u; i U;' U;.

kettlecatelbowcornerΤεχνίτη Νοημοσύνη και Ρομποτική

7 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

104 εμφανίσεις











A

Generalization

o
f

the

Noisy
-
Or

Model







Knowledg
e

System
s

Laboratory
Compute
r

Scienc
e

Department Stanfor
d

University,

C
A

94305



Abstract


Th
e

Noisy
-
Or

mode
l

i
s

convenient

fo
r
de-
scribing

a

clas
s

o
f

uncertain

relationships
i
n

Bayesia
n

network
s

[Pearl

1988]
.

Pearl
describes

th
e

Noisy
-
Or

model

fo
r

Boolean
variables
.

Her
e

w
e

generalize

th
e

model

to
nar
y

inpu
t

and

outpu
t

variables

an
d

t
o

ar-
bitrar
y

function
s

other

tha
n

th
e

Boolean
O
R

function.

Thi
s

generalization

i
s a useful

modelin
g

ai
d

fo
r

constructio
n

of Bayesian

networks
.
W
e

illustrat
e

with
som
e

example
s

includin
g

digital

circui
t

di-

agnosis

an
d

network

reliabilit
y

analysis.

u;

=

f
.


Whe
n

a

lin
e

failur
e

doe
s

no
t

occu
r

o
n

line
i

th
e

devic
e

just

transmits

it
s

inpu
t

t
o

it
s

output,
i.e.
,

U;
'

=

U;.

This

non
-
failur
e

event

occurs

with
probabilit
y

1

-

q;.

This

overall

structure

induces a

probability
distributio
n

P(XIU1
,

U
2

..
.
,

Un
)

which

i
s

easily
computable[Pearl1988]

.

When

eac
h

U
;

i
s

interpreted

as

a

"cause
"

o
f

the

"effect
"

X
,

th
e

Boolean

Noisy
-
Or

models

dis
j
unctive

interactio
n

o
f

th
e

causes.

Eac
h

caus
e

i
s

"inhibited"

with

probabilit
y

q;
,

i.e.,

ther
e

i
s

a

probability

q
;

that
eve
n

when
th
e

caus
e

U
;

i
s

active
,

i
t

wil
l

no
t

affect
X.



1

INTRODUCTION


Th
e

Boolea
n

Noisy
-
O
r

structur
e

serves

as

a

use-
fu
l

mode
l

fo
r

capturin
g

non
-
deterministi
c

disjunc-
tive

interactions

between

th
e

cause
s

o
f

a
n

effect
[Pear
l

1988
]

.

Th
e

Boolean

Noisy
-
O
r

ca
n

b
e

explaine
d

as

fol-

lows
.

Conside
r

a

Boolea
n

O
R

gat
e

wit
h

multiple
input
s

U1
,

U2
,

..
.
,

U
n

an
d

a
n

outpu
t

X
.

Now

c

n-
side
r

som
e

non
-
determinism

associate
d

wit
h

eac
h

m-
put

define
d

as

follows:

O
n

eac
h

inpu
t

lin
e

U
;

a

non-

deterministi
c

lin
e

failur
e

functio
n

M

i
s

introduced
(see

Fi
g

1
,

considering

F

t
o

b
e

a

Boolean

?
R

gate).
Th
e

lin
e

failur
e

function

M

take
s

U;

as

mpu
t

and
ha
s

a

Boolea
n

outpu
t

u;.

Instea
d

o
f

U
;

bein
g

con-

necte
d

t
o

th
e

O
R

gate

w
e

no
w

hav
e

u
:

connecte
d

to
the

O
R

gat
e

instead.

Th
e

line

failure

functio
n

ca
n

b
e

conceptualized
a
s

a

non
-
deterministi
c

devic
e

-

there

i
s

a

probabil-
it
y

q
;


(calle
d

th
e

inhibito
r

probability
)

that

th
e

l

ne

failure

function

causes

a

'lin
e

failure'.

Whe
n

a

lme
failur
e

occur
s

o
n

lin
e

i
,

th
e

outpu
t

o
f

th
e

devic
e

is
f

(i.
e.
,

false)

irrespective

o
f

wha
t

th
e

input

is
,

i.e.,




Als
o

wit
h

Rockwell

International

Science

Center,
Palo

Alt
o

Laboratory
,

Palo

Alto
,

C
A

94301.

I
n

a

Bayesian

networ
k

interpretation,

eac
h

of
th
e

variable
s

U;

ca
n

b
e

considere
d

a
s

a

predeces-
so
r

nod
e

o
f

th
e

variabl
e

X.

Th
e

conditional

proba-
bilit
y

distributio
n

P(XIU1
,

U
2

..
.
,

Un
)

i
s

computed
fro
m

th
e

probabilities

q
;.


In

domains

where

suc
h

dis-
junctiv
e

interaction
s

occur
,

instead

o
f

full
y

specify-

in
g

opaqu
e

conditional

probability

distributions
,

the
Noisy
-
Or

mode
l

can

b
e

use
d

instead.

Th
e

inhibitor
probabilitie
s

ar
e

fe
w

in

number

(on
e

associat.ed

with

eac
h

predecessor

U
;

o
f

X)

and

woul
d

b
e

intuitively

easie
r

t
o

specif
y

becaus
e

o
f

thei
r

direc
t
relatio
n

to

th
e

underlyin
g

mechanis
m

o
f

causation.

This

pape
r

generalizes

th
e

Noisy
-
O
r
mode
l

to
th
e

cas
e

wher
e

bot
h

th
e

'c
a
use
'

variable
s
U;

an
d

'ef-
fect'

variable

X

nee
d

no
t

b
e

Boolean.

Instead,

they

ca
n

b
e

discrete

variable
s

wit
h

an
y

numbe
r

o
f

states.
Furthermore

th
e
underlyin
g

deterministic

function
i
s

no
t

restri
c
t
e
d

to
b
e

the

Boolea
n

O
R

function,

it

can

b
e

an
y

discret
e

function.

I
n

othe
r

yvords
,

in

Fig

1
,

F

ca
n

b
e

an
y

discrete

function.

Seen

as

a

modelin
g

tool
,

thi
s

generalizatio
n

pro-

vides

a

framework

t
o

move

fro
m

a
n

underlying

ap-

proximat
e

deterministic

mode
l

(th
e

function

F
)

t
o

a

more

realisti
c

probabilistic

mode
l
(th
e

distribution
P(XIU1,

U
2

..
.
,

Un))

wit
h

th
e

specificatio
n

o
f

onl
y

a
fe
w

probabilisti
c

parameters

(th
e

inhibitor

probabil-
ities).





n

A

Generalization

o
f

th
e

Noisy
-
Or

Model

209



u
1


N
t

u


u
2

N
2

u


X

F



L_...J

Un

N
n

u
'
n



Figure

1
:

Th
e

generalized

Noisy
-
Or

model.



I
n

domains

where

th
e

generalized

Noisy
-
Or
is
applicable,

i
t

make
s

th
e

modelin
g

tas
k

muc
h

easier
when

compared

t
o

th
e

alternative

o
f

direc
t

specifi-

catio
n

o
f

th
e

probabilistic

mode
l

P(XjU
1
,
U
2

.

