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Causes and coincidences

Causes and coincidences. Tom Griffiths Cognitive and Linguistic Sciences Brown University.

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Causes and coincidences

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  1. Causes and coincidences Tom Griffiths Cognitive and Linguistic Sciences Brown University

  2. “It could be that, collectively, the people in New York caused those lottery numbers to come up 9-1-1… If enough people all are thinking the same thing, at the same time, they can cause events to happen… It's called psychokinesis.”

  3. 76 years 75 years (Halley, 1752)

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

  5. Outline • A Bayesian approach to causal induction • Coincidences • what makes a coincidence? • rationality and irrationality • the paradox of coincidences • Explaining inductive leaps

  6. Outline • A Bayesian approach to causal induction • Coincidences • what makes a coincidence? • rationality and irrationality • the paradox of coincidences • Explaining inductive leaps

  7. Causal induction • Inferring causal structure from data • A task we perform every day … • does caffeine increase productivity? • … and throughout science • three comets or one?

  8. Reverend Thomas Bayes

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

  10. Bayesian causal induction causal structures Hypotheses: Priors: Data: Likelihoods:

  11. Causal graphical models(Pearl, 2000; Spirtes et al., 1993) • Variables X Y Z

  12. Causal graphical models(Pearl, 2000; Spirtes et al., 1993) • Variables • Structure X Y Z

  13. Causal graphical models(Pearl, 2000; Spirtes et al., 1993) • Variables • Structure • Conditional probabilities p(y) p(x) X Y Z p(z|x,y) Defines probability distribution over variables (for both observation, and intervention)

  14. Bayesian causal induction Hypotheses: causal structures a priori plausibility of structures Priors: Data: observations of variables probability distribution over variables Likelihoods:

  15. Causal induction from contingencies C present (c+) C absent (c-) a c E present (e+) d b E absent (e-) “Does C cause E?” (rate on a scale from 0 to 100)

  16. Buehner & Cheng (1997) Chemical C present (c+) C absent (c-) 6 4 E present (e+) Gene 4 2 E absent (e-) “Does the chemical cause gene expression?” (rate on a scale from 0 to 100)

  17. People Buehner & Cheng (1997) Examined human judgments for all values of P(e+|c+) and P(e+|c-) in increments of 0.25 How can we explain these judgments? Causal rating

  18. C B C B E E Bayesian causal induction chance cause Hypotheses: B B p 1 - p Priors: frequency of cause-effect co-occurrence Data: each cause has an independent opportunity to produce the effect Likelihoods:

  19. C B C B E E Bayesian causal induction chance cause Hypotheses: B B

  20. C B C B E E evidence for a causal relationship Bayesian causal induction chance cause Hypotheses: B B

  21. Buehner and Cheng (1997) People Bayes (r = 0.97)

  22. DP (r = 0.89) Power (r = 0.88) Buehner and Cheng (1997) People Bayes (r = 0.97)

  23. Other predictions • Causal induction from contingency data • sample size effects • judgments for incomplete contingency tables (Griffiths & Tenenbaum, in press) • More complex cases • detectors (Tenenbaum & Griffiths, 2003) • explosions (Griffiths, Baraff, & Tenenbaum, 2004) • simple mechanical devices

  24. The stick-ball machine A B (Kushnir, Schulz, Gopnik, & Danks, 2003)

  25. Outline • A Bayesian approach to causal induction • Coincidences • what makes a coincidence? • rationality and irrationality • the paradox of coincidences • Explaining inductive leaps

  26. What makes a coincidence?

  27. “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

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

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

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

  31. 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

  32. 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

  33. prior odds are low 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 posterior odds are middling

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

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

  36. 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

  37. prior odds are low likelihood ratio is high posterior odds are middling prior odds are low prior odds are low likelihood ratio is middling likelihood ratio is very high posterior odds are low posterior odds are high mere coincidence HHHH HHHHHHHHHH suspicious coincidence HHHHHHHHHHHHHHHHHH cause

  38. Mere and suspicious coincidences • Transition produced by • increase in likelihood ratio (e.g., coinflipping) • increase in prior odds (e.g., genetics vs. ESP) suspicious coincidence evidence for a causal relation mere coincidence

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

  40. 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

  41. Prior odds low high high low Rationality and irrationality coincidence cause Likelihood ratio (evidence) ? chance

  42. The bombing of London (Gilovich, 1991)

  43. Change in... People Number Ratio Location Spread (uniform)

  44. T T X X X X Bayesian causal induction chance cause Hypotheses: T T T T X X X X p 1 - p Priors: Data: bomb locations uniform + regularity Likelihoods: uniform

  45. Change in... People Bayes Number Ratio Location Spread (uniform) r = 0.98

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