Transcript Inductive inference in perception and cognition

Priors and predictions in
everyday cognition
Tom Griffiths
Cognitive and Linguistic Sciences
data
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behavior
What computational problem is the brain solving?
Does human behavior correspond to an
optimal solution to that problem?
Inductive problems
• Inferring structure from data
• Perception
– e.g. structure of 3D world from 2D visual data
data
hypotheses
cube
shaded hexagon
Inductive problems
• Inferring structure from data
• Perception
– e.g. structure of 3D world from 2D data
• Cognition
– e.g. relationship between variables from samples
data
hypotheses
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Reverend Thomas Bayes
Bayes’ theorem
Posterior
probability
Likelihood
Prior
probability
p(d | h) p(h)
p(h | d ) 
 p(d | h) p(h)
hH
h: hypothesis
d: data
Sum over space
of hypotheses

Bayes’ theorem
p(h | d)  p(d | h) p(h)
h: hypothesis
d: data
Perception is optimal
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Körding & Wolpert (2004)
Cognition is not
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Do people use priors?
p(h | d)  p(d | h) p(h)
Standard answer: no

(Tversky & Kahneman, 1974)
This talk: yes
What are people’s priors?
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?
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?
Prior knowledge matters when…
• …using a single datapoint
– predicting the future
– joint work
• …using secondhand data
– effects of priors on cultural transmission
Outline
• …using a single datapoint
– predicting the future
– joint work with Josh Tenenbaum (MIT)
• …using secondhand data
– effects of priors on cultural transmission
– joint work with Mike Kalish (Louisiana)
• Conclusions
Outline
• …using a single datapoint
– predicting the future
– joint work with Josh Tenenbaum (MIT)
• …using secondhand data
– effects of priors on cultural transmission
– joint work with Mike Kalish (Louisiana)
• Conclusions
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?
Making predictions
• 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?)
• We could replace “time” with any other
variable that ranges from 0 to some
unknown upper limit
Everyday prediction problems
• 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?
• 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?
• You see taxicab #107 pull up to the curb in front of
the train station. How many cabs in this city?
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)
Bayesian inference
p(ttotal|t)  p(t|ttotal) p(ttotal)
posterior
probability
likelihood
prior
p(ttotal|t)  1/ttotal 1/ttotal
assume “uninformative”
random
prior
sample
(0 < t < ttotal)
Bayesian inference
p(ttotal|t)  1/ttotal 1/ttotal
posterior
probability
random
sampling
“uninformative”
prior
What is the best guess for ttotal?
How about maximal value of p(ttotal|t)?
p(ttotal|t)
ttotal = t
ttotal
Bayesian inference
p(ttotal|t)  1/ttotal 1/ttotal
posterior
probability
random
sampling
“uninformative”
prior
What is the best guess for ttotal?
Instead, compute t* such that p(ttotal > t*|t) = 0.5:
p(ttotal|t)
ttotal
Bayesian inference
p(ttotal|t)  1/ttotal 1/ttotal
posterior
probability
random
sampling
“uninformative”
prior
What is the best guess for ttotal?
Instead, compute t* such that p(ttotal > t*|t) = 0.5.
Yields Gott’s Rule: P(ttotal > t*|t) = 0.5 when t* = 2t
i.e., best guess for ttotal = 2t
Applying Gott’s rule
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t  4000 years, t*  8000 years
Applying Gott’s rule
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t  130,000 years, t*  260,000 years
Predicting everyday events
• You meet someone who is 35 years old. How long
will they live?
– “70 years” seems reasonable
• Not so simple:
– 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?
The effects of priors
• Different kinds of priors p(ttotal) are
appropriate in different domains.
Uninformative: p(ttotal) 1/ttotal
The effects of priors
• Different kinds of priors p(ttotal) are
appropriate in different domains.
e.g. wealth
e.g. height
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
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?
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?
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?
How long did the typical
pharaoh reign in ancient
Egypt?
…using a single datapoint
• 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)
– non-parametric distribution when necessary
• Reveals a surprising correspondence between
probabilities in the mind and in the world
Outline
• …using a single datapoint
– predicting the future
– joint work with Josh Tenenbaum (MIT)
• …using secondhand data
– effects of priors on cultural transmission
– joint work with Mike Kalish (Louisiana)
• Conclusions
Cultural transmission
• Most knowledge is based on secondhand data
• Some things can only be learned from others
– cultural objects transmitted across generations
• Cultural transmission provides an opportunity
for priors to influence cultural objects
Iterated learning
(Briscoe, 1998; Kirby, 2001)
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• Each learner sees data, forms a hypothesis,
produces the data given to the next learner
• c.f. the playground game “telephone”
Objects of iterated learning
• Languages
• Religious concepts
• Social norms
• Myths and legends
• Causal theories
Explaining linguistic universals
• Human languages are a subset of all logically
possible communication schemes
– universal properties common to all languages
(Comrie, 1981; Greenberg, 1963; Hawkins, 1988)
• Two questions:
– why do linguistic universals exist?
– why are particular properties universal?
Explaining linguistic universals
• Traditional answer:
– linguistic universals reflect innate constraints
specific to a system for acquiring language
• Alternative answer:
– iterated learning imposes “information bottleneck”
– universal properties survive this bottleneck
(Briscoe, 1998; Kirby, 2001)
Analyzing iterated learning
What are the consequences of iterated learning?
Complex
algorithms
Analytic results
?
Kirby (2001)
Simulations
Smith, Kirby, &
Brighton (2003)
Simple
algorithms
Komarova, Niyogi,
& Nowak (2002)
Brighton (2002)
Iterated Bayesian learning
p(h|d)
p(h|d)
p(d|h)
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p(d|h)
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Learners are rational Bayesian agents
(covers a wide range of learning algorithms)
Markov chains
x
x
x
x
x
x
x
x
Transition matrix
P(x(t+1)|x(t))
• Variables x(t+1) independent of history given x(t)
• Converges to a stationary distribution under
easily checked conditions
Markov chain Monte Carlo
• A strategy for sampling from complex
probability distributions
• Key idea: construct a Markov chain which
converges to target distribution
– e.g. Metropolis algorithm
– e.g. Gibbs sampling
Gibbs sampling
For variables x = x1, x2, …, xn
Draw xi(t+1) from P(xi|x-i)
x-i = x1(t+1), x2(t+1),…, xi-1(t+1), xi+1(t), …, xn(t)
(Geman & Geman, 1984)
(a.k.a. the heat bath algorithm in statistical physics)
Gibbs sampling
(MacKay, 2002)
Iterated Bayesian learning
• Defines a Markov chain on (h,d)
Iterated Bayesian learning
• Defines a Markov chain on (h,d)
• This Markov chain is a Gibbs sampler for
p(d,h)  p(d | h) p(h)