.
.
Un)∙

similarl
y

define
d

quantitie
s

u

(j)
,
u;
,

I
;

associated
wit
h

th
e

variabl
e

u
;

0

Th
e

lin
e

failur
e

functio
n

M

associates

a

prob-

abilit
y

value

P
/

h
(
j
)


with

every

inde
x

0




j

<

m;.

Thi
s

quantity

can

b
e

rea
d

as

th
e

inhibito
r

probabil-

it
y

for

th
e

jt
h

stat
e

o
f

inpu
t

u
i

0

Th
e

line

failur
e

functio
n

can

b
e

conceptualized

as

a

non
-
deterministic

devic
e

that

take
s

th
e

value

of
U;

as

th
e

input

an
d

outputs

a

valu
e

for

u;
.

Thi
s

de-
vice

fails

with

probability

prh(j)

i
n

stat
e

j
.

When
a

failure

i
n

state

j

occurs
,

th
e

outpu
t

o
f

th
e

device
is

u;(j
)

regardless

o
f

th
e

input
.

Whe
n

n
o

failur
e

oc-
curs
,

i
f

th
e

inpu
t

is

u;(j
)

th
e

outpu
t

i
s

u
;
(j
)

-

this

ca
n

b
e

v
iewe
d

as

"passin
g

th
e

inpu
t

throug
h

t
o

the
output"

(not
e

tha
t

th
e

inde
x

j

o
f

th
e

outpu
t

state
an
d

th
e

inpu
t

state

ar
e

sam
e

in

this

case)
.

Th
e

prob
-

ability

o
f

n
o

failur
e

occurin
g

i
s

denote
d

b
y

P
t
o
f
ai
l
.
W
e

se
e

that:

I
n

suc
h

domains,

th
e

tas
k

o
f

creating

a

Bayesia
n

net
-

P

o
j
ai
l

=

1

_

L

n
h




wor
k

woul
d

procee
d

a
s

follows:

t

P/

O

j<m;

(
j
)



Variable
s

an
d

deterministic

function
s

that

re-
lat
e

the
m

and

approximate
th
e

non
-
deterministic

behaviour

o
f

th
e

domain
ar
e

identified.




A

networ
k

i
s

created

wit
h

thi
s

informatio
n

with
a

nod
e

fo
r

eac
h

variable
,

and

a

link

fro
m

each
o
f

U1,

U2
,

..
.
,

Un

t
o

X

fo
r

eac
h

relatio
n
of
form

X

=

F(
U
t
,
U2
,
..
.
,
Un)



(Th
e

networ
k

is
assume
d

t
o

be

acyclic).



Inhibitor

probabilitie
s

for

eac
h

lin
k

i
n

th
e

net-

wor
k

ar
e

elicited.



Th
e

generalize
d

Noisy
-
Or

mode
l

i
s

use
d

t
o
au-
tomaticall
y

'lift'

th
e

network

fro
m

th
e

previ-
ou
s

ste
p

int
o

a

fully

specifie
d

Bayesian

network
whic
h

ha
s

th
e

sam
e

topolog
y

a
s

th
e

network.



2

TH
E

GENERALIZE
D

MODEL

Th
e

generalize
d

Noisy
-
Or

model

i
s

illustrate
d

i
n

Fig

Th
e

outpu
t

X

i
s

a

discrete

rando
m

variabl
e

with
m.,

states
.

W
e

wil
l

refe
r

t
o

th
e

jth

state

o
f

X

as

x(j)
and

us
e

x

t
o

refe
r

t
o

"any

stat
e

o
f

X".

F

(se
e

Fi
g

1
)

ca
n

b
e

conceptualize
d

a
s

a

de-
terministic

devic
e

tha
t

output
s

som
e

valu
e

x

o
f

X
for

eac
h

possible

joint

stat
e

u

,

u

,

..
.
,

u



o
f

the

inputs

U

,
U

,
..
.
,
U

.


I
n

othe
r

words

F

i
s

a

dis-
crete

functio
n

tha
t

map
s

th
e

spac
e

o
f

join
t

states

of
U



X

U



X



.



X

U


int
o

th
e

se
t

o
f

state
s

o
f

X
.

W
e

note

that

th
e

mode
l

describe
d

abov
e

induces

a
n

uncertai
n

relationship

between

th
e

output

X

and
th
e

variables

Ui.

Thi
s

relationship

i
s

captured
b
y

the
conditiona
l

distribution

P(XIU1
,
U2
,
..
.
,
Un)∙

I
n

th
e

nex
t

section

w
e

procee
d

t
o

show

how
thi
s

conditiona
l

distributio
n

i
s

compute
d

from

the
function

F

and

th
e

inhibito
r

probabilities
.

W
e

will
us
e

th
e

notatio
n

U

t
o

denot
e

th
e

vecto
r

o
f

vari-
able
s

[U1
,
U2
,
..
.
,
Un]
.


Similarly
,

w
e

wil
l

us
e

u

to
denot
e

an
y

join
t

state

[
u

1
,

U2
1







,

Un
]

o
f

U
.

U
'

and

'

u

ar
e

define
d

similarl
y

wit
h

respec
t

t
o

th
e

variables

1.


Eac
h

U
;

i
s

a

discrete

rando
m

variable.
Eac
h

u;
i
s

a

discret
e

rando
m

variable

with

th
e

sam
e

number
o
f

states

a
s

U;.

W
e

wil
l

refer

t
o

th
e

number

o
f

state
s

o
f

U;

and

u
;

a
s

m;.

W
e

wil
l

refe
r

.t
o

th
e

jth

stat
e

o
f

U;

as
u
;
(j
)

wher
e

0



j

<

m;.

W
e
cal
l

j

th
e

inde
x

o
f

state
u;(j).

W
e

wil
l

us
e

u
;

t
o

denot
e

"any

stat
e

o
f

U;"
.

As
a
n

example

o
f

th
e

us
e

o
f

u;,

conside
r

th
e

statement,
"Ever
y

stat
e

u
;

o
f

U;

ha
s

a

uniqu
e

index

associated

with

it"

.

W
e

define

I;

t
o

b
e

th
e

function

tha
t

returns

the

inde
x

o
f

a

stat
e

u
;

o
f

U;
,

i.e.
,

I;(u;)

=

j

wher
e

j

is
th
e

index

o
f

state

u
;

o
f

variabl
e

U;
.

W
e

als
o

have

u;
.

Not
e

tha
t

P(XIUt,U2
,
.
.
.
,Un
)

abbreviates

to

P(XIU).

I
n

th
e

sp
e
cial

case where

ever
y

inhibito
r

proba-

bilit
y

is

zer
o

eac
h

variable

u;

always

ha
s

th
e

"same"

value

as

U
;

(i.e.
,

th
e

state

o
f

u
;

has

th
e

sam
e

index
a
s

th
e

stat
e

o
f

U;)
.

I
n

this

specia
l

cas
e

th
e

variables
U
;
I


become

su
p
erfluous
,

w
e

could

J


US
t

a
s

wel
l

remove

th
e

line

failure

function
s

an
d

connec
t

th
e

each

input

U;

directl
y

throug
h

t
o

F.

I
n

thi
s

specia
l

case,

th
e

overal
l

mode
l

degener-

ate
s

t
o

a

deterministic

function

wher
e

th
e

valu
e

of
outpu
t

X

i
s

determine
d

fro
m

th
e

value
s

o
f

th
e

input
variable
s

U;

b
y

th
e

functio
n

F
.

