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

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causes and coincidences

Causes and coincidences

Tom Griffiths

Cognitive and Linguistic Sciences

Brown University

slide4

“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.”

slide6

76 years

75 years

(Halley, 1752)

the paradox of coincidences
The paradox of coincidences

How can coincidences simultaneously lead us to

irrational conclusions and significant discoveries?

outline
Outline
  • A Bayesian approach to causal induction
  • Coincidences
    • what makes a coincidence?
    • rationality and irrationality
    • the paradox of coincidences
  • Explaining inductive leaps
outline1
Outline
  • A Bayesian approach to causal induction
  • Coincidences
    • what makes a coincidence?
    • rationality and irrationality
    • the paradox of coincidences
  • Explaining inductive leaps
causal induction
Causal induction
  • Inferring causal structure from data
  • A task we perform every day …
    • does caffeine increase productivity?
  • … and throughout science
    • three comets or one?
bayes theorem

Likelihood

Prior

probability

Posterior

probability

Sum over space

of hypotheses

Bayes’ theorem

h: hypothesis

d: data

bayesian causal induction
Bayesian causal induction

causal structures

Hypotheses:

Priors:

Data:

Likelihoods:

causal graphical models pearl 2000 spirtes et al 19932
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)

bayesian causal induction1
Bayesian causal induction

Hypotheses:

causal structures

a priori plausibility of structures

Priors:

Data:

observations of variables

probability distribution over variables

Likelihoods:

causal induction from contingencies
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)

buehner cheng 1997
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)

buehner cheng 19971

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

bayesian causal induction2

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:

bayesian causal induction3

C

B

C

B

E

E

Bayesian causal induction

chance

cause

Hypotheses:

B

B

bayesian causal induction4

C

B

C

B

E

E

evidence for a

causal relationship

Bayesian causal induction

chance

cause

Hypotheses:

B

B

buehner and cheng 1997
Buehner and Cheng (1997)

People

Bayes (r = 0.97)

buehner and cheng 19971

DP (r = 0.89)

Power (r = 0.88)

Buehner and Cheng (1997)

People

Bayes (r = 0.97)

other predictions
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
slide27

The stick-ball machine

A

B

(Kushnir, Schulz, Gopnik, & Danks, 2003)

outline2
Outline
  • A Bayesian approach to causal induction
  • Coincidences
    • what makes a coincidence?
    • rationality and irrationality
    • the paradox of coincidences
  • Explaining inductive leaps
a common definition coincidences are unlikely events

“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
bayesian causal induction5

Prior odds

low

high

?

high

Likelihood ratio

(evidence)

?

low

Bayesian causal induction

cause

chance

bayesian causal induction6

Prior odds

low

high

high

low

Bayesian causal induction

coincidence

cause

Likelihood ratio

(evidence)

?

chance

what makes a coincidence
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

what makes a coincidence1
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

what makes a coincidence2

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

slide38
HHHHHHHHHH

HHTHTHTTHT

prior odds

are low

likelihood ratio

is high

posterior odds

are middling

bayesian causal induction7
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

slide40

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

slide41

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

mere and suspicious coincidences
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

testing the definition
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
slide44

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

rationality and irrationality

Prior odds

low

high

high

low

Rationality and irrationality

coincidence

cause

Likelihood ratio

(evidence)

?

chance

slide46

The bombing of London

(Gilovich, 1991)

slide48

Change in...

People

Number

Ratio

Location

Spread

(uniform)

bayesian causal induction8

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

slide50

Change in...

People

Bayes

Number

Ratio

Location

Spread

(uniform)

r = 0.98

coincidences in date
Coincidences in date

May 14, July 8, August 21, December 25

vs.

