1. Truth-conduciveness Without Reliability: A Skeptical Derivation of Ockham’s Razor Kevin T. Kelly Department of Philosophy Carnegie Mellon University www.cmu.edu

2. Naivete Lo! An apple.

3. Skeptical Hypothesis Lo! An apple. Maybe you are a brain in a vat. Everything would look the same.

4. poof Maybe you are a brain in a vat. Everything would look the same. Skeptical Hypothesis

5. Retrenchment That’s not a serious possibility You have the burden of proof. It’s remote. It’s implausible. It’s distant from the actual world. You’re not in my community. Who cares about the worst case?

6. Retrenchment That’s not a serious possibility You have the burden of proof. It’s remote. It’s implausible. It’s distant from the actual world. You’re not in my community. Who cares about the worst case?

7. Unsatisfying Possibilities delimited a priori: circular account. Possibilities delimited a priori: circular account. Possibilities delimited a posteriori: how do we seek knowledge? Possibilities delimited a posteriori: how do we seek knowledge? So there!

8. Zen Approach Don’t rush to defeat the demon. Don’t rush to defeat the demon. Grrrr!

9. Zen Approach Don’t rush to defeat the demon. Don’t rush to defeat the demon. Get to know him extremely well. Get to know him extremely well. Justification may be located in the demon’s power rather than in his weakness. Justification may be located in the demon’s power rather than in his weakness.

10. The Zen of Computation Algorithms are justified by efficiency. Algorithms are justified by efficiency. Efficiency means you couldn’t do better. Efficiency means you couldn’t do better. You couldn’t do better due to a demonic argument (the halting problem, etc). You couldn’t do better due to a demonic argument (the halting problem, etc).

11. Scientific Theory Choice Which theory is true?

12. Ockham Says: Choose the Simplest!

13. Skeptical Hypothesis Maybe a complex theory is true but the data are simple

14. Puzzle An indicator must be sensitive to what it indicates. An indicator must be sensitive to what it indicates. simple

15. Puzzle An indicator must be sensitive to what it indicates. An indicator must be sensitive to what it indicates. complex

16. Puzzle But Ockham’s razor always points at simplicity. But Ockham’s razor always points at simplicity. simple

17. Puzzle But Ockham’s razor always points at simplicity. But Ockham’s razor always points at simplicity. complex

18. Meno If we know that the truth is simple, we don’t need Ockham’s razor. If we know that the truth is simple, we don’t need Ockham’s razor. simple

19. Meno If we don’t know that the truth is simple, what good is Ockam’s razor? If we don’t know that the truth is simple, what good is Ockam’s razor? complex

20. Some Standard Responses

21. Simple Theories are Virtuous Testable (Popper, Glymour) Testable (Popper, Glymour) Unified (Friedman, Kitcher) Unified (Friedman, Kitcher) Explanatory (Harman) Explanatory (Harman) Symmetrical (Malament) Symmetrical (Malament) Compress data (Rissanen) Compress data (Rissanen) Interesting (Vitanyi) Interesting (Vitanyi)

22. But the Truth Might Not be Virtuous To conclude that a theory is true because it is virtuous is wishful thinking (van Fraassen). To conclude that a theory is true because it is virtuous is wishful thinking (van Fraassen).

23. Overfitting (Akaike, Sober, Forster) Overfitting (Akaike, Sober, Forster) Empirical estimates based on complex models have greater mean squared distance from the truth Empirical estimates based on complex models have greater mean squared distance from the truth Truth

24. Overfitting (Akaike, Sober, Forster) Overfitting (Akaike, Sober, Forster) Empirical estimates based on complex models have greater mean squared distance from the truth. Empirical estimates based on complex models have greater mean squared distance from the truth. Pop!

25. Overfitting (Akaike, Sober, Forster) Overfitting (Akaike, Sober, Forster) Empirical estimates based on complex models have greater mean squared distance from the truth. Empirical estimates based on complex models have greater mean squared distance from the truth. clamp Truth

26. Overfitting (Akaike, Sober, Forster) Overfitting (Akaike, Sober, Forster) Empirical estimates based on complex models have greater mean squared distance from the truth. Empirical estimates based on complex models have greater mean squared distance from the truth. clamp Truth Pop!