Thu
s

th
e

general
-











a

a

a

P
(

)

{

t

u

i
f

u

=

;

u
i


=

q
;

=

N;

I

U

L

=

210

Srinivas



ize
d

Noisy
-
Or

mode
l

can

b
e

viewe
d

as

starting

with
a

deterministi
c

mode
l

(th
e

functio
n

F
)

an
d

the
n

in-
troducin
g

failure
s

i
n

th
e
inputs
,

viz
,

th
e

inhibitor
probabilities,

resulting
finally

i
n

a

non
-
deterministic
model.


3

CHAR
A

CTERIZIN
G

P(XIU)

W
e

not
e

tha
t

w
e

hav
e

already

defined

P
(
u
;

j

u
;

)

i
n
term
s

o
f

th
e

inhibito
r

probabilities.

Th
e

abov
e

equation

is

easil
y

converted

to

a
n

al-

gorith
m

(describe
d

later
)

to

generat
e

a

conditional

probabilit
y

tabl
e

give
n
th
e

inhibito
r

probabilities

an
d

th
e

function

F.


3.
1

BOOLEA
N

NOISY
-
O
R

A
S

A

Each

lin
e

failure

function

M

defines

a

probability

SPEC
I

A
L

C
A

S
E

distribution

P;(U;jU;
)

relatin
g

u;

an
d

U;
.

Fro
m

th
e

Th
e

generalized

Noisy
-
O
r

collapses

t
o

b
e

the
mode
l

fo
r
.M

w
e

se
e

that

th
e

distributio
n

P
;

i
s

cal
-


Boolean

Noisy
-
O
r

[Pear
l
1988]

whe
n

al
l

th
e

variables
culate
d

as:


ar
e

Boolea
n
1
,


th
e

function

F

i
s

th
e

Boolean

OR,


,

_

P

ofail

+


i
n
h(I;
(


;)
)

I;
(

;)

I

(

)

inh
(
O
)


an
d

P
jn
h
(
1
)


0
.
I
n

other

words,

i

U;

u;

-


pfnh(l
(
u
1.))


otherwise

can

fai
l

with

probability

q
;

wit
h

th
e
output

being

(
1)

"false"

bu
t

it

canno
t

fai
l

with

output

being

"true
"

.
Th
e

equation

abov
e

summarize
s

th
e

following

Le
t

/
;


n
d


t
;

de
n

t
e
.

th
e

"true"

and

"false"

facts
:

i
f

th
e

th
e output

u
;

o
f

.M

i
s

th
e

"same"

as

state
s

?
f

vanabl
e

U;
.

Sim
i
!
arl
y

w
e

hav
e

f
x

an
d

.
t
x

t
h
e

I


np
u
t

u
;

(
1
'

.e.
,

t
h
e

m



d
I
'

ce
s

o
f

b
o
t
h

ar
e

t
h
e

same
)
,

fo
r

variabl
e
.

X
.

Th
e

followmg

can

b
e

show
n

easily

the
n

either

th
e

device

M

i
s

workin
g

normally

o
r

i
t


fro
m

equatiO
n

2

above:

ha
s

faile
d

i
n

th
e

stat
e

u

.

I
f

th
e output

u
;

is

not

th
e

"same
"

as

inpu
t

u;,

the
n

th
e

devic
e

ha
s

faile
d

in

stat
e

u;.

W
e

no
w

characterize

th
e

distribution

P
(X
j
U
)

P
(
f
x

l
u
)


=

IT

q;

{ilu;=ti}

i
n

term
s

o
f

th
e

inhibitor

probabilities

fo
r

eac
h

U;

an
d

th
e

functio
n

F.

W
e

not
e

that:

P(xju)

=

L

P(xju1
,

u)P(u1ju)

ul

1
-


IT

{ilu;=t;}


4


INTERESTIN
G

SP
E

CIAL
CASES

W
e

not
e

that

once

w
e

kno
w

th
e

stat
e

1

o
f

U

1

,

4.
1

CHOIC
E

O
F

A

FUNCTIO
N

F

w
e

kno
w

th
e

value

x

o
f

X
,

since

x

=

F(u')
.

I
n

other
words
,

X

i
s

independen
t

o
f

U

onc
e

U
1

is

known.
Th
e

above

equatio
n

therefor
e

simplifie
s

to:

P(xju)

=

L

P(xju')P(u1ju)

ul

W
e

not
e

tha
t

P(xju1
)

=

1

whe
n

x

=

F(u1)

and
P(xju1
)
=

0

when

x

-
f:.

F(u1).

This

simplifie
s

the
defining

equatio
n

to:

Th
e

generalized

model

described

abov
e

allow
s

the
use

o
f

an
y

discrete

function

F

relatin
g

U

to

X
.

We
no
w

sugges
t

a

particula
r

for
m

o
f

F

tha
t

is

'compat-
ible'

with

th
e

Boolean

Noisy
-
Or
,

i
.
e.
,

F

degenerates

t
o

th
e

Boolean

O
R

function

whe
n

th
e

input
s

and

output
s

ar
e

Boolean23:

P(xju)

=



{U1Ix=F(
U
1
)}

P(
u
I

ju)

I
n

essence,

thi
s

functio
n

i
s

a

weighte
d

average

-

w
e

ar
e

finding

the

fractio
n

o
f

eac
h

input's

state's

index

ove
r

th
e

maximu
m

possibl
e

inde
x

o
f

that

in-

Now

w
e

not
e
tha
t

th
e

dependence

o
f

U
1

=
[u1,

u2,

...

,

u
n

]

on

u

=

(
u1,

u2
,

..
.

,

Un
]

can

b
e

split
int
o

n

pairwis
e

dependence
s

o
f

u
;

on

u;.

This

i
s

be-
caus
e

th
e

value

o
f

a

variabl
e

U
/

depend
s

solel
y

on
U
;

an
d

no
t

on

an
y

other

variabl
e

U
j

where

i
-
f:
.

j.

Thu
s

w
e

can

simplify

th
e

equatio
n

to:

P(xju)

=


P(ulju)

{
U
1
I
x
=
F(U1)}

put,

averaging

thes
e

fractions
,

scaling

thi
s

quantity

to

th
e

maximu
m

inde
x

o
f

th
e

output
,

and

mapping

bac
k

to

an

actua
l

stat
e

o
f

th
e

output

afte
r

converting

th
e

scaled

result

to

a
n

integer.


1

Fo
r

Boolean

variable
s

w
e

define

the

inde
x

o
f

the
"
f
alse"

state

t
o

b
e

0

and

th
e

inde
x

of

th
e

"true
"

state

to
b
e

1.

2We

us
e

the

syntax

rl

for

th
e

Ceiling

function.

Fo
r

a
real

numbe
r

x,

r
X

l

i
s

the

smallest

intege
r

i

tha
t

satisfies
i

>

x.

IT

P
;
(
u;ju
;
)

(2)

-

3In

th
e

f
o
llowing

equation
,

not
e

agai
n

tha
t

x(j)

de-

{
U1Ix=F(U1)}

u'

notes

th
e

jth

stat
e

o
f

X.





'

A

Generalizatio
n

o
f

th
e

Noisy
-
Or

Mode
l

211



Thi
s

additive

function

wil
l

hav
e

th
e

characteris- tic

tha
t

as
an
y

inpu
t

goe
s

'higher
'

i
t
wil
l
ten
d

t
o

drive
th
e

outpu
t

'higher'.

