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NOISY OR GATEJavier Turek and Eyal Regev
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Several causes sharing a common effect.
YZ
Ui
e+
e-
X
Polytrees
U1 Un
… …
Like a tree, but with multiple parents.
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Link matrix Tx|u
Tx|u contains the conditional probabilities:P(X=x | U1=u1,…,Un=un)
1. The table is huge (contains 2n entries).
2. Who can know such information anyway?You cannot expect to find the P(X | U1,…,Un) table.
However, you may know how every Ui influences X separately.
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The OR-gateHaven’t done
homework
Caught cheating
Failed an exam More likely!
OR
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Inhibitors
Failed an exam
Caught cheating
OR
AND
Paid the TA
Haven’t done
homework
AND
Re-doing the
course
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Inhibitors
U1
X
U2
Un
I1
I2
In
OR
AND
AND
AND
Inhibitors are independent
Associate probability to an inhibitor.
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Noisy OR-Gate
U1
X
U2
Un
I1
I2
In
OR
AND
AND
AND
| , :i k iP x u u k i q
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Noisy OR-Gate
U1
X
U2
Un
I1
I2
In
OR
AND
AND
AND
:u iT i U True
Tu
if
|1 if +
u
u
ii T
ii T
q x
P x uq x
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Message Passing – updating X
YZ
U
e+
e-
X
| , | |Belief x P x e e P e x P x e D x A x
Y ZD x U x U x |x u Xu
A x T C u
YU x
y
XC u
ZU x
u
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Message Passing – updating X
YZ
U
e+
W
e-
X
| , | | ABelief x P x e e P e x P x e D x x
YU x
y
XC u u
ZU x
XC w
Y ZD x U x U x | ,,
x u Xw
Xwu
T C CuA x w
| , | |Belief x P x e e P e x P x e D x A x
|x u Xu
A x T C u
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Message Passing – updating U,Y,Z
YZ
U
e+
e-
X
Y ZD x U x U x |x u Xu
A x T C u
XU u
YC x
ZC x
|xXx
uTU u D x ZY A xC x x U
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| ,x u wX Xx w
T CU D x wu
YZ
U
e+
W
e-
X
XU u
YC x
ZC x
Message Passing – updating W,U,Y,Z
XU w
Y ZD x U x U x | ,,
x u Xw
Xwu
T C CuA x w
ZY A xC x x U ZY A xC x x U
|x u Xu
A x T C u
|xXx
uTU u D x
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YZ
Ui
e+
e-
X
X iU u
YC x
ZC x
Message Passing – many parents
X nU u
|:k
i u ku k i k i
X Xx
xU D T uCu x
ZY A xC x x U
| X ku k
x uT CA x u Y ZD x U x U x
Link matrix
U1 Un
… …
1XU u
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Belief Update – Noisy OR-Gate
if
1 if +
u
u
i X ku i T k
i X ku i T k
D x q C u x
Bel xD x q C u x
u u
i X i X ku i T k T
qC u C u
u u
i X i X ku i T k T
qC u C u
1u u
i X i X ku i T k T
qC u C u
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Belief Update – Noisy OR-Gate
1 1 if
1 1 1 if +
i X ii
i X ii
D x q C u x
Bel xD x q C u x
1u u
i X i X ku i T k T
qC u C u
1i X i X ii
qC u C u 1 1 i X i
i
q C u
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Update messages
Where
The message to the child is the same:
1 if +
1 if
i i i i i
X i
i i i
D x q D x q uU u
D x D x u
1 1i k X kk i
q C u
kY
ii
kXC Ay U x x
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Example
Windows Vista
No electricity
Virus 2K bug
Does not start
Wrong date
Stolen Password
Lost data
D1 D2 D3 D4
M1 M2 M3 M4
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Example
D1 D2 D3 D4
M1 M2 M3 M4
0.01\0.99 0.1\0.9 0.2\0.8 0.2\0.8
ijq0.8
0.1 0.9
0.2 0.5
0.1 0.7
0.80.2
|W , forhere j iij kP m d d k iq
\ (1 )i ip p
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Conjunction query
Conjunction query q: find the belief that several events happen simultaneously.
Applying the chain rule:
Product of m belief updates
Q
i ii I
q X x
1 2 2 1
1 2 1 1 2 1
1 2 2 1
, , , , ,
| | , ,
| , , , ,
m m m
m m
m m m
P q P x x x x x
P x P x x P x x x
P x x x x x
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Answering a query
In our example:
Computing P(q):
1 1 2 3 4q d m m m m
1 1 1 2 1 1
3 2 1 1
4 3 2 1 1
| | ,
| , ,
| , , ,
P q P d P m d P m m d
P m m m d
P m m m m d
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Example Update the belief on M1
Update likelihoods and priors:
1 11 21 1 2| 1 1 1 1 1 1 0.272P m q qd p
1 1p 2p
1 2MU d
4 2MC d
1 2 11 21 111 ,1 0.92,0.2MU d q q q
4 1 12 2 2 2 2, 1 0.338,0.662M M MC d U d p U d p
D1 D2 D3 D4
M1 M2 M3 M4
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Example Update the belief on M2
Update likelihoods and priors:
2 2 121 421 41| | 1 1 1 1 1 0.81,m d dP m P m q q p
1 1p
2 4 12 42 12, 0.45,0.9MU d q q q
4 2 24 4 4 4 4, 1 0.111,0.889M M MC d U d p U d p
2 4MU d
4p
4 4MC d
D1 D2 D3 D4
M1 M2 M3 M4
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Example Update the belief on M3
Update likelihoods and priors:
3 32 1 1
13 33 3
1| |
1 1 1 1 1 1 0.836
, ,P m P m
q q p
m m d d
1 1p
3 3 13 33 131 ,1 0.98,0.8MU d q q q
4 3 33 3 3 3 3, 1 0.23,0.77M M MC d U d p U d p
3 3MU d
3p 4 3MC d
D1 D2 D3 D4
M1 M2 M3 M4
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Example Update the belief on M4
4
3 2 1 14
42,3,4
|
1 1 0
,
.
,
7
,
81i M ii
m m m dP m
q C d
4 3MC d
4 4MC d
4 2MC d
D1 D2 D3 D4
M1 M2 M3 M4
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Example
And the final solution is…
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![Page 27: Javier Turek and Eyal Regev. Several causes sharing a common effect. Y Z UiUi e+e+ e-e- X Polytrees U1U1 UnUn … … Like a tree, but with multiple parents](https://reader033.vdocument.in/reader033/viewer/2022051614/5519b91555034660578b48dd/html5/thumbnails/27.jpg)
Example
And the final solution is…
1 1 1 2 1 1
3 2 1 1
4 3 2 1 1
3
| | ,
| , ,
| , , ,
0.01 0.272 0.81 0.836 0.781 1.439 10
P q P d P m d P m m d
P m m m d
P m m m m d
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Thank You!