Normative models of human inductive inference

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Normative models of human inductive inference

• Tom Griffiths
• Department of Psychology
• Cognitive Science Program
• University of California, Berkeley

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Perception is optimal
Körding Wolpert (2004)
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Cognition is not
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Optimality and cognition

• Can optimal solutions to computational problems
  shed light on human cognition?

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Optimality and cognition

• Can optimal solutions to computational problems
  shed light on human cognition?
• Can we explain aspects of cognition as the result
  of sensitivity to natural statistics?
• What kind of representations are extracted from
  those statistics?

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Optimality and cognition

• Can optimal solutions to computational problems
  shed light on human cognition?
• Can we explain aspects of cognition as the result
  of sensitivity to natural statistics?
• What kind of representations are extracted from
  those statistics?

Joint work with Josh Tenenbaum
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Natural statistics
Neural representation
Images of natural scenes
sparse coding
(Olshausen Field, 1996)
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Predicting the future

• How often is Google News updated?
• t time since last update
• ttotal time between updates
• What should we guess for ttotal given t?

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Reverend Thomas Bayes
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Bayes theorem
h hypothesis d data
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Bayes theorem
h hypothesis d data
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Bayesian inference

• p(ttotalt) ? p(tttotal) p(ttotal)

posterior probability
likelihood
prior
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Bayesian inference

• p(ttotalt) ? p(tttotal) p(ttotal)
• p(ttotalt) ? 1/ttotal p(ttotal)

posterior probability
likelihood
prior
assume random sample (0 lt t lt ttotal)
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The effects of priors
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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 for 11 years
  Erlang
• Prior distributions derived from actual data
• Use 5 values of t for each
• People predict ttotal

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people
empirical prior
parametric prior
Gotts rule
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Predicting the future

• People produce accurate predictions for the
  duration and extent of everyday events
• People are sensitive to the statistics of their
  environment in making these predictions
• form of the prior (power-law or exponential)
• distribution given that form (parameters)

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Optimality and cognition

• Can optimal solutions to computational problems
  shed light on human cognition?
• Can we explain aspects of cognition as the result
  of sensitivity to natural statistics?
• What kind of representations are extracted from
  those statistics?

Joint work with Adam Sanborn
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Categories are central to cognition
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Sampling from categories
Frog distribution P(xc)
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Markov chain Monte Carlo

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

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Metropolis-Hastings algorithm(Metropolis et al.,
1953 Hastings, 1970)

• Step 1 propose a state (we assume
  symmetrically)
• Q(x(t1)x(t)) Q(x(t))x(t1))
• Step 2 decide whether to accept, with
  probability

Metropolis acceptance function
Barker acceptance function
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Metropolis-Hastings algorithm
p(x)
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Metropolis-Hastings algorithm
p(x)
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Metropolis-Hastings algorithm
p(x)
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Metropolis-Hastings algorithm
p(x)
A(x(t), x(t1)) 0.5
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Metropolis-Hastings algorithm
p(x)
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Metropolis-Hastings algorithm
p(x)
A(x(t), x(t1)) 1
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A task

• Ask subjects which of two alternatives comes
  from a target category

Which animal is a frog?
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A Bayesian analysis of the task
Assume
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Response probabilities

• If people probability match to the posterior,
  response probability is equivalent to the Barker
  acceptance function for target distribution p(xc)

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Collecting the samples
Which is the frog?
Trial 1
Trial 2
Trial 3
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Verifying the method
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Training

• Subjects were shown schematic fish of
  different sizes and trained on whether they came
  from the ocean (uniform) or a fish farm (Gaussian)

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Between-subject conditions
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Choice task

• Subjects judged which of the two fish came
  from the fish farm (Gaussian) distribution

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Examples of subject MCMC chains
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Estimates from all subjects

• Estimated means and standard deviations are
  significantly different across groups
• Estimated means are accurate, but standard
  deviation estimates are high
• result could be due to perceptual noise or
  response gain

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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)
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Choice task
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Samples from Subject 3(projected onto plane from
LDA)
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Mean animals by subject
S1
S2
S3
S4
S5
S6
S7
S8
giraffe
horse
cat
dog
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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

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Markov chain Monte Carlo with people

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

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Conclusion

• Optimal solutions to computational problems can
  shed light on human cognition
• We can explain aspects of cognition as the result
  of sensitivity to natural statistics
• We can use optimality to explore representations
  extracted from those statistics

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Relative volume of categories
Convex Hull
Minimum Enclosing Hypercube
Convex hull content divided by enclosing
hypercube content
Giraffe Horse Cat Dog
0.00004 0.00006 0.00003 0.00002

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Discrimination method(Olman Kersten, 2004)
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Parameter space for discrimination

• Restricted so that most random draws were
  animal-like

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MCMC and discrimination means
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Iterated learning(Kirby, 2001)
Each learner sees data, forms a hypothesis,
produces the data given to the next learner
With Bayesian learners, the distribution over
hypotheses converges to the prior (Griffiths
Kalish, 2005)
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Explaining convergence to the prior
PL(hd)
PL(hd)
PP(dh)
PP(dh)

• Intuitively data acts once, prior many times
• Formally iterated learning with Bayesian agents
  is a Gibbs sampler on P(d,h)

(Griffiths Kalish, in press)
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Iterated function learning(Kalish, Griffiths,
Lewandowsky, in press)

• Each learner sees a set of (x,y) pairs
• Makes predictions of y for new x values
• Predictions are data for the next learner

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Function learning experiments
Examine iterated learning with different initial
data
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