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Exploring subjective probability
distributions using Bayesian statistics
Tom Griffiths
Department of Psychology
Cognitive Science Program
University of California, Berkeley
Perception is optimal
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Körding & Wolpert (2004)
Cognition is not
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Bayesian models of cognition
• What would an “ideal cognizer” do?
– “rational analysis” (Anderson, 1990)
• How can structured representations be
combined with statistical inference?
– graphical models, probabilistic grammars, etc.
• What knowledge guides human inferences?
– questions about priors and likelihoods
Exploring subjective distributions
Natural statistics in cognition
(joint work with Josh Tenenbaum)
Markov chain Monte Carlo with people
(joint work with Adam Sanborn)
Exploring subjective distributions
Natural statistics in cognition
(joint work with Josh Tenenbaum)
Markov chain Monte Carlo with people
(joint work with Adam Sanborn)
Natural statistics
Images of natural scenes
Prior distributions
p(x)
Predicting the future
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How often is Google News updated?
t = time since last update
ttotal = time between updates
What should we guess for ttotal given t?
Bayesian inference
p(ttotal|t)  p(t|ttotal) p(ttotal)
posterior
probability
likelihood
prior
Bayesian inference
p(ttotal|t)  p(t|ttotal) p(ttotal)
posterior
probability
likelihood
prior
p(ttotal|t)  1/ttotal p(ttotal)
assume
random
sample
(0 < t < ttotal)
The effects of priors
Evaluating human predictions
• Different domains with different priors:
–
–
–
–
–
a movie has made $60 million
your friend quotes from line 17 of a poem
you meet a 78 year old man
a movie has been running for 55 minutes
a U.S. congressman has served for 11 years
[power-law]
[power-law]
• Prior distributions derived from actual data
• Use 5 values of t for each
• People predict ttotal
[Gaussian]
[Gaussian]
[Erlang]
people
Gott’s rule
empirical prior
parametric prior
Exploring subjective distributions
Natural statistics in cognition
(joint work with Josh Tenenbaum)
Markov chain Monte Carlo with people
(joint work with Adam Sanborn)
Markov chain Monte Carlo
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Metropolis-Hastings algorithm
(Metropolis et al., 1953; Hastings, 1970)
Step 1: propose a state (we assume symmetrically)
Q(x(t+1)|x(t)) = Q(x(t))|x(t+1))
Step 2: decide whether to accept, with probability
Metropolis acceptance
function
Barker acceptance
function
A task
Ask subjects which of two alternatives
comes from a target category
Which animal is a frog?
A Bayesian analysis of the task
Assume:
Response probabilities
If people probability match to the posterior,
response probability is equivalent to the Barker
acceptance function for target distribution p(x|c)
Collecting the samples
Which is the frog?
Which is the frog?
Which is the frog?
Trial 1
Trial 2
Trial 3
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)
Choice task
Samples from Subject 3
(projected onto a plane)
Mean animals by subject
S1
giraffe
horse
cat
dog
S2
S3
S4
S5
S6
S7
S8
Marginal densities
(aggregated across subjects)
Giraffes are
distinguished by
neck length,
body height and
body tilt
Horses are like
giraffes, but with
shorter bodies
and nearly
uniform necks
Cats have longer
tails than dogs
Markov chain Monte Carlo with people
• Probabilistic models can guide the design of
experiments to measure psychological variables
• Markov chain Monte Carlo can be used to
sample from subjective probability distributions
– category distributions (Metropolis-Hastings)
– prior distributions (Gibbs sampling)
• Effective for exploring large stimulus spaces,
with distributions on a small part of the space
Conclusion
• Probabilistic models give us a way to explore the
knowledge that guides people’s inferences
• Basic problem for both cognition and perception:
identifying subjective probability distributions
• Two strategies:
– natural statistics
– Markov chain Monte Carlo with people