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Dutch books and epistemic events

ILLC 2005 Interfacing Probabilistic and Epistemic Update. Dutch books and epistemic events. Jan-Willem Romeijn Psychological Methods University of Amsterdam. Outline.  Updating by conditioning  Violations of conditioning  External shocks to the probability

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Dutch books and epistemic events

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  1. ILLC 2005 Interfacing Probabilistic and Epistemic Update Dutch books and epistemic events Jan-Willem Romeijn Psychological Methods University of Amsterdam

  2. Outline  Updating by conditioning  Violations of conditioning  External shocks to the probability  Meaning shifts in epistemic updates  A Bayesian model of epistemic updates  No-representation theorem  Concluding remarks - 2 -

  3.  Updating by conditioning Updating by conditioning is a consistency constraint for incorporating new facts in a probability assignment. probability assignment p events A, B, C, ... conditioning on events probabilistic conclusions p(  | ABC...) If probability theory is seen as a logic, updating functions like a deductive inference rule. - 3 -

  4. Muddy Venn diagrams Conditioning on the fact that A is like zooming in on the probability assignment p within the set of possible worlds A. A p p(  | A)  A Probability is represented by the size of rectangulars. Apart from normalising the probability of A, no changes are induced by the update operation. - 4 -

  5. Violating conditioning Bayesian conditioning is violated if, in the course of the update, we also change the probabilities within A. A p(  | A ) pA(  )  B B  B  B B B The updated probability is pA(B) < p(B|A). This difference makes vulnerable for a Dutch book. - 5 -

  6. Rational violations? • In particular cases, violations of conditioning may seem rational. •  Violations of the likelihood principle in classical statistics, model selection problems. •  Epistemic updates: incorporating facts about knowledge states. • Can we make sense of such violations from within a Bayesian perspective? - 6 -

  7.  B'  B B' B  Possible resolution Violations are understandable if they result from changes in meaning. On learning A we may reinterpret B as B'. p( B | A ) p( B' | A ) p( B | A' ) ?  B B Can we represent such a meaning shift as Bayesian update, saying that we actually learned A' ? - 7 -

  8.  Probability shocks Violations of conditioning can be understood as an external shock to the probability assignment p. A A  B  B B B p= 1/4 p= 1/4 p'= 3/8 p'= 1/8 p= 1/4 p= 1/4 p'= 3/8 p'= 1/8 The events are associated with the same possible worlds, denoted •, but these worlds are assigned probabilities p', according to a new constraint . - 8 -

  9. Restricting the shock External shocks to the probability assignment may be governed by further formal criteria, such as minimal distance  between p and p'.    p p' Such criteria may be conservative, but they are not consistent. - 9 -

  10. Choosing premises From a logical point of view, the update procedure comes down to choosing new premises. premise p premise p' events A, B, C, ... events A, B, C, ... conclusion p( | ABC...) conclusion p'( | ABC...) This is the extra-logical domain of objective Bayesianism: formally constrained prior probabilities. - 10 -

  11.  Meaning shifts The update operation can also be seen as a change to the semantics: p(B' | A) < p(B | A). A A  B  B' B B' p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 The probabilities of possible worlds remain the same, but the update induces an implicit change of the facts involved. - 11 -

  12. Epistemic updates Consider two research groups, 1 and 2, that try to discover which of A, B, orC holds:  D1 D1 A B C p= 1/3 p= 1/3 p= 1/3  D2 D2 The groups use different methods, delivering doubt or certainty in differing sets of possible worlds. - 12 -

  13. Conditional probability According to the standard definition of conditional probability, we have p( D2 | D1) = 1/2: D1  D1 D1 A B A B C p= 1/2 p= 1/2 p= 1/3 p= 1/3 p= 1/3  D2  D2 D2 D2 But is this also the appropriate updated probability? - 13 -

  14. Updated probability It seems that after an update with D1, the second research group has very little to doubt about: D1 D1 A B A B p= 1/2 p= 1/2 p= 1/2 p= 1/2  D2 D2  D'2 Updating induces a meaning shift D2D'2 , and the correct updated probability is p(D'2 | D1)= 0. - 14 -

  15.  Epistemic events The meaning shift D2D'2 can be understood by including epistemic states into the semantics. C B D1 A B C B  2  A D2 A B C 1  The diagram shows the accessible epistemic states in the world-state B. - 15 -

  16. External states After learning that D1, we may exclude world-state C from the state space. C C B B C C 2  B B 2    A A W A A W C A B C A B 1  1  - 16 -

  17. Epistemic update But a full update also comprises conditioning on the accessible epistemic states of both research groups. C C B B B B 2  2  A A A A A B C A B C 1  1  This latter step brings about the event change D2D'2. - 17 -

  18. Bayesian conditioning There is no violation of conditioning in the example. D1 D'1 C C ? B B C C B B 2  2    A A A A W W C A B C A B 1  1  It is simply unclear which event we are supposed to update with upon learning that group 1 is in doubt: D1 or D'1. - 18 -

  19.  Choosing semantics Many puzzles on the applicability of Bayesian updating can be dealt with by making explicit the exact events we update upon. A A'  B B  B B ? p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 p= 1/4 We must choose the semantics so as to include all these events. Is that always possible? - 19 -

  20. Judy Benjamin updates In updating a probability p to p by distance minimisation under a partition of constraints , we may have for some B and all . Now suppose that we can associate the constraints with a partition of events G: - 20 -

  21. No-representation theorem In Bayesian conditioning on events A from a partition, the prior is always a convex combination of the posteriors: But because p(B|G)> p(B) for all but one , we have It thus seems that there is no set of events G that can mimic distance minimisation on the constraints . - 21 -

  22.  In closing • Some considerations for further research: • There is a large gap between the epistemic puzzles and cases like model selection. • It is unclear what kind of event is behind violations of the likelihood principle, as in the stopping rule. • Probabilistic consistency may not be the only virtue if we object to a principled distinction between epistemology and logic. - 22 -

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