Transcript Author-Topic Modeling from Large Document Collections

Optimal predictions in everyday cognition
Tom Griffiths
Brown University
Predicting the future
The effects of prior knowledge
Strategy:
examine the influence of prior knowledge in an
inductive problem we solve every day
What should we use as the prior, p(ttotal)?
Optimality and Bayesian inference
Many people believe that perception is optimal…
Josh Tenenbaum
MIT
Gott (1993): use the uninformative prior
p(ttotal) 1/ttotal
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Yields a simple prediction rule:
t* = 2t
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How often is Google News updated?
t = time since last update
ttotal= time between updates
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…but cognition is not.
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What should we guess for ttotal given t?
More generally…
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t  4000 years, t*  8000 years
• You encounter a phenomenon that has existed for t
units of time. How long will it continue into the
future? (i.e. what’s ttotal?)
Predicting everyday events
• This seems like a good strategy…
– You meet someone who is 35 years old. How long will
they live?
– “70 years” seems reasonable
• We could replace “time” with any other variable that
ranges from 0 to some unknown upper limit
In particular, there is controversy over whether
people’s inferences follow Bayes’ rule
Posterior
probability
Likelihood
Prior
probability
p(d | h) p(h)
p(h | d ) 
 p(d | h) p(h)
hH
Sum over space
of hypotheses
h: hypothesis
d: data
which indicates how a rational agent should update
beliefs about hypotheses h in light of data d.
Several results suggest people do not combine
prior probabilities with data correctly.
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predict the future
infer causal relationships
identify the work of chance
assess similarity and make generalizations
learn languages and concepts
…and solve other inductive problems?
Drawing strong conclusions from limited
data requires using prior knowledge
– You meet someone who is 78 years old. How long will
they live?
– You meet someone who is 6 years old. How long will
they live?
• You read about a movie that has made $60 million to
date. How much money will it make in total?
• You see that something has been baking in the oven
for 34 minutes. How long until it’s ready?
The effects of priors
• You meet someone who is 78 years old. How long
will they live?
• Your friend quotes to you from line 17 of his favorite
poem. How long is the poem?
p(ttotal|t)  p(t|ttotal) p(ttotal)
Evaluating human predictions
likelihood prior
• Different domains with different priors:
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–
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assuming random sampling, the likelihood is
p(t|ttotal) = 1/ttotal
What is the best guess for ttotal? (call it t*)
Not the maximal value of p(ttotal|t)
(that’s just t* = t)
We use the posterior median
P(ttotal < t*|t) = 0.5
t*t
p(ttotal|t)
p(ttotal|t)
ttotal = t
ttotal
ttotal
empirical prior
parametric prior
Nonparametric priors
You arrive at a friend’s house,
and see that a cake has been in
the oven for 34 minutes. How
long will it be in the oven?
People make good predictions
despite the complex distribution
No direct experience
You learn that in ancient Egypt,
there was a great flood in the
11th year of a pharaoh’s reign.
How long did he reign?
People identify the form, but are
mistaken about the parameters
Bayesian inference
posterior
probability
people
Gott’s rule
How long did the typical pharaoh
reign in ancient Egypt?
• You see taxicab #107 pull up to the curb in front of
the train station. How many cabs in this city?
A puzzle
If they do not use priors, how do people…
• But, it’s not so simple:
Everyday prediction problems
(e.g., Tversky & Kahneman, 1974)
Results
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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 for 11 years [Erlang]
Prior distributions derived from actual data
Use 5 values of t for each
People predict ttotal
A total of 350 participants and ten scenarios
Conclusions
• People produce accurate predictions for the
duration and extent of everyday events
• People have strong prior knowledge
– form of the prior (power-law or exponential)
– distribution given that form (parameters)
– non-parametric distribution when necessary
• Reveals a surprising correspondence between
probabilities in the mind and in the world, and
suggests that people do use prior probabilities in
making inductive inferences