on the revision of probabilistic beliefs using uncertain evidence hei chan and adnan darwiche ucla...
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On the Revision of Probabilistic Beliefs using Uncertain Evidence
Hei Chan and Adnan Darwiche
UCLA
Presented by: Valerie Sessions
October 6, 2004
Overview
• Jeffrey’s Rule / Probability Kinematics
• Virtual Evidence Method
• Switching between methods
• Interpreting evidential statements
• Commutativity of Revisions
• Bounding Belief Change
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?
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
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
Example 1
)Pr(
)(rP),Pr()|(rP
g
ggg c
ccscs
3.0
7.012.0
= 0.28
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(
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
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
Jeffrey’s Rule -> Virtual Evidence
• Divide new Prob. by old Prob. for ratio
41
4
)Pr(
)(rP
a
aa
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)
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
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
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
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
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,