August 3, August 3, August 3, August 3

bayesian causal induction9
Bayesian causal induction

chance

cause

Hypotheses:

B

B

B

P

P

P

P

P

P

P

P

p 1 - p

Priors:

Data:

birthdays of those present

uniform

uniform + regularity

Likelihoods:

August

slide54

Bayes

People

slide55

Rationality and irrationality

  • People’s sense of the strength of coincidences gives a close match to the likelihood ratio
    • bombing and birthdays
slide56

Rationality and irrationality

  • People’s sense of the strength of coincidences gives a close match to the likelihood ratio
    • bombing and birthdays
  • Suggests that we accept false conclusions when our prior odds are insufficiently low
rationality and irrationality1

Prior odds

low

high

high

low

Rationality and irrationality

coincidence

cause

Likelihood ratio

(evidence)

?

chance

the paradox of coincidences1

Reason

Consequence

Significant discovery

Incorrect current theory

Correct current theory

False conclusion

The paradox of coincidences

Prior odds can be low for two reasons

Attending to coincidences makes

more sense the less you know

coincidences
Coincidences
  • Provide evidence for causal structure, but not enough to make us believe that structure exists
  • Intimately related to causal induction
    • an opportunity to discover a theory is wrong
  • Guided by a well calibrated sense of when an event provides evidence of causal structure
outline3
Outline
  • A Bayesian approach to causal induction
  • Coincidences
    • what makes a coincidence?
    • rationality and irrationality
    • the paradox of coincidences
  • Explaining inductive leaps
explaining inductive leaps
Explaining inductive leaps
  • How do people
    • infer causal relationships
    • identify the work of chance
    • predict the future
    • assess similarity and make generalizations
    • learn functions, languages, and concepts

. . . from such limited data?

  • What knowledge guides human inferences?
which sequence seems more random
Which sequence seems more random?

HHHHHHHHHH

vs.

HHTHTHTTHT

subjective randomness

evidence for a random

generating process

Subjective randomness
  • Typically evaluated in terms of p(d | chance)
  • Assessing randomness is part of causal induction
randomness and coincidences

strength of coincidence

evidence for a random

generating process

Randomness and coincidences
randomness and coincidences1
Randomness and coincidences

r = -0.96

r = -0.94

pick a random number

People

Bayes

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

Pick a random number…
bayes theorem2
Bayes’ theorem

inference = f(data,knowledge)

bayes theorem3
Bayes’ theorem

inference = f(data,knowledge)

slide70

Predicting the future

Human predictions match optimal predictions from empirical prior

iterated learning briscoe 1998 kirby 2001

inference

sampling

sampling

inference

d0

h1

d1

h2

p(h|d)

p(d|h)

p(h|d)

p(d|h)

(Griffiths & Kalish, submitted)

Iterated learning(Briscoe, 1998; Kirby, 2001)

production

production

learning

learning

data

hypothesis

data

hypothesis

slide72

Iteration

1 2 3 4 5 6 7 8 9

conclusion
Conclusion
  • Many cognitive judgments are the result of challenging problems of induction
  • Bayesian statistics provides a formal framework for exploring how people solve these problems
  • Makes it possible to ask…
    • how do we make surprising discoveries?
    • how do we learn so much from so little?
    • what knowledge guides our judgments?
collaborators
Collaborators
  • Causal induction
    • Josh Tenenbaum (MIT)
    • Liz Baraff (MIT)
  • Iterated learning
    • Mike Kalish (University of Louisiana)
causes and coincidences1
Causes and coincidences

“coincidence” appears in 13/60 cases

p(“cause”) = 0.01

p(“cause”|“coincidence”) = 0.26

a reformulation unlikely kinds
A reformulation: unlikely kinds
  • Coincidences are events of an unlikely kind
    • e.g. a sequence with that number of heads
  • Deals with the obvious problem...

p(10 heads) < p(5 heads, 5 tails)

problems with unlikely kinds
Problems with unlikely kinds
  • Defining kinds

August 3, August 3, August 3, August 3

January 12, March 22, March 22, July 19, October 1, December 8

problems with unlikely kinds1
Problems with unlikely kinds
  • Defining kinds
  • Counterexamples

HHHH>HHTT

P(4 heads) < P(2 heads, 2 tails)

HHHH>HHHHTHTTHHHTHTHHTHTTHHH

P(4 heads) > P(15 heads, 8 tails)