Further
,

th
e

input
s

ar
e

'equally
weighted
'

regardles
s

o
f

thei
r

arity
.

So
,

fo
r

example,
a

chang
e

fro
m

stat
e

0

t
o
stat
e

1

i
n

a

Boolea
n

input
wil
l

hav
e

jus
t

th
e

sam
e

effec
t

a
s

a

change

fro
m

0

to

5

i
n

a
n

inpu
t

wit
h

6

states.

F

inally
,

th
e

output

i
s

0

i
f

an
d

onl
y

i
f

al
l

th
e

input
s

ar
e

0.

W
e

not
e

that

thi
s

functio
n

reduce
s

t
o

the

Boolea
n

O
R

functio
n

i
n

th
e

cas
e

wher
e

al
l

inputs

ar
e

Boolea
n

and

th
e

output

i
s

Boolean.


4.2

CAS
E

O
F

BOOLEA
N

OUTPUT

AND

nAR
Y

INPUTS

Conside
r

th
e

cas
e

wher
e

X

i
s

a

Boolea
n

variable
an
d

th
e

input
s

U
;

ar
e

nary
.

Th
e

functio
n

F

i
s

de-
fine
d

a
s

i
n

th
e

previou
s

section
.

Further,

w
e

define

Finally
,

w
e

not
e

tha
t

th
e

Boolea
n

Noisy
-
O
r

for-
mulatio
n
o
f

[Pearl

1988
]

and

it
s

generalizatio
n

to
nar
y

inputs

describe
d

i
n

Sectio
n

4.2

alway
s

result
i
n

a

distributio
n

whic
h

is

not

strictl
y

positiv
e

since
P
(
t
x
i
f
)

=

0.


5

COMPUTIN
G

P(XIU)

W
e

conside
r

th
e

complexit
y

o
f

generatin
g

th
e

prob-

abilitie
s

i
n

th
e

tabl
e

P(XIU).

Le
t

S

=

IJ;

m
;

b
e

th
e

siz
e

o
f

th
e

join
t

state
spac
e

o
f

al
l

th
e

inputs

U;
.

W
e

firs
t

not
e

that
P;(u

lui
)

ca
n

b
e

compute
d

i
n

e(1
)

tim
e

fro
m

the

inhibitor

pro
b
abilities.

Thi
s

lead
s

to:

P(
u
'
ju
)

=

II

P;(
u

iu;
)

=

e(n)


Therefore:

Pinh(O
)

=

q
;


an
d

Plnh(j
)

=

0

fo
r

j

f.

0
.

W
e

se
e

that

w
e

hav
e

a

restricted

generalizatio
n

o
f

th
e

Boolean

Noisy
-
Or.

P(xju)

=



{
x
l
x
=F(U'
)
}

P(u'ju)

=

e(S
n
)

This

specia
l

ca
s
e

o
f

nar
y

input
s

and

Boolean outpu
t

i
s

interestin
g

sinc
e

i
t

ha
s

bette
r

computa-
tional

propertie
s

than

th
e

general

cas
e

while

be-
in
g

mor
e

general

tha
n

th
e

Boolea
n

Noisy
-
O
r

(see
Se
c

5.2).


4.
3


O
B
TAININ
G

STRICTL
Y

POSITIVE
D
I

STR
I

BU
T
IONS

I
n

som
e

situations

i
t

i
s

desirabl
e

fo
r

th
e

condi-
tiona
l

distribution

o
f

a

Bayesian

network

nod
e

X
with

predecessors

U

t
o

b
e

strictly

positive
,

i.e.

,
VxVuP(xju
)

>

0.

Fo
r

th
e

generalized

Noisy
-
O
r

model
,

th
e

defini-

tion

o
f
P(xju
)

i
s

i
n

Equatio
n
2
.

Fro
m

thi
s

definition

w
e

not
e

tha
t

th
e

following

condition

is
necessar
y

to

ensur
e

a

strictly

positiv
e

distribution:


Fo
r

al
l
state
s

x

o
f

X
,

th
e

se
t

{u'j
x

=
F(u')
}

i
s

no
t

empty
.

I
n

othe
r

words
,

F
should

b
e

a

function

tha
t

map
s

ont
o

X.


Thi
s

conditio
n

i
s

a

natura
l

restrictio
n

-

i
f

F
doe
s

no
t

satisf
y

this

condition,

th
e

variable

X
,

i
n

ef-
fect
,

ha
s

superfluou
s

states
.

Fo
r

example
,

th
e

func-
tion

define
d

i
n

Section

4
.
1
satisfie
s

this

restriction.

Assuming

that

th
e

abov
e

conditio
n

i
s

satisfied,

Thi
s

is

because,

fo
r

a
give
n

x

an
d

u

w
e

hav
e

to

traverse

th
e

entire

stat
e

spac
e

o
f

u'

t
o

chec
k

which

u
'

satisf
y

x

=

F(u').

T
o

comput
e

th
e

entir
e

tabl
e

w
e

ca
n

naively

comput
e

each

entr
y

independentl
y

i
n

which

case

we

have:

P(XIU
)

=

mxSe(Sn
)

=

e(mxnS2)

Howeve
r

th
e

followin
g

algorithm

compute
s

the
tabl
e

i
n

e(nS2):

Begi
n

Algorithm

Fo
r

eac
h

stat
e

u

o
f

U:



Fo
r

al
l

state
s

x

o
f

X

set

P(xju
)

t
o

0.



Fo
r

eac
h

state

u
'

o
f

u

:

-

Se
t

x

=

F(u').

-

Increment

P(xju
)

b
y

P(u'ju).

En
d

Algorithm


5.1

BOOLEA
N

NOI
S

Y
-
OR

I
n

th
e

cas
e

o
f

th
e

Boolea
n

Noisy
-
Or
,

al
l

U
;

and

X

ar
e

Boolea
n

variables.

W
e

se
e

fro
m

Se
c

3.1

that:

P(f
x
i
u
)

II

q
;

=

e(n)

{
ilu
;
=
t
;
}

Fo
r

computing

th
e

table
,

w
e

se
e

tha
t
since

th
e

following

condition

i
s

sufficien
t

(though

no
t

nec-

P(txlu)

=

1
-

P(fxlu)
,

w
e

ca
n

comput
e

bot
h

prob-

essar
y
)

t
o

ensur
e

a

strictly

positiv
e

distribution:

Fo
r

an
y

u
'

and

u
,

P(u'ju
)

>

0
,

i.e.,

Ti
u
'

P;(u;ju;
)

> 0.

This

second

conditio
n

i
s

a

stronger

restriction.

Fro
m

Equatio
n

1

w
e

not
e that

thi
s

restriction
is

equivalen
t

t
o

requirin
g

tha
t

al
l

inhibitor
probabil-
ities

b
e

strictl
y

positive,

i.e.
,

that

prh(j)

>

0

for
allOs;j<m;.

abilitie
s

fo
r

a

particular

u

i
n

e
(
n
)

time
.
S
o

the
time

require
d

t
o

calculate

th
e

entir
e

tabl
e

P(XIU)
i
s

e(Sn).

W
e

se
e

that

i
n

th
e

cas
e

o
f
th
e

Boolea
n

Noisy
-
Or
there

i
s

a

substantia
l

savin
g

ove
r

th
e

genera
l

cas
e

in

computing

probabilities.

