#### Transcript Inductive Bayesian Logic

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 Meaning shifts in epistemic updates A Bayesian model of epistemic updates No-representation theorem Concluding remarks -2- 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- 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- Violating conditioning Bayesian conditioning is violated if, in the course of the update, we also change the probabilities within A. p( | A ) A B B B B pA( ) B B The updated probability is pA(B) < p(B|A). This difference makes vulnerable for a Dutch book. -5- 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- Possible resolution Violations are understandable if they result from changes in meaning. On learning A we may reinterpret B as B'. p( B | A ) B B p ( B' | A ) B' B' p( B | A' ) ? B B Can we represent such a meaning shift as Bayesian update, saying that we actually learned A' ? -7- Probability shocks Violations of conditioning can be understood as an external shock to the probability assignment p. A B p= 1/4 p= 1/4 B p'= 3/8 p'= 3/8 B p= 1/4 p= 1/4 A B p'= 1/8 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- 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- 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 - Meaning shifts The update operation can also be seen as a change to the semantics: p (B' | A) < p (B | A). A B p= 1/4 p= 1/4 B' p= 1/4 p= 1/4 B p= 1/4 p= 1/4 B' p= 1/4 p= 1/4 A The probabilities of possible worlds remain the same, but the update induces an implicit change of the facts involved. - 11 - Epistemic updates Consider two research groups, 1 and 2, that try to discover which of A, B, or C holds: D1 D1 A p= 1/3 B p= 1/3 D2 C p= 1/3 D2 The groups use different methods, delivering doubt or certainty in differing sets of possible worlds. - 12 - Conditional probability According to the standard definition of conditional probability, we have p( D2 | D1) = 1/2: D1 D1 A p= 1/3 D2 B p= 1/3 C p= 1/3 D2 D1 A p= 1/2 D2 But is this also the appropriate updated probability? - 13 - B p= 1/2 D2 Updated probability It seems that after an update with D1, the second research group has very little to doubt about: D1 A p= 1/2 D2 D1 A p= 1/2 B p= 1/2 B p= 1/2 D'2 D2 Updating induces a meaning shift D2 D'2 , and the correct updated probability is p ( D'2 | D1) = 0. - 14 - Epistemic events The meaning shift D2 D'2 can be understood by including epistemic states into the semantics. C D1 A B C B B 2 A D2 A B 1 C The diagram shows the accessible epistemic states in the world-state B. - 15 - External states After learning that D1, we may exclude world-state C from the state space. C C B B C 2 2 B A A A B 1 C C B A A W A - 16 - B 1 C W Epistemic update But a full update also comprises conditioning on the accessible epistemic states of both research groups. C C B B 2 2 B A A A B 1 B A A C A B 1 C This latter step brings about the event change D2 D'2. - 17 - Bayesian conditioning There is no violation of conditioning in the example. D1 D'1 C C ? B B C 2 2 B A A A B 1 C C B A A W A B 1 C W 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 - Choosing semantics Many puzzles on the applicability of Bayesian updating can be dealt with by making explicit the exact events we update upon. A B p= 1/4 p= 1/4 B p= 1/4 p= 1/4 ? B p= 1/4 p= 1/4 B p= 1/4 p= 1/4 We must choose the semantics so as to include all these events. Is that always possible? - 19 - A' Judy Benjamin updates In updating a probability p to p by distance minimisation under a partition of constraints , we may have p p ( B) p( B) p p ( B) p( B) for some B and all . Now suppose that we can associate the constraints with a partition of events G: p ( ) p( | G ), - 20 - p(G ) d 1. No-representation theorem In Bayesian conditioning on events A from a partition, the prior is always a convex combination of the posteriors: p( B) p( A ) p( B | A ) d . But because p(B|G) > p(B) for all but one , we have p( B) p(G ) p( B | G ) d . It thus seems that there is no set of events G that can mimic distance minimisation on the constraints . - 21 - 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 -