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Everyday inductive leaps Making predictions and detecting coincidences

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### 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)

hypotheses

cube

shaded hexagon

Inductive problems- Inferring structure from data
- Perception
- e.g. structure of 3D world from 2D visual data

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

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

Predicting the future

How often is Google News updated?

t = time since last update

ttotal = time between updates

What should we guess forttotalgivent?

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)

The effects of priors

Different kinds of priorsp(ttotal) are appropriate in different domains

e.g. wealth

e.g. height

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

Probability matching

p(ttotal|tpast)

Proportion of judgments below predicted value

ttotal

Quantile of Bayesian posterior distribution

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

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)

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

"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."

(Gilovich, 1991)

(Gilovich, 1991)

(Snow, 1855)

The paradox of coincidences

How can coincidences simultaneously lead us to irrational conclusions and significant discoveries?

“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 eventschance

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 inductionWhat 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 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 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

cause

C

C

E

E

1 -

0 < p(E) < 1

p(E) = 0.5

Bayesian causal inductionHypotheses:

Priors:

frequency of effect in presence of cause

Data:

Likelihoods:

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

Empirical tests

- Is this definition correct?
- from coincidence to evidence
- How do people assess complex coincidences?
- the bombing of London
- coincidences in date

Empirical tests

- Is this definition correct?
- from coincidence to evidence
- How do people assess complex coincidences?
- the bombing of London
- coincidences in date

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

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

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 51 55 59 63 70 87 99

Number of heads/males

Posterior probability

47 51 55 59 63 70 87 99

r = -0.98

Empirical tests

- Is this definition correct?
- from coincidence to evidence
- How do people assess complex coincidences?
- the bombing of London
- coincidences in date

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

cause

T

T

T

T

T

T

X

X

X

X

X

X

X

X

uniform

+

regularity

uniform

Bayesian causal inductionHypotheses:

1 -

Priors:

Data:

bomb locations

Likelihoods:

75 years

cause

B

B

B

B

B

B

B

B

P

P

P

P

P

P

P

P

uniform

uniform + regularity

August

Bayesian causal inductionHypotheses:

1 -

Priors:

Data:

birthdays of those present

Likelihoods:

Coincidences

- Provide evidence for causal structure, but not enough to make us believe that structure exists
- Intimately related to causal induction
- an opportunity to revise a theory
- a window on the process of discovery
- Guided by a well calibrated sense of when an event provides evidence of causal structure

Consequence

significant discovery

false

true

false conclusion

The paradox of coincidencesThe utility of attending to coincidences

depends upon how much you know already

Subjective randomness

- View randomness as an inference about generating processes behind data
- Analysis similar (but inverse) to coincidences
- randomness is evidence against a regular generating process

(Griffiths & Tenenbaum, 2003)

Aspects of language acquisition

(Goldwater, Griffiths, & Johnson, 2006)

Conclusions

- We can learn about cognition (and not just perception) by thinking about optimal solutions to computational problems
- We can study induction using the inferences that people make every day
- Bayesian inference offers a way to understand these inductive inferences

Magic tricks

Magic tricks are regularly used to identify infants’ ontological commitments

Can we use a similar method with adults?

(Wynn, 1992)

Ontological commitments

(Keil, 1981)

What’s a better magic trick?

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

- Participants rate the quality of 45 transformations, 10 appearances, and 10 disappearances
- direction of transformation is randomized between subjects
- A second group rates similarity
- Objects are chosen to lie at different points in a hierarchy

Applicable predicates

What’s a better magic trick?

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

Ontological asymmetries

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

Analyzing asymmetry

milk

water

a brick

a vase

a rose

a daffodil

a dove

a blackbird

a man

a girl

- Build a regression model:
- similarity
- appearing object
- disappearing object
- contains people
- direction in hierarchy (-1,0,1)
- All factors significant
- Explains 90.9% of variance

Applicable predicates

Summary: magic tricks

- Certain factors reliably influence the estimated quality of a magic trick
- Magic tricks might be a way to investigate our ontological assumptions
- inviolable laws that are otherwise hard to assess
- A Bayesian theory of magic tricks?
- strong evidence for a novel causal force
- causal force is given low prior probability

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

- 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 kinds

- Defining kinds
- Counterexamples

HHHH>HHTT

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

HHHH>HHHHTHTTHHHTHTHHTHTTHHH

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

Sampling from categories

Frog distribution P(x|c)

Markov chain Monte Carlo

- Sample from a target distributionP(x) by constructing Markov chain for whichP(x) is the stationary distribution
- Markov chain converges to its stationary distribution, providing outcomes that can be used similarly to samples

Sampling from natural categories

Examined distributions for four natural categories: giraffes, horses, cats, and dogs

Presented stimuli with nine-parameter stick figures (Olman & Kersten, 2004)

Markov chain Monte Carlo with people

- Rational models can guide the design of psychological experiments
- Markov chain Monte Carlo (and other methods) can be used to sample from subjective probability distributions
- category distributions
- prior distributions

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