Thi
s

savin
g

is
achieve
d

by
takin
g

int
o

accoun
t

th
e

specia
l

characteristic
s

o
f

the
Boolea
n

O
R

functio
n

an
d
th
e

inhibito
r

probabilities
when

computing

th
e

distribution.







21
2

Srinivas



5.
2



BOO
L

EA
N

OUTPUT

AN
D

nARY
INPUTS


Fro
m

an

analysi
s

simila
r

t
o

th
e

previou
s

section

we
note

tha
t

computation

o
f

P
(
X
I
U
)

take
s

0

(
S
n
)

time
i
n

this

cas
e

too.


5.
3

STORAG

E

COMPLEXITY

Fo
r

th
e

genera
l

cas
e

w
e

need

t
o

stor
e

mi

inhibitor
probabilitie
s

pe
r

predecessor
.

Therefor
e

i
n

this
cas
e

0
(
n
mm
a
x)


storag
e

i
s

require
d
wher
e

m
m
a
x

=

mru
q

(
mi
)
.

Thi
s

contrast
s

with

O(mxm

a
x
)

fo
r

stor-

ing

th
e
whol
e

probability

table.

Fo
r

th
e

Boolea
n

Noisy
-
O
r

w
e

nee
d

t
o

store

on
e

inhibito
r

probability

pe
r

predecessor

and

this

is

e(n)
.

Usin
g

table
s

instea
d

woul
d

cos
t

0(
2

X

2n)

=

0(2n).

I
n

th
e

cas
e

o
f

nar
y

input
s

an
d

Boolean

output
(
a
s

described

abov
e
)

on
e

inhibito
r probability

per
predecessor

i
s

stored.

Thus

storag
e

requiremen
t

is

0(n)
.

Usin
g

a
tabl
e

woul
d

cos
t

O(
m

a
x
)



5.4

REDUCING

COM
P
UT
A
TION
COMP
L

EXITY


I
n

general
,

on
e

coul
d

reduc
e

the

complexit
y

o
f

com-
puting

P(:z:lu
)

i
f

one

coul
d

tak
e

advantag
e

o
f

special
propertie.s

o
f

the

functio
n

F

t
o

efficientl
y

generate

thos
e

u
'

tha
t

satisf
y

x

=

F(u'
)

fo
r

a

particula
r

x.

Give
n

a

function

F
,

w
e

thu
s

need

a
n

efficient
algorithm

Invert

suc
h

tha
t

lnver
t

(
x
)

=

{u
l

x

=

F(u)}.

By

choosing

F
carefully

one

can

devis
e

ef-

ficient

Invert

algorithms
.

However
,

t
o

b
e

usefu
l

as
a

modeling

device,

th
e

choic
e

o
f

F

ha
s

als
o

t
o

be
guide
d

b
y

the

more

importan
t

criterio
n

o
f

whether
F

doe
s

indee
d

mode
l

a

frequentl
y

occurrin
g

clas
s

of

phenomena.

Thi
s

Noisy
-
Or

generalizatio
n

ha
s

hig
h

complex-

it
y

fo
r

computing

probabilit
y

table
s

fro
m

the

in-

hibitor

probabilitie
s
4
.


I
f

the

generalizatio
n

i
s

seen

mostl
y

as

a

useful

modeling

paradigm,

the
n

this

complexity

i
s

not

a

problem,

sinc
e

the

inhibitor

probabilitie
s

can

b
e

pre
-
compil
ed

int
o

probability

table
s
befor
e

inferenc
e

takes

place.

Inferenc
e

can

b
e

the
n

performe
d

wit
h

standar
d

Bayesia
n

network

propagation

algorithms.

I
f

this

generalization
,

however
,

i
s

seen

as

a

metho
d

o
f

saving

storag
e

by

restricting

th
e

models

t
o

a

specifi
c

kin
d

o
f

interaction
,

the

cos
t

o
f

com-

puting

th
e

probabilities

o
n

the

fl
y

ma
y

outweigh

the

gain
s

o
f

savin
g

space.



4

However
,

th
e

Boolean

Noisy

O
r

doe
s

no
t

suffe
r

from
thi
s

p
r
ob
l
e
m


sinc
e

th
e

s
pecia
l


structure

o
f

the
F

f
u
nction

an
d

th
e

fact

tha
t

th
e

input
s

an
d

outputs

ar
e

Boolean

reduc
e

t
h
e

c
o
mpl
e
xit
y


dramatically

b
y

a

f
a
cto
r

of

S.


A



B

F




c


Each

line

ha
s

lh
e

probabilit
y

o
f

failure

marke
d

o
n

il.


Figur
e

2
:

A

digital

circuit












For

every

link

th
e

failure

function

N

hazJ

th
e

following

inhibitor
probabilities

(wher
e

X

i
s

th
e

predecesso
r

variabl
e

of

th
e

link):

p;rh(f
)

=

O.Q
l

an
d

p;rh(t)

=

0


Figur
e

3
:

A

generalized
Nois
y

o
r

mode
l

o
f

th
e

circuit




6

EXAMPLES


6.1

DIGITA
L

CI
R
CUI
T

DIAGN
O

SIS

Th
e

generalized

Noisy
-
Or

provid
e
s

a

straight-
forward

metho
d

fo
r

doing

digital circuit

diagnosis.
Consider

the

circuit

i
n

Fi
g

2
.

Let

u
s

assum
e

that

eac
h

lin
e

(
i.e.,

wir
e
)

i
n

the

circuit

ha
s

a

probability

o
f

failur
e

o
f

0.0
1

an
d

that
whe
n

a

lin
e

fails,

th
e
input

t
o

th
e

device
s

downstrea
m

o
f

th
e

lin
e

i
s

false.

Each

o
f

th
e

input
s

t
o

th
e

devices

i
n

the

circuit

i
s

now

modele
d

with

a

state

variable

i
n

a

Noisy
-
Or
model

(
se
e

Fi
g

3).

Th
e

functio
n

F

fo
r

th
e

general-
ize
d

Noisy
-
O
r

whic
h

i
s

associate
d

with

eac
h

nod
e

is

th
e

trut
h

table

o
f

th
e

digita
l

devic
e

whose

output
th
e

nod
e

represents.

W
e

have

a
n

inhibito
r

probabil-
it
y

o
f

0.0
1

associate
d

with

th
e

fals
e

stat
e

alon
g

each
lin
k

and

a
n

inhibito
r

probabilit
y

o
f

0

associate
d

with

th
e

true

stat
e

(
sinc
e

the

lines

canno
t

fai
l

i
n

the

true

stat
e

i
n

our

faul
t

mode
l
)
.

A

Bayesian

networ
k

i
s

no
w

constructe
d

from
th
e

Noisy
-
Or

mode
l

(
se
e

Fi
g

4)

usin
g

the

algorithm

described

in

Section

5.

Not
e

that

t
o
complete

the
Bayesia
n

network

one

need
s

the

marginal

distribu-
tion
s
o
n

the

inputs

t
o

the

circuit.