27. Does Not Aim at True Theory Does Not Aim at True Theory...even if the simple theory is known to be false…...even if the simple theory is known to be false… clamp Four eyes!

28. Simple data would be a miracle in a complex world. Simple data would be a miracle in a complex world. Simple data would be expected in a simple world. Simple data would be expected in a simple world. Miracle Argument (Putnam, Rosenkrantz)

29. Planetary retrograde motion Mars Earth Sun Miracle Argument

30. Complex theory  Simple theory Simple data would be a miracle in a complex world. Simple data would be a miracle in a complex world. Simple data would be expected in a simple world. Simple data would be expected in a simple world. epicycle lapping

31. Miracle Argument ’’ Simple theory lapping Simple data would be a miracle in a complex world. Simple data would be a miracle in a complex world. Simple data would be expected in a simple world. Simple data would be expected in a simple world. Complex theory epicycle

32. However… Simple data would not be a miracle if the complex theory’s parameter were set near  ; Simple data would not be a miracle if the complex theory’s parameter were set near  ; Complex theory  Simple theory epicycle lapping

33. The Real Miracle Ignorance about model: p(S)  p(C); + Ignorance about parameter settings within theories: p(C(  ) | C)  p(C(  ’ ) | C). = Knowledge about parameter settings across theories p(C(  )) << p(S). Is it knognorance or Ignoredge? CP        

34. The Ellsberg Paradox 1/3 ?? 3 ball colors with these frequencies Urn

35. The Ellsberg Paradox pqr Human betting preferences p > q 1/3 ??

36. The Ellsberg Paradox p > q Human betting preferences r < pqr ! pqr 1/3 ??

37. Diagnosis pqr ?? ignoranceknowledge

38. Robust Bayesianism (Levi, Kadane, Seidenfeld) 1/3?? knowledgeignorance 1/3 02/3 1/3 2/30 Choose the act with highest worst-case expected value.... pqr Credence is range of probs.

39. Worst-case Expected Values 1/3?? > < p qr > 0 0 2/3 1/3 ??

40. Whither Ockham? Since you don’t really know that complex worlds won’t produce simple data, shouldn’t your ignorance include distributions concentrated on such possibilities? I prefer ignoredge.

41. In Any Event The coherentist foundations of Bayesianism have nothing to do with short-run truth-conduciveness.

42. Temptation If only the probabilities p(C(q’ ) | C) were chances rather than opinions. Then the alleged miracle would be a proper miracle.

43. Proof of God (R. Koons 1999) 1.Natural chance is determined by the fundamental theory of natural chance. 2.If Ockham’s razor reliably infers the theory of natural chance, the chance that a complex theory of natural chance would have its parameters set to produce simple data must be low. 3.But since natural chance is determined by the free parameters of the fundamental theory of natural chance, the parameter setting is not governed by natural chance. 4.Hence, it must be governed by non-natural chance. 5.Holy water is available at the exit.

44. Moral The basic point is right. Solution: 1.Keep naturalism 2.Keep fundamental scientific knowledge 3.Dump short-run reliability as explication of truth-conduciveness.

45. Externalist Magic Externalist Magic Simplicity informs via hidden causes or tracking mechanisms. Simplicity informs via hidden causes or tracking mechanisms. G SimpleB(Simple) SimpleB(Simple) SimpleB(Simple) Leibniz, evolution Kant Ouija board

46. Practice and data are the same. Practice and data are the same. Knowledge vs. non-knowledge depends on hidden causes. Knowledge vs. non-knowledge depends on hidden causes. By Ockham’s razor, better to explain Ockham’s razor without the hidden causes. By Ockham’s razor, better to explain Ockham’s razor without the hidden causes. Metaphysicians for Ockham ? With Friends Like Those… With Friends Like Those…

47. The Last Gasp: Convergence Complexity truth Bayes (washing out of the prior) BIC (Schwarz) Structural Risk Minimization (Vapnik, Harman) TETRAD (Spirtes, Glymour, Scheines)

