1 / 98

Everyday inductive leaps Making predictions and detecting coincidences

Everyday inductive leaps Making predictions and detecting coincidences. Tom Griffiths Department of Psychology Program in Cognitive Science University of California, Berkeley (joint work with Josh Tenenbaum, MIT). data. hypotheses. cube. shaded hexagon. Inductive problems.

saddam
Download Presentation

Everyday inductive leaps Making predictions and detecting coincidences

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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)

  28. 76 years 75 years (Halley, 1752)

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

  30. “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

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

  32. 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

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

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

  35. 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

  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 likelihood ratio is high

  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 prior odds are low likelihood ratio is high posterior odds are middling

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

  39. 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:

  40. 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

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

  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. 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

  44. 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

  45. 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

  46. 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

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

  48. 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

  49. The bombing of London

More Related