Everyday inductive leaps Making predictions and detecting coincidences

1. Everyday inductive leapsMaking predictions and detecting coincidences Tom Griffiths Department of Psychology Program in Cognitive Science University of California, Berkeley (joint work with Josh Tenenbaum, MIT)

2. data hypotheses cube shaded hexagon Inductive problems • Inferring structure from data • Perception • e.g. structure of 3D world from 2D visual data

3. hypotheses fair coin data two heads HHHHH Inductive problems • Inferring structure from data • Perception • e.g. structure of 3D world from 2D visual data • Cognition • e.g. whether a process is random

4. Perception is optimal

5. Cognition is not

6. Everyday inductive leaps • Inferences we make effortlessly every day • making predictions • detecting coincidences • evaluating randomness • learning causal relationships • identifying categories • picking out regularities in language • A chance to study induction in microcosm, and compare cognition to optimal solutions

7. Two everyday inductive leaps Predicting the future Detecting coincidences

8. Two everyday inductive leaps Predicting the future Detecting coincidences

9. Predicting the future How often is Google News updated? t = time since last update ttotal = time between updates What should we guess forttotalgivent?

10. Reverend Thomas Bayes

11. Likelihood Prior probability Posterior probability Sum over space of hypotheses Bayes’ theorem h: hypothesis d: data

12. Bayes’ theorem h: hypothesis d: data

13. Bayesian inference p(ttotal|t)  p(t|ttotal) p(ttotal) posterior probability likelihood prior

14. Bayesian inference p(ttotal|t)  p(t|ttotal) p(ttotal) p(ttotal|t)  1/ttotal p(ttotal) posterior probability likelihood prior assume random sample (0 < t < ttotal)

15. The effects of priors Different kinds of priorsp(ttotal) are appropriate in different domains e.g. wealth e.g. height

16. The effects of priors

17. Evaluating human predictions • Different domains with different priors: • a movie has made $60 million[power-law] • your friend quotes from line 17 of a poem[power-law] • you meet a 78 year old man[Gaussian] • a movie has been running for 55 minutes[Gaussian] • a U.S. congressman has served 11 years[Erlang] • Prior distributions derived from actual data • Use 5 values oftfor each • People predictttotal

18. people empirical prior Gott’s rule parametric prior

19. Probability matching p(ttotal|tpast) Proportion of judgments below predicted value ttotal Quantile of Bayesian posterior distribution

20. Probability matching p(ttotal|tpast) ttotal Proportion of judgments below predicted value • Average over all • prediction tasks: • movie run times • movie grosses • poem lengths • life spans • terms in congress • cake baking times Quantile of Bayesian posterior distribution

21. Predicting the future • People produce accurate predictions for the duration and extent of everyday events • Strong prior knowledge • form of the prior (power-law or exponential) • distribution given that form (parameters) • Contrast with “base rate neglect” (Kahneman & Tversky, 1973)

22. Two everyday inductive leaps Predicting the future Detecting coincidences

23. November 12, 2001: New Jersey lottery results were 5-8-7, the same day that American Airlines flight 587 crashed

24. "It could be that, collectively, the people in New York caused those lottery numbers to come up 911," says Henry Reed. A psychologist who specializes in intuition, he teaches seminars at the Edgar Cayce Association for Research and Enlightenment in Virginia Beach, VA. "If enough people all are thinking the same thing, at the same time, they can cause events to happen," he says. "It's called psychokinesis."

25. The bombing of London (Gilovich, 1991)

26. The bombing of London (Gilovich, 1991)

27. John Snow and cholera (Snow, 1855)

29. 76 years 75 years (Halley, 1752)

30. The paradox of coincidences How can coincidences simultaneously lead us to irrational conclusions and significant discoveries?

31. “an event which seems so unlikely that it is worth telling a story about” “we sense that it is too unlikely to have been the result of luck or mere chance” A common definition: Coincidences are unlikely events

32. Coincidences are not just unlikely... HHHHHHHHHH vs. HHTHTHTTHT

33. cause chance Hypotheses: a novel causal relationship exists no such relationship exists p(cause) p(chance) Priors: Data: d p(d|cause) p(d|chance) Likelihoods: Bayesian causal induction

34. Prior odds low high ? high Likelihood ratio (evidence) ? low Bayesian causal induction cause chance

35. Prior odds low high high low Bayesian causal induction coincidence cause Likelihood ratio (evidence) ? chance

36. What makes a coincidence? A coincidence is an event that provides evidence for causal structure, but not enough evidence to make us believe that structure exists

37. What makes a coincidence? A coincidence is an event that provides evidence for causal structure, but not enough evidence to make us believe that structure exists likelihood ratio is high

38. What makes a coincidence? A coincidence is an event that provides evidence for causal structure, but not enough evidence to make us believe that structure exists prior odds are low likelihood ratio is high posterior odds are middling

39. prior odds are low likelihood ratio is high posterior odds are middling HHHHHHHHHH HHTHTHTTHT

40. chance cause C C E E  1 -  0 < p(E) < 1 p(E) = 0.5 Bayesian causal induction Hypotheses: Priors: frequency of effect in presence of cause Data: Likelihoods:

41. prior odds are low likelihood ratio is high posterior odds are middling prior odds are low likelihood ratio is low posterior odds are low coincidence HHHHHHHHHH HHTHTHTTHT chance

42. Empirical tests • Is this definition correct? • from coincidence to evidence • How do people assess complex coincidences? • the bombing of London • coincidences in date

43. Empirical tests • Is this definition correct? • from coincidence to evidence • How do people assess complex coincidences? • the bombing of London • coincidences in date

44. prior odds are low likelihood ratio is high posterior odds are middling prior odds are low likelihood ratio is very high posterior odds are high coincidence HHHHHHHHHH cause HHHHHHHHHHHHHHHHHHHHHH

45. From coincidence to evidence • Transition produced by • increase in likelihood ratio (e.g., coin flipping) • increase in prior odds (e.g., genetics vs.ESP) coincidence evidence for a causal relation

46. Testing the definition • Provide participants with data from experiments • Manipulate: • cover story: genetics vs. ESP (prior) • data: number of heads/males (likelihood) • task: “coincidence or evidence?” vs. “how likely?” • Predictions: • coincidences affected by prior and likelihood • relationship between coincidence and posterior

47. Proportion “coincidence” 47 51 55 59 63 70 87 99 Number of heads/males Posterior probability 47 51 55 59 63 70 87 99 r = -0.98

48. Empirical tests • Is this definition correct? • from coincidence to evidence • How do people assess complex coincidences? • the bombing of London • coincidences in date

49. Complex coincidences • Many coincidences involve structure hidden in a sea of noise (e.g., bombing of London) • How well do people detect such structure? • Strategy: examine correspondence between strength of coincidence and likelihood ratio

50. The bombing of London