1. Inductive Amnesia The Reliability of Iterated Belief Revision

2. Even Odd Straight Crooked Reliability “Confirmation” Performance “Primitive norms” Correctness “Coherence” Classical statistics Bayesianism Learning theory Belief Revision Theory A Table of Opposites

3. The Idea • Belief revision is inductive reasoning • A restrictive norm prevents us from finding truths we could have found by other means • Some proposed belief revision methods are restrictive • The restrictiveness is expressed as inductive amnesia

4. Inductive Amnesia • No restriction on memory... • No restriction on predictive power... • But prediction causes memory loss... • And perfect memory precludes prediction! • Fundamental dilemma

5. Outline • I. Seven belief revision methods • II. Belief revision as learning • III. Properties of the methods • IV. The Goodman hierarchy • V. Negative results • VI. Positive results • VII. Discussion

6. Points of Interest • Strong negative and positive results • Short run advice from limiting analysis • 2 is magic for reliable belief revision • Learning as cube rotation • Grue

7. Part I Iterated Belief Revision

8. Bayesian (Vanilla) Updating B Propositions are sets of “possible worlds”

9. Bayesian (Vanilla) Updating E new evidence B

10. Bayesian (Vanilla) Updating • Perfect memory • No inductive leaps E B B’ B’ = B *E = BÇE

11. Epistemic Hell B

12. Epistemic Hell E Surprise! B Epistemic hell

13. Epistemic Hell • Scientific revolutions • Suppositional reasoning • Conditional pragmatics • Decision theory • Game theory • Data bases E B Epistemic hell

14. Ordinal EntrenchmentSpohn 88 • Epistemic state S maps worlds to ordinals • Belief state of S = b (S ) = S -1(0) • Determines “centrality” of beliefs • Model: orders of infinitesimal probability w + 1 w 2 1 B = b (S) 0 S

15. Belief Revision Methods * takes an epistemic state and a proposition to an epistemic state S b(S) S’ E * b (S *E )

16. Spohn Conditioning *CSpohn 88 b (S ) S

17. E Spohn Conditioning *CSpohn 88 new evidence contradicting b (S ) E b (S ) S

18. E B’ Spohn Conditioning *CSpohn 88 E *C b (S ) S S *C E

19. E B’ Spohn Conditioning *CSpohn 88 • Conditions an entire entrenchment ordering • Perfect memory • Inductive leaps • No epistemic hell on consistent sequences • Epistemic hell on inconsistent sequences E *C b (S ) S S *C E

20. Lexicographic Updating *LSpohn 88, Nayak 94 S

21. E Lexicographic Updating *LSpohn 88, Nayak 94 S

22. E B’ Lexicographic Updating *LSpohn 88, Nayak 94 • Lift refuted possibilities above non-refuted possibilities preserving order. • Perfect memory on consistent sequences • Inductive leaps • No epistemic hell *L S S *L E

23. Minimal or “Natural” Updating *MSpohn 88, Boutilier 93 B S

24. Minimal or “Natural” Updating *MSpohn 88, Boutilier 93 E B S

25. Minimal or “Natural” Updating *MSpohn 88, Boutilier 93 • Drop the lowest possibilities consistent with the data to the bottom and raise everything else up one notch • inductive leaps • No epistemic hell • But... E *M S S *ME

26. Amnesia • What goes up can come down • Belief no longer entails past data E

27. Amnesia • What goes up can come down • Belief no longer entails past data E’ E *M

28. Amnesia • What goes up can come down • Belief no longer entails past data E’ E *M *M

29. The Flush-to-a Method *F,a Goldszmidt and Pearl 94 B S

30. E The Flush-to-a Method *F,a Goldszmidt and Pearl 94 a E E S

31. E The Flush-to-a Method *F,a Goldszmidt and Pearl 94 • Send non-E worlds to a fixed level a and drop E -worlds rigidly to the bottom • Perfect memory on sequentially consistent data ifa is high enough • Inductive leaps • No epistemic hell a E E *F,a S S *F,aE

32. Ordinal Jeffrey Conditioning *J,a Spohn 88 S

33. Ordinal Jeffrey Conditioning *J,a Spohn 88 E E E S

34. Ordinal Jeffrey Conditioning *J,a Spohn 88 E E E a S

35. Ordinal Jeffrey Conditioning *J,a Spohn 88 • Drop E worlds to the bottom. Drop non-E worlds to the bottom and then jack them up to level a • Perfect memory on consistent sequences if a is large enough • No epistemic hell • Reversible • But... *J,a E E E a B B’ S S *J,aE

36. Empirical Backsliding

37. Empirical Backsliding E a

38. Empirical Backsliding • Ordinal Jeffrey conditioning can increase the plausibility of a refuted possibility E a

39. The Ratchet Method *R,a Darwiche and Pearl 97 S

40. The Ratchet Method *R,a Darwiche and Pearl 97 b + a b E S

41. The Ratchet Method *R,a Darwiche and Pearl 97 • Like ordinal Jeffrey conditioning except refuted possibilities move up by a from their current positions • Perfect memory if a is large enough • Inductive leaps • No epistemic hell b + a b E *R,a B B’ S S *R,aE

42. Part II Belief Revision as Learning

43. Iterated Belief Revision • (S0 * ()) = S0 • (S0 * (E0, ..., En, En+1)) = (S0 * (E0, ..., En, )) * En+1 S0 S0 S1 S2 E0 E1 * b (S0) b (S1) b (S2)

44. A Very Simple Learning Paradigm outcome sequence mysterious system 0 0 1 0 0 possible infinite trajectories e n e|n

45. Empirical Propositions • Empirical propositions are sets of possible trajectories • Some special cases: e “fan” k n s [s] = the proposition that shas occurred [k, n] = the proposition that k occurs at stage n {e} = the proposition that the future trajectory is exactly e

46. Trajectory Identification • (*, S0) identifies eÛ for all but finitely many n, b(S0 * ([0, e(0)], ..., [n, e(n)]) = {e}

47. Trajectory Identification • (*, S0) identifies eÛ for all but finitely many n, b(S0 * ([0, e(0)], ..., [n, e(n)]) = {e} possible trajectories e

48. Trajectory Identification • (*, S0) identifies eÛ for all but finitely many n, b(S0 * ([0, e(0)], ..., [n, e(n)]) = {e} b (S 0) e

49. Trajectory Identification • (*, S0) identifies eÛ for all but finitely many n, b(S0 * ([0, e(0)], ..., [n, e(n)]) = {e} b (S 1)

50. Trajectory Identification • (*, S0) identifies eÛ for all but finitely many n, b(S0 * ([0, e(0)], ..., [n, e(n)]) = {e} b (S 2)