Markov chain Monte Carlo with people

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Markov chain Monte Carlowith people

• Tom Griffiths
• Department of Psychology
• Cognitive Science Program
• UC Berkeley

with Mike Kalish, Stephan Lewandowsky, and Adam
Sanborn
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Inductive problems
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Computational cognitive science

• Identify the underlying computational problem
• Find the optimal solution to that problem
• Compare human cognition to that solution
• For inductive problems, solutions come from
  statistics

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Statistics and inductive problems
Cognitive science Categorization Causal
learning Function learning Language
Statistics Density estimation Graphical
models Regression Probabilistic grammars
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Statistics and human cognition

• How can we use statistics to understand
  cognition?
• How can cognition inspire new statistical models?
• applications of Dirichlet process and Pitman-Yor
  process models to natural language
• exchangeable distributions on infinite binary
  matrices via the Indian buffet process (priors on
  causal structure)
• nonparametric Bayesian models for relational data

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Statistics and human cognition

• How can we use statistics to understand
  cognition?
• How can cognition inspire new statistical models?
• applications of Dirichlet process and Pitman-Yor
  process models to natural language
• exchangeable distributions on infinite binary
  matrices via the Indian buffet process (priors on
  causal structure)
• nonparametric Bayesian models for relational data

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Statistics and human cognition

• How can we use statistics to understand
  cognition?
• How can cognition inspire new statistical models?
• applications of Dirichlet process and Pitman-Yor
  process models to natural language
• exchangeable distributions on infinite binary
  matrices via the Indian buffet process
• nonparametric Bayesian models for relational data

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Are people Bayesian?
Reverend Thomas Bayes
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Bayes theorem
h hypothesis d data
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People are stupid
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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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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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A different approach

• Instead of asking whether people are rational,
  use assumption of rationality to investigate
  cognition
• If we can predict peoples responses, we can
  design experiments that measure psychological
  variables

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Two deep questions

• What are the biases that guide human learning?
• prior probability distribution P(h)
• What do mental representations look like?
• category distribution P(xc)

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Two deep questions

• What are the biases that guide human learning?
• prior probability distribution on hypotheses,
  P(h)
• What do mental representations look like?
• distribution over objects x in category c, P(xc)

Develop ways to sample from these distributions
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Outline

• Markov chain Monte Carlo
• Sampling from the prior
• Sampling from category distributions

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Outline

• Markov chain Monte Carlo
• Sampling from the prior
• Sampling from category distributions

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Markov chains
x
x
x
x
x
x
x
x
Transition matrix T P(x(t1)x(t))

• Variables x(t1) independent of history given
  x(t)
• Converges to a stationary distribution under
  easily checked conditions (i.e., if it is ergodic)

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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
• Two main schemes
• Gibbs sampling
• Metropolis-Hastings algorithm

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Gibbs sampling

• For variables x x1, x2, , xn and target P(x)
• Draw xi(t1) from P(xix-i)
• x-i x1(t1), x2(t1),, xi-1(t1), xi1(t), ,
  xn(t)

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Gibbs sampling
(MacKay, 2002)
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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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Outline

• Markov chain Monte Carlo
• Sampling from the prior
• Sampling from category distributions

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Iterated learning(Kirby, 2001)
What are the consequences of learners learning
from other learners?
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Analyzing iterated learning
PL(hd)
PL(hd)
PP(dh)
PP(dh)
PL(hd) probability of inferring hypothesis h
from data d PP(dh) probability of generating
data d from hypothesis h
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Iterated Bayesian learning
PL(hd)
PL(hd)
PP(dh)
PP(dh)
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Analyzing iterated learning
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Stationary distributions

• Markov chain on h converges to the prior, P(h)
• Markov chain on d converges to the prior
  predictive distribution

(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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Revealing inductive biases

• Many problems in cognitive science can be
  formulated as problems of induction
• learning languages, concepts, and causal
  relations
• Such problems are not solvable without bias
• (e.g., Goodman, 1955 Kearns Vazirani, 1994
  Vapnik, 1995)
• What biases guide human inductive inferences?
• If iterated learning converges to the prior,
  then it may provide a method for investigating
  biases

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Serial reproduction(Bartlett, 1932)

• Participants see stimuli, then reproduce them
  from memory
• Reproductions of one participant are stimuli for
  the next
• Stimuli were interesting, rather than controlled
• e.g., War of the Ghosts

