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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

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Overview Jeffrey’s Rule / Probability Kinematics Virtual Evidence Method Switching between methods Interpreting evidential statements Commutativity of Revisions Bounding Belief Change

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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?

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Probability Kinematics Two probability distributions disagree on probabilities for a set of events, but agree on how that event affects another event.

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Jeffrey’s Rule Uses Probability Kinetics Given a probability distribution and some uncertain evidence bearing on this we have…

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Example 1 = 0.28

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Virtual Evidence Method Given PR and new evidence n we have

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Example 2

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Virtual Evidence -> Jeffrey’s Rule Virtual Evidence To Jeffrey’s:

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Jeffrey’s Rule -> Virtual Evidence Divide new Prob. by old Prob. for ratio

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Virtual Evidence and Jeffrey’s Rule in Belief Networks Virtual Evidence was built for this P(B) P(A) P(n|A) For Jeffrey’s Rule -> Convert to Virtual Evidence and then put in belief network (cheat)

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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

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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

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Commutativity of Iterated Revisions Jeffrey’s Rule is not commutative Wagner suggests Bayes Factors Odd of a given b are defined by: Bayes factor given by:

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Bounding Belief Change Chan and Darwiche present a distance measure to bind belief revisions

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Bounding Belief Change Using these theorems with Jeffrey’s Rule and the Virtual Evidence Method Jeffrey’s Rule Virtual Evidence Method

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