48. The Last Gasp: Convergence Complexity truth Plink! Blam!

49. The Last Gasp: Convergence Complexity truth Plink! Blam!

50. The Last Gasp: Convergence Complexity truth Plink! Blam!

51. Logic is Backwards Ockham methods are sufficient for convergence. Ockham methods are sufficient for convergence. But every finite variant of a convergent method converges (Salmon). But every finite variant of a convergent method converges (Salmon). So Ockham’s razor is not necessary for convergence. So Ockham’s razor is not necessary for convergence. Alternative ranking truth

52. Truth Conduciveness Reliability Reliability Too strong: Too strong: Circles or magic required. Circles or magic required. Convergence Convergence Too weak Too weak Doesn’t single out simplicity Doesn’t single out simplicity Complex Simple ComplexSimple

53. Truth Conduciveness Indication or tracking Indication or tracking Too strong: Too strong: Circles or magic required. Circles or magic required. Convergence Convergence Too weak Too weak Doesn’t single out simplicity Doesn’t single out simplicity “Straightest” convergence “Straightest” convergence Just right? Just right? Complex Simple ComplexSimple

54. Truth-conduciveness as Straightest Convergence ComplexSimple

55. Ancient Roots "Living in the midst of ignorance and considering themselves intelligent and enlightened, the senseless people go round and round, following crooked courses, just like the blind led by the blind." Katha Upanishad, I. ii. 5, c. 600 BCE.

56. Retraction New output does not entail previous output. New output does not entail previous output. tt + 1 Retracted Content

57. Eliminate Needless Retractions Truth

58. Necessary Retractions are Virtuous Truth

59. Demon’s Role as Justifier Truth I can force every convergent method to retract this often, so your retractions are justified by my power.

60. Eliminate Needless Delays to Retractions theory

61. application corollary application theory application corollary application corollary Eliminate Needless Delays to Retractions

62. Easy Comparisons at least as bad = at least as many retractions at least as late time retractions

63. Worst-case Retraction Time Bounds... (1, 2, ∞)...

64. Empirical Complexity Hopeless ideas: Syntactic length Computational incompressibility By what miracle do notational conventions indicate truth?

65. Empirical Complexity Close but no cigar: Free parameters Broken symmetries Meno, I want simplicity itself, not parts of simplicity.

66. Empirical Complexity Empirical complexity of T in  = the length of the maximum path (T1, …, Tn, T) of answers in  the demon can force from an arbitrary convergent method. T T3 T2 T1 Keep up!

67. Polynomial Order Data = open intervals around Y at rational values of X. Data = open intervals around Y at rational values of X.

68. Polynomial Order Demon shows flat line until convergent method takes bait. Demon shows flat line until convergent method takes bait. Zero degree curve

69. Polynomial Order Demon shows flat line until convergent method takes bait. Demon shows flat line until convergent method takes bait. Zero degree curve

70. Polynomial Order Then switches to tilted line until convergent method takes the bait. Then switches to tilted line until convergent method takes the bait. First degree curve

71. Polynomial Order Then switches to parabola until convergent method takes the bait … Then switches to parabola until convergent method takes the bait … Second degree curve

72. Complexity can be Complex T3 T8T5 T7T4 T2 0 1 2 3 Complexity given e:

73. Complexity Relative to Data T3 T8T5 T7T4 T2 0 1 2 3 Complexity given e + e’:

74. Complexity Relative to Data T5T4 T2 0 1 2 3 T7 Complexity given e + e’:

75. Timed Retraction Bounds r(M, e, n) = the least timed retraction bound for worlds satisfying theories of complexity n and producing finite input history e. r(M, e, n) = the least timed retraction bound for worlds satisfying theories of complexity n and producing finite input history e. Empirical Complexity0123... M

76. M is Efficient at e For each convergent M’ that agrees with M along finite input history e, For each convergent M’ that agrees with M along finite input history e, for each complexity n: for each complexity n: r(M, e, n)  r(M’, e, n) Empirical Complexity0123... MM’

77. M is Strongly Beaten at e There exists convergent M’ that agrees with M up to the end of e, such that There exists convergent M’ that agrees with M up to the end of e, such that for each complexity n: for each complexity n: r(M, e, n) > r(M’, e, n). Empirical Complexity0123... MM’