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General strategy

• Use well-studied and simple stimuli for which
  peoples inductive biases are known
• function learning
• concept learning
• color words
• Examine dynamics of iterated learning
• convergence to state reflecting biases
• predictable path to convergence

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Iterated function learning

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

(Kalish, Griffiths, Lewandowsky, in press)
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Function learning experiments
Examine iterated learning with different initial
data
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Initial data
Iteration
1 2 3 4
5 6 7 8 9
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Identifying inductive biases

• Formal analysis suggests that iterated learning
  provides a way to determine inductive biases
• Experiments with human learners support this idea
• when stimuli for which biases are well understood
  are used, those biases are revealed by iterated
  learning
• What do inductive biases look like in other
  cases?
• continuous categories
• causal structure
• word learning
• language learning

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Statistics and cultural evolution

• Iterated learning for MAP learners reduces to a
  form of the stochastic EM algorithm
• Monte Carlo EM with a single sample
• Provides connections between cultural evolution
  and classic models used in population genetics
• MAP learning of multinomials Wright-Fisher
• More generally, an account of how products of
  cultural evolution relate to the biases of
  learners

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Outline

• Markov chain Monte Carlo
• Sampling from the prior
• Sampling from category distributions

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Categories are central to cognition
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Sampling from categories
Frog distribution P(xc)
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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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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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Conclusion

• Markov chain Monte Carlo provides a way to sample
  from subjective probability distributions
• Many interesting questions can be framed in terms
  of subjective probability distributions
• inductive biases (priors)
• mental representations (category distributions)
• Other MCMC methods may provide further empirical
  methods
• Gibbs for categories, adaptive MCMC,

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A different approach

• Instead of asking whether people are rational,
  use assumption of rationality to investigate
  cognition
• If we can predict peoples responses, we can
  design experiments that measure psychological
  variables

Randomized algorithms ? Psychological experiments
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From sampling to maximizing
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From sampling to maximizing

• General analytic results are hard to obtain
• (r ? is Monte Carlo EM with a single sample)
• For certain classes of languages, it is possible
  to show that the stationary distribution gives
  each hypothesis h probability proportional to
  P(h)r
• the ordering identified by the prior is
  preserved, but not the corresponding probabilities

(Kirby, Dowman, Griffiths, in press)
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Implications for linguistic universals

• When learners sample from P(hd), the
  distribution over languages converges to the
  prior
• identifies a one-to-one correspondence between
  inductive biases and linguistic universals
• As learners move towards maximizing, the
  influence of the prior is exaggerated
• weak biases can produce strong universals
• cultural evolution is a viable alternative to
  traditional explanations for linguistic
  universals

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Iterated concept learning

• Each learner sees examples from a species
• Identifies species of four amoebae
• Iterated learning is run within-subjects

hypotheses
data
(Griffiths, Christian, Kalish, in press)
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Two positive examples
data (d)
hypotheses (h)
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Bayesian model(Tenenbaum, 1999 Tenenbaum
Griffiths, 2001)
d 2 amoebae h set of 4 amoebae
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Classes of concepts(Shepard, Hovland, Jenkins,
1958)
color
size
shape
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
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Experiment design (for each subject)
6 iterated learning chains
6 independent learning chains
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Estimating the prior
data (d)
hypotheses (h)
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Estimating the prior
Prior
Bayesian model
Human subjects
0.861
Class 1
Class 2
0.087
0.009
Class 3
0.002
Class 4
0.013
Class 5
Class 6
0.028
r 0.952
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Two positive examples(n 20)
Human learners
Bayesian model
Probability
Probability
Iteration
Iteration
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Two positive examples(n 20)
Human learners
Probability
Bayesian model
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Three positive examples
data (d)
hypotheses (h)
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Three positive examples(n 20)
Human learners
Bayesian model
Probability
Probability
Iteration
Iteration
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Three positive examples(n 20)
Human learners
Bayesian model
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Classification objects
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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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Problems with classification objects
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Problems with classification objects
Minimum Enclosing Hypercube
Convex Hull
Convex hull content divided by enclosing
hypercube content
Giraffe Horse Cat Dog
0.00004 0.00006 0.00003 0.00002

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Allowing a Wider Range of Behavior

• An exponentiated choice rule results in a
  Markov chain with stationary distribution
  corresponding to an exponentiated version of the
  category distribution, proportional to p(xc)?

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Category drift

• For fragile categories, the MCMC procedure could
  influence the category representation
• Interleaved training and test blocks in the
  training experiments

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