Her
e

w
e

have

mad
e

a

choic
e

o
f

uniform
distribution
s

fo
r

these







P

D

F=

D,E

P

A

D

B

tA,B

Prob

B


c

t

B,C

Prob

I

I

0.0198

I

I

0.9801

I

I

0.9999

I

f

0.9900

I

f

0.0000

I

f

0.9900

f

I

0.9900

f

I

0.0000

f

I

0.9900

f

f

0.0000

f

f

0.0000

f

f

0.0000


Ul

Un

A

Generalizatio
n

o
f

th
e

Noisy
-
Or

Mode
l

213




Gf


"
U
2
'
g}
--


G

X

U2

Un

-
---
-
a
:
-







L

-
----



Fi
gure

5:

Modeling

devic
e

failure

wit
h

a
n

'extended'
device.





Th
e
node


A,

B

an
d

C

ar
e

as::
n
g
ne
d

umform

ma.r
g
1na.l

dtstnbution

.

P(A"'

t)"'

P(B"'

t)"'

P(C"'

t)

"'0.5.


Figur
e
4
:

Bayesia
n

networ
k

fo
r

digita
l

circui
t

exam-

ple.



marginals
.
5

A
s

a
n

exampl
e

of

the

us
e

of

th
e

resulting

Bayesia
n

network
,

conside
r

t
he

diagnostic

question

"What

i
s the

distribution

o
f

D

given

F

is

fals
e

and

B

i
s

true

?".

Th
e

evidenc
e

B

=

t

an
d

F

=

f

i
s

declare
d

i
n

th
e

Bayesia
n

networ
k

an
d

an
y

stan-

dar
d

update

algorith
m

lik
e

the

Jensen
-
Spiegelhalter

[Jense
n

1989
,

Lauritze
n

1988
]

algorith
m

i
s

use
d

to

yiel
d

the

distributio
n

P(D

=

t!
F

=

j,

B

=

t
)

=

0.984

an
d

P(
D

=!IF
=

j,

B

=

t
)

=

0.016.

Note

that

thi
s

exampl
e
doe
s

no
t

include

a

model

fo
r

devic
e

failure

-

only

lin
e

failure
s

are

considered.

However

th
e

metho
d

ca
n

b
e

extende
d

easily

t
o

han-

dl
e

devic
e

failure

b
y

replacin
g

ever
y

devic
e
G

i
n

the

circui
t

with

th
e

'extended
'

devic
e

c'

as

show
n

in
Fi
g

5
.

I
n

thi
s

figure
,

th
e

input

(variable
)
G
1


ha
s

a
margina
l

distribution

whic
h

reflect
s

th
e

probability
o
f

failur
e

o
f

the

device
.
Al
l

the

inhibitor

probabilities
o
n

th
e

lin
e

G
1

ar
e

se
t

t
o

0
.

Note

tha
t

the

particu-
la
r

faul
t

mode
l

illustrate
d

here

i
s

a

'faile
d

a
t

false'
model,

i.e.
,

when

th
e

devic
e

i
s

broken,

its

output
i
s

false
.
On
e

nice

feature

of

th
e

method

described
above

i
s
tha
t

i
t

i
s

incremental.

I
f

a

device

i
s

added
o
r
remove
d

fro
m

th
e

underlying

circui
t

a

correspond-
in
g

nod
e

ca
n

b
e

added

or

remove
d

from

th
e

Bayesian


5Thes
e

marginals

ca
n

b
e

seen

as

th
e

distribution

over
the

input
s

provide
d

b
y

the

enVironmen
t

outsid
e

th
e

cir
-

Ea.ch

link ha.s

the

probability

o
f

failure

marke
d

o
n

it.


F
i
gur
e

6
:

A

networ
k

wit
h

unreliabl
e

links.




network

-

ther
e

is

no

nee
d

to

construc
t

a

complete
diagnostic

mode
l

from

scratch.

Thi
s

method
relate
s

ver
y

wel
l

t
o

the

model
based

reasonin
g

approac
h

i
n

thi
s

particular

do-
main
[deKlee
r

1987
,

deKlee
r

1989
,

Geffne
r

1987].
W
e

describe

a

probabilisti
c

approac
h

t
o

model-
base
d

diagnosi
s

using

Bayesia
n

network
s

i
n

detail
i
n

[Srinivas

1993b
,

Srinivas

1993a].


6.
2

NETWORK

CONNECT
I
VITY

Th
e

followin
g

exampl
e
use
s

the
Boolea
n

Noisy
-
Or
an
d

th
e

followin
g
exampl
e

generalize
s

i
t

t
o

us
e

the
generalize
d

Noisy
-
Or.

Consider

the

network

show
n

i
n

Fig

6
.

Sa
y

each

lin
k

is

unreliable
-

whe
n

th
e

lin
k

i
s

'down
'

th
e

link

i
s

no
t

traversable
.

Th
e

reliabilit
y

of

eac
h

lin
k

L

is

quantifie
d

b
y

a

probabilit
y

o
f

failur
e

I

(marke
d

on

the

lin
k

i
n

the

network)
.

No
w

consider

the

question

"Wha
t
i
s

the

probabilit
y

tha
t

a

pat
h

exists

from

A

toG?".

Consider

th
e

subse
t

o
f

the

networ
k

consisting

of

A

an
d

it
s

descenda
n
t
s

(i
n

ou
r

example,

fo
r

sim-

plicity
,

thi
s

i
s

the

whol
e

network).

W
e

first

asso-

cuit
.

Such

a

distribution

i
s

no
t

usually

available.

But

ciate

eac
h

nod
e

with

th
e

Boolea
n

O
R

as

the

F

func-

whe
n

th
e

distribution

i
s

not

available,

al
l

diagnosis

is

perforce

carrie
d

ou
t

wit
h

the

assumption

tha
t

al
l

inputs

ar
e

known.

Furthermore,

whe
n

all

the

input
s

ar
e

known,
i
t

i
s

t
o

b
e

note
d

tha
t

th
e

answe
r

t
o

an
y

diagnostic

ques-
tion

i
s

no
t

affecte
d

by

th
e

actual

choic
e

of

margina
l

as
lon
g

a
s

the

marginal

i
s

an
y

strictly

positive

distribution.

tion
.

Eac
h

of

th
e

lin
k

failur
e

probabilities

translates
directly
int
o

th
e

inhibitor

probability

fo
r

the

false
stat
e

along

eac
h

link
.

Th
e

inhibito
r

probability

for
th
e

true

stat
e

i
s

0.

Thi
s

networ
k

i
s
no
w

use
d

t
o

creat
e

a

Bayesian

network

usin
g

th
e

algorith
m

o
f

Se
c

5.

Th
e

Bayesian





networ
k

ha
s

the

sam
e

topolog
y

as

the

network i
n

Fi
g

6
.

T
o

complet
e

th
e

distribution

o
f

the
Bayesian

networ
k

th
e

roo
t

nod
e

A

ha
s

t
o

b
e

as-
signe
d

a

margina
l

distribution
.

W
e

assig
n

a
n
arbi- trar
y

strictl
y

positiv
e

distributio
n

t
o

the

roo
t

node



Fo
r
roo
t

nod
e

A
,

set

n
A

=

1.7



Fo
r

ever
y

no
n

roo
t
nod
e

U

in

the

graph
considere
d

i
n

graph

orde
r

(
with

ances-
tors

before

descendant
s
)

:

=

L
p
e
P
a
rents(U)

n
p

A

(
since

evidenc
e

i
s

going

t
o

b
e

declare
d

fo
r

th
e

root

node
,

th
e

actual

distributio
n

i
s

irrelevan
t
)
.

