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).
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