# 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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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.

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

Example 1 = 0.28

Virtual Evidence Method Given PR and new evidence n we have

Example 2

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

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

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)

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 Odd of a given b are defined by: Bayes factor given by:

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

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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