Th
e

answe
r

t
o

th
e

questio
n

asked

originall
y

is

no
w

obtained

as

follows:

Declar
e

the

evidenc
e

A

=

t
(
and

no

othe
r

evidenc
e
)

,

d
o

evidenc
e

propagation
and

loo
k

at

th
e

update
d

belief
o
f

G
.

I
n

thi
s

example,

we

get

Bel(
G

=

t
)

=

0.787
4

and
Bel(G

=

/)

=

0.2126.6

Thes
e

belief
s

ar
e

precisely

th
e

probabilities

tha
t

a

pat
h

exist
s

o
r

doe
s

not

exis
t

respectively

from

A

t
o

G.

T
o

se
e

why,

consider

the

cas
e

wher
e

link

failures

canno
t

happe
n

(
i.e.
,

link

failur
e

probabilit
y

i
s
zer
o
)

.

The
n

i
f

an
y

variable

i
n

the

network

i
s

declared

to

b
e

true

then

every

downstream

variabl
e

t
o

which

it

ha
s

some

pat
h

wil
l

als
o

b
e

tru
e

du
e

t
o

th
e

natur
e

of

the

Boolea
n

O
R

function
.

Onc
e

the

failur
e

proba-

bilitie
s

ar
e

introduced
,

belief

propagatio
n

give
s

us,

i
n

essence
,

the

probability

that

a

connecte
d

se
t

of

link
s

existe
d

between

A

and

G

forcin
g

the

O
R

gate

a
t

G

t
o

have

th
e

outpu
t

true.

Furthermore
,

i
t

i
s

t
o

b
e

note
d

that

becaus
e

be-

lie
f

propagatio
n

update
s

belief
s

a
t

every

node
,

the

probabilit
y

o
f

a

pat
h

existin
g

fro
m

A

t
o

any

node

X

downstream

o
f

i
t

i
s

available

as

Bel
(
X

=

t).

Thi
s

metho
d

can

b
e

extende
d

wit
h

som
e

minor

variation
s

t
o

answer

more

genera
l

questions

o
f

the

for
m

"Wha
t

i
s

the

probabilit
y

that

ther
e

exist
s

a

pat
h

fro
m

an
y

node

in

a

set

o
f

node
s

S
t
o

a

target

nod
e

T

?".


6.
3


NETWOR
K

CONNECTIVITY
EX
T

ENDED

Conside
r

the

exac
t

sam
e

networ
k

as

i
n

the

previ-
ou
s

example.

Th
e

question

no
w

aske
d

i
s

"Wha
t

is
the

probability

distributio
n

ove
r

th
e

number

o
f

paths
existing

fro
m

A

t
o

G

?".

Conside
r

the

subse
t

o
f

th
e

networ
k

consistin
g

of
A

an
d

it
s

descendants.

Fo
r

ever
y

nod
e

U

w
e

make
the

numbe
r

o
f

state
s

b
e

n
u

+

1

wher
e

nu

i
s

the
numbe
r

o
f

path
s

fro
m

roo
t

node

A

t
o

th
e

node

U.

Th
e

state
s

o
f

U

ar
e

numbere
d

fro
m

0

throug
h

nu.

W
e

wil
l

refe
r

t
o

the

ith

stat
e

o
f

nod
e

U

as

u
(
i
)

.

Th
e

number

nu

can

b
e

obtaine
d

fo
r

eac
h

node

i
n

th
e

network

throug
h

th
e

followin
g

simpl
e

graph

traversal

algorithm:


Begin

Algorithm


6The

updated

belie
f

Bel(X

=

x)

o
f

a

variable

X

is
the

conditiona
l

probability

P(
X

=

xiE)

where

E

i
s

all

th
e

available

evidence.

nu

En
d

Algorithm


To

buil
d

the

Noisy
-
O
r

model
,

w
e

no
w

associate
intege
r

addition

as

the

fun
ctio
n

F

associate
d

with
eac
h

node
.
Fo
r

example
,

i
f

R

and

S
ar
e

parent
s

of
T

and

the

stat
e

o
f
R

i
s

know
n

t
o

b
e

r2

an
d

th
e

state
o
f

Si
s

know
n

t
o

b
e

s3,

the
n

th
e

functio
n

map
s

this
stat
e

o
f

the

parent
s

t
o

stat
e

t(2+3
)

=

t
5

o
f

th
e

child

T.

W
e

no
w

set

th
e

inhibito
r

probabilities

as

fol-
lows
:

Sa
y

th
e

predecessor

nod
e

o
f
som
e

lin
k

L

in

the
grap
h

i
s

a

nod
e

U
.

W
e

set
th
e

inhibito
r

probabil-
it
y

fo
r

stat
e

u(O
)

t
o

b
e

the

lin
k

failure

probability

l

an
d

al
l

othe
r

inhibito
r

probabilities

t
o

b
e

0
.

Tha
t

is
P{ph(O
)

=

l
,

wher
e

l

i
s

the

lin
k

failure

probability
and

P{ph
(
i
)

=

0

fo
r

i

=

1
,

2

.
.
.

,

nu.

W
e

no
w

construc
t

the

Bayesia
n

network

from
the

network

describe
d

above.

Th
e

marginal

proba-
bilit
y

fo
r

the

roo
t

node

i
s

again

se
t

arbitrarily

t
o

any
strictly

positiv
e

distributio
n

sinc
e

i
t

ha
s

n
o

effect

on
t
.
h
e

result.

Th
e

answe
r

t
o

the

question

posed

abov
e

i
s

ob-

taine
d

b
y

declarin
g

the

evidence

A

=

1

and

then

doin
g

belief

propagatio
n

t
o

ge
t

th
e

update
d

beliefs

fo
r

G
.

Th
e

update
d

belie
f

distributio
n

obtained

for

G

i
s

precisely

the

distribution

ove
r

th
e

numbe
r

of

path
s

fro
m

A

t
o

G.

T
o

se
e

why
,

conside
r

the

cas
e

where

ther
e

ar
e

no

link

failures
.

The
n

whe
n

A

i
s

declare
d

t
o

have

the

valu
e

1
,

the

additio
n

functio
n

a
t

each

downstream

nodes

count
s
exactly

th
e

number

o
f

paths

fro
m

A

t
o

itself
.

Onc
e

the

failures

ar
e

introduced

the

ex-

ac
t

coun
t

becomes

a

distributio
n

ove
r

the

numbe
r

of

active

paths.

I
n

thi
s

example
,

w
e

get

th
e

distribution:

Bel(
G
=

0
)

=

0.2126
,

Bel(
G

==

1
)

=

0.3466,
Bel(
G

=

2
)

=

0.2576,

Bel(
G

=

3
)

=

0.1326
and
Bel(G

=

4
)

=

0.0506
.

W
e

see

that

Bel(
G

=

0
)

is
th
e

same

probability

as

Bel(G

=

f
)


i
n

the

previ-
ou
s

example
,

viz
,

th
e

probability

tha
t

n
o

pat
h

exists
fro
m
A

t
o

G.

Not
e

tha
t

afte
r

belief

updating,

th
e

distribution
o
f

numbe
r

o
f

paths

fro
m

A

t
o

an
y

nod
e

X

down-
strea
m

o
f

i
t

i
s

av
a
ilabl
e

as

the
distribution

Bel(X)
afte
r

belief

propagation.

