27

X3

X1

X3

X3

X2

X3

X2

X1

X2

X1

X2

X2

X3

X1

X1

For all nodes n of a strategy s wehave dened:

evidence e n , i.e. outcomes ofsteps performed to get to noden,

probability P(e n ) of getting to

node n, and utility f (e n ) being a real num-

ber.

Let L( s ) be the set of terminalnodes of strategy s .Expected utility of strategy isE f (s ) = L (s ) P(e ) f (e ).

14


15/27

X3

X1

X3

X3

X2

X3

X2

X1

X2

X1

X2

X2

X3

X1

X1

Strategy s is optimal iff it maxi-mizes its expected utility.

Strategy s is myopically optimal iffeach step of strategy s is selectedso that it maximizes expected utilityafter the selected step is performed(one step look ahead ).

15


16/27

Application 3 : Adaptive test of basicoperations with fractions

Examples of tasks:

T 1:34

56

18 =

1524

18 =

58

18 =

48 =

12

T 2: 16 +1

12 =2

12 +1

12 =3

12 =14

T 3: 14 112 =

14

32 =

38

T 4: 12 12 13 + 13 = 14 23 = 212 = 16 .

16


17/27

Elementary and operational skillsCP Comparison (common nu-

merator or denominator)

12

> 13 , 23 > 13

AD Addition (comm. denom.) 17 +27 =

1+ 27 =

37

SB Subtract. (comm. denom.) 25 15 =

2 15 =

15

MT Multiplication 12 35 = 310

CD Common denominator 12 ,23 =

36 ,

46

CL Cancelling out 46 =2223 =

23

CIM Conv. to mixed numbers 72 =32+ 1

2 = 312

CMI Conv. to improp. fractions 3 12 =32+ 1

2 =72

17


18/27

MisconceptionsLabel Description Occurrence

MAD ab +cd =

a+ cb+ d 14.8%

MSB ab cd =

a cb d 9.4%

MMT1 ab cb = acb 14.1%MMT2 ab

cb =

a+ cbb 8.1%

MMT3 ab cd =

adbc 15.4%

MMT4 ab cd =

acb+ d 8.1%

MC abc =

abc 4.0%

18


19/27

Student model

CP

ACD

HV2 HV1

AD SB CMI CIM CL CD MT

MMT1 MMT2 MMT3 MMT4MCMAD MSB

ACMI ACIM ACL

19


20/27

Evidence model for task T 134

56

18 =

1524

18 =

58

18 =

48 =

12

T 1 MT & CL & ACL & SB & MMT 3 & MMT 4 & MSB

CL

MMT4

MSB

SB

MMT3

ACL MT

T1

X1

P(X1 | T1 )

Hugin: model-hv-2.net

20


21/27

Using information gain as the utility function

The lower the entropy of a probability distribution the more we know.

H (P(X )) = x

P(X = x ) log P(X = x )

0

0.5

1

0 0.5 1

e n t r o p

y

robabilit

Information gain in a node n of a strategy

IG(e n) = H (P(S )) H (P(S | e n ))

21


22/27

Skill Prediction Quality

74

76

78

80

82

84

86

88

90

92

0 2 4 6 8 10 12 14 16 18 20

Q u a

l i t y o

f s k

i l l p r e

d i c t i o n s

Number of answered uestions

adaptiveaveragedescending

ascending

22


23/27

Application 4: Troubleshooting

Dezide Advisor customized to a specic portal, seen from the usersperspective through a web browser.

23


24/27

Application 2: Troubleshooting - Light print problem

FF3

F2

F1

F4

FaultsActions

A3

A2

A1

Q1

Problem

Questions

Problems: F1 Distribution problem , F2 Defective toner , F3Corrupted dataow , and F4 Wrong driver setting .

Actions:A1

Remove, shake and reseat toner ,A2

Try another toner , and A3 Cycle power .

Questions: Q1 Is the conguration page printed light?

24


25/27

Troubleshooting strategy

A1 = no

A2 = yes

Q1 = no

A1 = yes A2 = yes

Q1 = yes

A1 = yes

A2 = no

A1 = no A2 = no

A2

Q1

A1

A2 A1

The task is to nd a strategy s S minimising expected cost of repair

ECR(s ) = L (s )

P(e ) ( t(e ) + c(e ) ) .

25


26/27

Expected cost of repair for a given strategy

A1 = no

A2 = yes

Q1 = no

A1 = yes A2 = yes

Q1 = yes

A1 = yes

A2 = no

A1 = no A2 = no

A2

Q1

A1

A2 A1

ECR(s ) =

P(Q1 = no, A1 = yes) cQ1 + c A1+ P(Q1 = no, A1 = no, A2 = yes) cQ1 + c A1 + c A2

+ P(Q1 = no, A1 = no, A2 = no) cQ1 + c A1 + c A2 + cCS+ P(Q1 = yes, A2 = yes) cQ1 + c A2+ P(Q1 = yes, A2 = no, A1 = yes) cQ1 + c A2 + c A1+ P(Q1 = yes, A2 = no, A1 = no) cQ1 + c A2 + c A1 + cCS

Demo: www.dezide.com Products/Demo/Try out expert mode

26


27/27

Commercial applications of Bayesian networks ineducational testing and troubleshooting

Hugin Expert A/S.software product: Hugin - a Bayesian network tool.http://www.hugin.com/

Educational Testing Service (ETS)the worlds largest private educational testing organizationResearch unit doing research on adaptive tests using Bayesiannetworks: http://www.ets.org/research/

SACSO ProjectSystems for Automatic Customer Support Operations

- research project of Hewlett Packard and Aalborg University.The troubleshooter offered as DezisionWorks by Dezide Ltd.http://www.dezide.com/

27

presentace karny

Documents