Iterated Bayesian learning
• Defines a Markov chain on (h,d)
• This Markov chain is a Gibbs sampler for
p(d,h)  p(d | h) p(h)
• Rate of convergence is geometric
– Gibbs sampler converges geometrically

(Liu, Wong, & Kong, 1995)
Analytic results
• Iterated Bayesian learning converges to
p(d,h)  p(d | h) p(h)
• Corollaries:

– distribution over hypotheses converges to p(h)
– distribution over data converges to p(d)
– the proportion of a population of iterated
learners with hypothesis h converges to p(h)
Implications for linguistic universals
• Two questions:
– why do linguistic universals exist?
– why are particular properties universal?
• Different answers:
– existence explained through iterated learning
– universal properties depend on the prior
• Focuses inquiry on the priors of the learners
– cultural objects reflect the human mind
A method for discovering priors
Iterated learning converges to the prior…
…evaluate prior by producing iterated learning
Iterated function learning
data
hypotheses
• Each learner sees a set of (x,y) pairs
• Makes predictions of y for new x values
• Predictions are data for the next learner
Function learning in the lab
Stimulus
Feedback
Response
Slider
Examine iterated learning with different initial data
Initial
data
Iteration
1
2
3
4
5
6
7
8
9
…using secondhand data
• Iterated Bayesian learning converges to the prior
• Constrains explanations of linguistic universals
• Open questions in Bayesian language evolution
– variation in priors
– other selective pressures
• Provides a method for evaluating priors
– concepts, causal relationships, languages, …
Outline
• …using a single datapoint
– predicting the future
• …using secondhand data
– effects of priors on cultural transmission
• Conclusions
Bayes’ theorem
p(h | d)  p(d | h) p(h)

A unifying principle for explaining
inductive inferences
Bayes’ theorem
behavior = f(data,knowledge)
data
Qui ckTi me™ and a
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behavior
Bayes’ theorem
behavior = f(data,knowledge)
knowledge
data
Qui ckTi me™ and a
TIFF (Uncompressed) decompr essor
are needed to see this pictur e.
behavior
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?
HHTHT
HHTHT
HHHHT
What’s the computational problem?
p(HHTHT|random)
p(random|HHTHT)
An inference about the structure of the world
An example: Gaussians
• If we assume…
– data, d, is a single real number, x
– hypotheses, h, are means of a Gaussian, 
– prior, p(), is Gaussian(0,02)
• …then p(xn+1|xn) is Gaussian(n, x2 + n2)
x n / x2  0 / 02
n 
2
2
1/ x  1/ 0
1
 
2
2
1/ x  1/ 0
2
n
0 = 0, 02 = 1, x0 = 20
Iterated learning results in rapid convergence to prior
An example: Linear regression
• Assume
– data, d, are pairs of real numbers (x, y)
– hypotheses, h, are functions
• An example: linear regression
– hypotheses have slope  and pass through origin
– p() is Gaussian(0,02)
y
}
x=1
y
0 = 1, 02 = 0.1, y0 = -1
}
x=1