Construction of Bayesian Networks

1

© 1998 HRL Laboratories, LLC. All Rights Reserved

Construction of Bayesian Networks
for Diagnostics

K. Wojtek Przytula: HRL Laboratories

&

Don Thompson: Pepperdine University



Malibu, California

2

© 1998 HRL Laboratories, LLC. All Rights Reserved

Diagnostics / Troubleshooting

Problem Definition

Given a set of system observations ( symptoms,

sensor readings, error codes, test results)

determine a root cause of system failure



Typical Techniques for Problem Solution

•
Decision Trees

•
Cased Based Reasoning

•
Bayesian Networks

3

© 1998 HRL Laboratories, LLC. All Rights Reserved

Bayesian

Networks

-

Definition



Ob
1

Ob
2

Ob
3

Ob
4

F1

F2

Au
x

Bayesian Networks*

are a class of probabilistic

models for knowledge representation

•
Nodes represent random variables

•
Edges represent causal dependencies

between variables

•
Annotations are prior and conditional

probabilities



* (also known as belief networks or causal networks)


4

© 1998 HRL Laboratories, LLC. All Rights Reserved

Bayesian Networks
-

Features

•

Bayesian networks can be constructed from domain

knowledge and/or learned from data


•

Network structure reflects the causal reality of the domain



•

Query: given state of some variables, compute the

probability of states of remaining variables


•

Computation: efficient implementation of probabilistic

calculations


•

Application: decision support in presence of uncertainty

e.g. diagnostics
-

tool assist human in finding a fault





5

© 1998 HRL Laboratories, LLC. All Rights Reserved

Problem Definition

•
Create a Bayesian network model for a diagnostic
support tool using diverse information sources
(manuals, test & repair procedures, repair statistics,
experts)


•
Balance fidelity with design cost


•
Refine the model by learning from experimental data

6

© 1998 HRL Laboratories, LLC. All Rights Reserved

Subsystem Definition Example


PLANT

SENSOR

COMPUTER

CONNECTION

INCORRECT

SIGNAL

INCORRECT

PHYSICAL

VALUE

SENSOR

RESISTANCE

7

© 1998 HRL Laboratories, LLC. All Rights Reserved

Model Development

•

Decompose modeled system into small subsystems


•

Define model granularity


•

Create simple models for subsystems and test performance


•

Gradually increase model complexity


•

Integrate subsystem models into a single system model


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© 1998 HRL Laboratories, LLC. All Rights Reserved

System

Decomposition

•
Determine

system

complexity

by

combining

–
Number

of

replaceable

components

or

faults

–
Number

of

tests,

symptoms,

error

messages


•
Subdivide

system

by

functional

parts


•
Identify

experts

from

–
System

Design/Engineering

–
Maintenance/Repair

9

© 1998 HRL Laboratories, LLC. All Rights Reserved

Subsystem

Definition

•
Fault

list


•
Rank

faults

by

failure

frequency


•
Observation

list

–
Failure

symptoms

–
Computer

error

messages

–
Built

in

test

results

–
Fault

troubleshooting

data

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© 1998 HRL Laboratories, LLC. All Rights Reserved

Simple

Subsystem

Model

•
One

fault,

conditionally

independent

observations


•
Causal

probability

determination

–
Only

necessary

for

fault
-
observation

pairs

–
All

others

zero


•
Thorough

testing

11

© 1998 HRL Laboratories, LLC. All Rights Reserved

INCORRECT

PHYSICAL

VALUE

INCORRECT

SIGNAL

FAULT NODE:


•

PLANT

•

SENSOR

•

CONNECTION

•

COMPUTER


Simple Network

Model Example



SENSOR

RESISTANCE

12

© 1998 HRL Laboratories, LLC. All Rights Reserved

INCORRECT

SIGNAL

INCORRECT

PHYSICAL

VALUE

SENSOR

RESISTANCE

PHYSICAL

VALUE

SIGNAL

PLANT

COMPUTER

CONNECTION

SENSOR

Complex Network

Model Example


13

© 1998 HRL Laboratories, LLC. All Rights Reserved

Probability Calculations


Goal: computation of the joint probability

distribution of all components influencing a

given test, i.e. calculation of the ensemble


{P(C
1
, C
2
, …, C
n
,T)}




for all Tests T and corresponding

adjacent components
C
i


C
1


T

C
2

C
n

14

© 1998 HRL Laboratories, LLC. All Rights Reserved

Probability Elicitation

•
Diagnostic Probability
–

Intuitive to Diagnostic Experts

–
conditional probability of the form P(C|T), indicating the
likelihood that a component fails given a particular test has
returned a failure condition

–
Example: P(Generator Defective | Alternator Light = On) = 0.65

•
Causal Probability
–

Counter
-
Intuitive to Diagnostic
Experts

–
conditional probability of the form P(T|C), indicating the
likelihood of a particular test outcome given a component has
failed

–
Example: P(Alternator Light = On | Generator Defective) = 0.8

•
Prior

Probability

–
unconditional

probability

of

component

failure

P(C)

–
Example
:

P(Generator

Defective)

=

0
.
25


15

© 1998 HRL Laboratories, LLC. All Rights Reserved

What Probability

Information is Sufficient?






•
Question
:

Given

the

prior

component

probability

distribution

{P(C)},

and

the

diagnostic

probability

distribution

{P(C|T)},

is

it

possible

to

uniquely

determine

the

causal

probability

distribution

{P(T|C)}

and

therefore

the

joint

distribution

{P(C,T)}?


•
Answer
:

NO
.

Prior

and

diagnostic

probability

information

does

not

characterize

causal

and

joint

probabilities
.

There

are

infinitely

many

causal

and

joint

probability

distributions

resulting

from

fixed

prior

and

diagnostic

probability

information
.



16

© 1998 HRL Laboratories, LLC. All Rights Reserved

Successful Elicitation





Given


•
{P(C
1
,

C
2
,

…
,

C
n
|T)}


•
(distribution

of

all

diagnostic

probabilities)

•

P(C
1
,

C
2
,

…
,

C
n
)


•
(single

prior)

•

P(C
1
,

C
2
,

…
,

C
n
|T’)


•
(single

nonfailure

diagnostic)


we

can

calculate


•
{P(C
1
,

C
2
,

…
,

C
n
,T)}


•
(joint

distribution)

•
{P(T| C
1
, C
2
, …, C
n
)}

•
(distribution of all causal probabilities)


•
Implementation: Matlab

C
1


T

C
2

C
n

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© 1998 HRL Laboratories, LLC. All Rights Reserved

Conclusion

•
Methodology of Bayesian Network Design

–
Iterative

–
Hierarchical

–
Model fidelity control

–
Simplified verification and testing


•
Probability Elicitation

–
Natural for diagnostic expert

–
Automatic re
-
computation of probabilities for the network