On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA...
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![Page 1: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/1.jpg)
On the Revision of Probabilistic Beliefs using Uncertain Evidence
Hei Chan and Adnan Darwiche
UCLA
Presented by: Valerie Sessions
October 6, 2004
![Page 2: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/2.jpg)
Overview
• Jeffrey’s Rule / Probability Kinematics
• Virtual Evidence Method
• Switching between methods
• Interpreting evidential statements
• Commutativity of Revisions
• Bounding Belief Change
![Page 3: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/3.jpg)
Questions to Keep in Mind(1) How should one specify uncertain evidence?(2) How should one revise a probability
distribution?(3) How should one interpret informal evidential
statements?(4) Should, and do, iterated belief revisions
commute?(5) What guarantees can be offered on the amount
of belief change induced by a particular revision?
![Page 4: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/4.jpg)
Probability Kinematics
• Two probability distributions disagree on probabilities for a set of events, but agree on how that event affects another event.
)(rP)Pr( gg cc
)|(rP)|Pr( gg cscs
![Page 5: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/5.jpg)
Jeffrey’s Rule• Uses Probability Kinetics
• Given a probability distribution and some uncertain evidence bearing on this we have…
)Pr(
),Pr()(rP
1 g
gn
ii c
csqs
)(rP gg cq
![Page 6: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/6.jpg)
Example 1
)Pr(
)(rP),Pr()|(rP
g
ggg c
ccscs
3.0
7.012.0
= 0.28
![Page 7: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/7.jpg)
Virtual Evidence Method
• Given PR and new evidence n we have
)|Pr(),|Pr(
)|Pr(
AnBAn
An
n
jj
n
ii
A
ABnB
1
1
)Pr(
),Pr()|Pr(
![Page 8: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/8.jpg)
Example 2
000369.0)|,Pr(
)989901.0*1()000005.0*1()009999.0*4()000095.0*4(
000095.0*4)|,Pr(
),Pr(
),Pr()|,Pr(
1
ba
ba
ba
baba
n
jj
a
![Page 9: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/9.jpg)
Virtual Evidence -> Jeffrey’s Rule
Virtual Evidence
aaaa :)|Pr(:)|Pr(
To Jeffrey’s:
)|Pr()(rP
)|Pr()(rP
aqa
aqa
a
a
)Pr()Pr(
)Pr()|Pr(
aa
aa
aa
a
![Page 10: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/10.jpg)
Jeffrey’s Rule -> Virtual Evidence
• Divide new Prob. by old Prob. for ratio
41
4
)Pr(
)(rP
a
aa
![Page 11: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/11.jpg)
Virtual Evidence and Jeffrey’s Rule in Belief Networks
• Virtual Evidence was built for this
Burglary TRUE FALSETRUE 0.95 0.01FALSE 0.05 0.99
TRUE 0.0001FALSE 0.9999
P(B) P(A)
Alarm TRUE FALSETRUE 0.4 0.1FALSE 0.6 0.9
P(n|A)
1:4)|Pr(:)|Pr( aa For Jeffrey’s Rule -> Convert to Virtual Evidence and then put in belief network (cheat)
![Page 12: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/12.jpg)
Interpreting Evidential Statements
• Looking at the evidence, I am willing to bet 2:1 that David is not the killer.
• Jeffrey’s Rule – “All things considered”– Pr'(killer) = 2/3
– Pr'(not killer) = 1/3
• Virtual Evidence – “Nothing else considered”– Pr(evidence|killer):Pr(evidence|not killer) = 2 : 1
![Page 13: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/13.jpg)
Process for Mapping Evidence(1) One must adopt a formal method for specifying
evidence (Jeffrey’s Rule or Virtual Evidence)
(2) One must interpret the informal evidence statement as a formal piece of evidence using the method chosen
(3) One must apply a revision, by mapping the original probability distribution and formal piece of evidence into a new distribution, according to a belief revision principle
![Page 14: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/14.jpg)
Commutativity of Iterated Revisions
• Jeffrey’s Rule is not commutative
• Wagner suggests Bayes Factors
)|Pr(
)|Pr()|(
ba
baba
Odd of a given b are defined by:
Bayes factor given by:
2
1
21
21
21
2121Pr,rP )Pr()Pr(
)(rP)(rP
):(
):():(
ee
ee
ee
eeeeF
![Page 15: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/15.jpg)
Bounding Belief Change
• Chan and Darwiche present a distance measure to bind belief revisions
)Pr(
)(rPminln
)Pr(
)(rPmaxln)rP(Pr,
D
)rP(Pr,)rP(Pr,
)|(
)|(
DD eBA
BAe
![Page 16: On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004.](https://reader036.fdocuments.us/reader036/viewer/2022072005/56649cda5503460f949a4836/html5/thumbnails/16.jpg)
Bounding Belief Change
• Using these theorems with Jeffrey’s Rule and the Virtual Evidence MethodJeffrey’s Rule
)Pr(minln
)Pr(maxln)rP(Pr,
11i
in
ii
in
i e
q
e
qD
Virtual Evidence Method
i
n
ii
n
iD
11minlnmaxln))|Pr((Pr,