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On the Revision of Probabilistic Beliefs using Uncertain Evidence

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

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On the Revision of Probabilistic Beliefs using Uncertain Evidence

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  1. On the Revision of Probabilistic Beliefs using Uncertain Evidence Hei Chan and Adnan Darwiche UCLA Presented by: Valerie Sessions October 6, 2004

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

  3. Questions to Keep in Mind • How should one specify uncertain evidence? • How should one revise a probability distribution? • How should one interpret informal evidential statements? • Should, and do, iterated belief revisions commute? • What guarantees can be offered on the amount of belief change induced by a particular revision?

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

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

  6. Example 1 = 0.28

  7. Virtual Evidence Method • Given PR and new evidence n we have

  8. Example 2

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

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

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

  12. 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

  13. Process for Mapping Evidence • One must adopt a formal method for specifying evidence (Jeffrey’s Rule or Virtual Evidence) • One must interpret the informal evidence statement as a formal piece of evidence using the method chosen • 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

  14. 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:

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

  16. 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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