78. M is Weakly Beaten at e There exists convergent M’ that agrees with M up to the end of e, such that There exists convergent M’ that agrees with M up to the end of e, such that For each n, r(M, e, n)  r(M’, e, n); For each n, r(M, e, n)  r(M’, e, n); Exists n, r(M, e, n) > r(M’, e, n). Exists n, r(M, e, n) > r(M’, e, n). Empirical Complexity0123... MM’

79. Demons for Ockham

80. Ockham’s Razor ? Don’t select a theory unless it is uniquely simplest in light of experience. Don’t select a theory unless it is uniquely simplest in light of experience. T5T4 T2 0 1 2 3 T7

81. Ockham’s Razor T7 Don’t select a theory unless it is uniquely simplest in light of experience. Don’t select a theory unless it is uniquely simplest in light of experience. T2 0 1 2 3 T7

82. Stalwartness Don’t retract your answer while it remains uniquely simplest Don’t retract your answer while it remains uniquely simplest T2 0 1 2 3 T7 T7,

83. Argument Sketch No matter what convergent M has done in the past, nature can force M to produce each answer down an arbitrary effect path, arbitrarily often. No matter what convergent M has done in the past, nature can force M to produce each answer down an arbitrary effect path, arbitrarily often. Nature can also force violators of Ockham’s razor or stalwartness either into an extra retraction or a late retraction in each complexity class. Nature can also force violators of Ockham’s razor or stalwartness either into an extra retraction or a late retraction in each complexity class.

84. Ockham Efficiency Theorem Let M converge to the true theory in problem P. The following are equivalent: Let M converge to the true theory in problem P. The following are equivalent: M is always Ockham and stalwart in P; M is always Ockham and stalwart in P; M is always efficient in P; M is always efficient in P; M is never weakly beaten in P. M is never weakly beaten in P.

85. Policy Retractions Many explanations have been offered to make sense of the here-today-gone-tomorrow nature of medical wisdom — what we are advised with confidence one year is reversed the next — but the simplest one is that it is the natural rhythm of science. Many explanations have been offered to make sense of the here-today-gone-tomorrow nature of medical wisdom — what we are advised with confidence one year is reversed the next — but the simplest one is that it is the natural rhythm of science. (Do We Really Know What Makes us Healthy, NY Times Magazine, Sept. 16, 2007). (Do We Really Know What Makes us Healthy, NY Times Magazine, Sept. 16, 2007).

86. Causal Inference Causal graph theory: more correlations  more causes. Causal graph theory: more correlations  more causes. Idealized data = list of conditional dependencies discovered so far. Idealized data = list of conditional dependencies discovered so far. Anomaly = the addition of a conditional dependency to the list. Anomaly = the addition of a conditional dependency to the list. partial correlations SG(S)G(S)

87. Causal Axioms (Pearl, Glymour) 1. Screening off: X is statistically independent of its non-descendents given its parents. 2. No invisible causes: The only true independence relations are those entailed by condition 1. P1P2 N1 N2 P2P1 X D

88. Forcible Sequence of Causal Theories X2X3WX1 Y1 Y2

89. Forcible Sequence of Causal Theories X2X3WX1 Y1 Y2 Y3 Y4

90. Forcible Sequence of Causal Theories X2X3WX1 Y1 Y2 Y3 Y4 Y5

91. Forcible Sequence of Causal Theories X2X3WX1 Y1 Y2 Y3 Y4 Y5Y4

92. Moral In counterfactual prediction, form of model matters and retractions are unavoidable. In counterfactual prediction, form of model matters and retractions are unavoidable. Ockham efficiency agrees very closely with best contemporary practice. Ockham efficiency agrees very closely with best contemporary practice. Maybe that’s all there is to it. Maybe that’s all there is to it.

93. Conclusions Ockham’s razor is necessary for staying on the straightest path to the truth Ockham’s razor is necessary for staying on the straightest path to the truth Does not reliably point at or indicate the truth. Does not reliably point at or indicate the truth. Demonstrably works without circles, evasions, or magic. Demonstrably works without circles, evasions, or magic. Such a theory is motivated in counterfactual inference and estimation. Such a theory is motivated in counterfactual inference and estimation.