Thi
s

method

can

b
e

extended

wit
h

t
o

answer

mor
e

genera
l

questions

o
f

the

for
m

"What

i
s

the

distribution

ove
r

the

numbe
r

o
f

path
s

tha
t

originate

7W
e

defin
e

the

root

nod
e

t
o have

a

single

pat
h

to
itself.



A

Generalizatio
n

o
f

th
e

Noisy
-
Or

Mode
l

215



i
n

an
y

node

i
n

a

set

o
f

nodes

S

an
d

terminate

in

a
target

node

T

?".

Anothe
r

interesting

exampl
e

which

ca
n be
solved

using

th
e

generalize
d

Noisy
-
Or

i
s

th
e

prob-
abilisti
c

minimu
m

cos
t

pat
h

problem:

Given

a

set
o
f

possibl
e

(positive
)

cost
s

o
n

eac
h

link

o
f

th
e

net-
wor
k

and

a

probabilit
y

distribution

ove
r

th
e

costs,
th
e
problem

i
s

t
o

determin
e

th
e
probabili
t
y

distri-
butio
n

over

minimu
m

cos
t

path
s

betwee
n

a

specified
pai
r

o
f

nodes.

Th
e

generalize
d

Noisy
-
Or
, in

fact
,

can

be
use
d

t
o

solv
e

a
n

entir
e

clas
s

o
f

networ
k

problems
[Sriniva
s

1993
c
]
.

Th
e

general

approac
h

i
s

a
s

in

the
examples

abov
e

-

th
e

proble
m
i
s

modele
d

usin
g

the
generalized

Noisy
-
O
r

and

the
n

Bayesia
n

propagation
i
s

use
d

i
n

th
e

resulting

Bayesian

networ
k

t
o

fin
d

the
answer.


Al
l

th
e

example
s

describe
d

abov
e

us
e

th
e

Noisy-
O
r

mode
l

a
t

ever
y

nod
e

in

th
e

network
.

However,
thi
s

i
s

no
t

necessary
.

Som
e

sections

o
f

a

Bayesian
networ
k

ca
n

b
e

constructed

'conventionally'
,

i.e.
,

by
direc
t

elicitation

o
f

topolog
y

and

inpu
t
o
f

probabil-
it
y

table
s

whil
e

othe
r

section
s

wher
e

th
e

Noisy
-
Or
mode
l

i
s

applicable
,

can

us
e

th
e

Noisy
-
O
r

formal-
Ism.


7

IMPLEMENTATION


This

generalized

Noisy
-
O
r

mode
l

ha
s

been

imple-
mente
d

i
n

th
e

IDEAL

[Sriniva
s

1990
]

system
.

When
creatin
g

a

Noisy
-
O
r

node
,

th
e

user

provide
s

th
e

in-
hibito
r

probabilitie
s

and

th
e

deterministi
c

function

F.

IDEA
L
ensures

tha
t

al
l

implemente
d

inference
algorithms

work

wit
h

Bayesian

networks

tha
t

con-
tai
n

Noisy
-
O
r

nodes
.

Thi
s

i
s

achieve
d

b
y

'compiling'
th
e

Noisy
-
O
r

informatio
n

o
f

eac
h

nod
e

int
o

a

con-
ditional

probability

distributio
n

fo
r

th
e

node
.

The

distribution

i
s

av
ailable

fo
r

all

inferenc
e

algorithms
t
o

use.


Acknowledgements

I

than
k

Richard

Fikes
,

Eri
c
Horvitz
,

Jac
k

Breese
an
d

Ken

Fertig

fo
r

invaluabl
e

discussions

an
d

sug-
gestions.


References


[deKlee
r

1987
]

d
e
Kleer
,

J
.

and
W
illiams,

B
.

C.
(1987
)

Diagnosing

multipl
e

faults.
Artifi
c

ia
l

Intelligence
,

Volume

32,
Numbe
r

1,

97
-
130.


[deKleer

1989]

d
e

Kleer
,

J
.

and

W
illiams,

B
.
C.
(1989
)

Diagnosis

wit
h
behavioral
modes
.

Proc
.

o
f

Eleventh

Interna
-

tiona
l

Join
t

Conferenc
e

o
n

AI
,

De-

troit
,

MI
.

1324
-
1330.

[Geffne
r

1987]

Geffner
,

H
. and

Pearl
,

J
.

(1987)
Distribute
d

Diagnosi
s

o
f

Systems
wit
h

Multipl
e
Faults
.

I
n

Proceed-
ings

o
f

the

3r
d

IEE
E

Conferenc
e

on
A
I

A
p
plications,

Kissimmee
,

FL,

Februar
y

1987
.

Als
o

i
n

Readings
i
n

Model

base
d
Diagnosis
,

Morgan
Kauffman.

[Jense
n

1989
]

Jensen
,

F
.

V.
,

Lauritze
n

S
.

L
.
and
Olese
n

K
.

G
.

{1989
)

Bayesia
n
up-
datin
g
in

recursiv
e

graphic
a
l

mod-
els

b
y

loca
l

computations
.

Repor
t

R

89
-
15,

Institut
e

fo
r

Electronic

Sys-

tems
,

Department

o
f

Mathematics

an
d

Compute
r

Science
,

University

o
f

Aalborg,

Denmark.

[Lauritze
n

1988]

Lauritzen
,

S
.

L
.

an
d

Spiegelhal-
ter
,

D
.

J
.
(1988)

Local

computa-
tion
s

wit
h

probabilities

o
n

graph-
ica
l

structure
s
an
d

thei
r

applica-
tion
s

t
o
exper
t

system
s

J.

R.
Statist.

So
c
.


B,

50
,

No
.

2,

157
-
224.

[Pear
l

1988
]

Pearl
,

J
.

(1988
)

Probabil
i
stic

Rea-
sonin
g
i
n

Intelligen
t

Systems
:

Net-
work
s

o
f

Plausibl
e

In
f
erence
.
Mor-
ga
n

Kaufman
n
Pu
b
lishers
,

Inc.
,
Sa
n

Mateo
,

Calif.

[Sriniva
s
1990
]

Srinivas,

S
.

and

Breese,

J
.

(1990)
IDEAL:

A

softwar
e

package for
analysis

o
f

influence

diagrams.
Proc
.

o
f

6t
h

Conf
.

o
n

Uncertainty
in

AI
,

Cambridge,

MA.

[Srinivas

1993a
]

Srinivas
,
S
.

(1993)

A

probabilistic
ATMS.

T
e
chnica
l

Report
,

Rockwell
International∙

Scienc
e

Center
,

Palo
Alt
o

Laborator
y
,

Pal
o

Alto
,

CA.

[Srinivas

1993b]

Srinivas
,

S
.

(1993)
Diagno
-

si
s

with

behavioural

modes us-
in
g

Bayesia
n

networks.
T
e
chni-
ca
l

Report,
Knowledg
e

Systems
Laborator
y
,

Compute
r

Scienc
e

De-
partment
,

Stanfor
d

University
.

(in
preparation).

[Srinivas

1993c
]

Srinivas,

S
.

(1993
)

Usin
g

th
e

gen-
eralize
d

Noisy
-
O
r

t
o

solv
e

proba-
bilistic

network

problems.

T
e
ch-
nica
l

Report
,

Knowledg
e
Systems
Laborator
y
,

Compute
r

Scienc
e

De-
partment
,

Stanfor
d

University
.

(in
preparation).