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Markov chain Monte Carlowith people
Tom GriffithsDepartment of Psychology
Cognitive Science Program
UC Berkeley
with Mike Kalish, Stephan Lewandowsky, and Adam Sanborn
Inductive problems
blicket toma
dax wug
blicket wug
S X Y
X {blicket,dax}
Y {toma, wug}
Learning languages from utterances
Learning functions from (x,y) pairs
Learning categories from instances of their members
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
Statistics and inductive problems
Cognitive science
Categorization
Causal learning
Function learning
Language
…
Statistics
Density estimation
Graphical models
Regression
Probabilistic grammars
…
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
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
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
Are people Bayesian?
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Reverend Thomas Bayes
Bayes’ theorem
€
P(h | d) =P(d | h)P(h)
P(d | ′ h )P( ′ h )′ h ∈H
∑
Posteriorprobability
Likelihood Priorprobability
Sum over space of hypothesesh: hypothesis
d: data
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People are stupid
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?
The effects of priors
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
peopleparametric priorempirical prior
Gott’s rule
A different approach…
Instead of asking whether people are rational, use assumption of rationality to investigate cognition
If we can predict people’s responses, we can design experiments that measure psychological variables
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(x|c)
€
limt →∞
P(x(t ) = i | x(0)) = π i
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(x|c)
Develop ways to sample from these distributions
Outline
Markov chain Monte Carlo
Sampling from the prior
Sampling from category distributions
Outline
Markov chain Monte Carlo
Sampling from the prior
Sampling from category distributions
• Variables x(t+1) independent of history given x(t)
• Converges to a stationary distribution under easily checked conditions (i.e., if it is ergodic)
x x x x x x x x
Transition matrixT = P(x(t+1)|x(t))
Markov chains
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
Gibbs sampling
For variables x = x1, x2, …, xn and target P(x)
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)
Gibbs sampling
(MacKay, 2002)
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
Metropolis-Hastings algorithm
p(x)
Metropolis-Hastings algorithm
p(x)
Metropolis-Hastings algorithm
p(x)
Metropolis-Hastings algorithm
A(x(t), x(t+1)) = 0.5
p(x)
Metropolis-Hastings algorithm
p(x)
Metropolis-Hastings algorithm
A(x(t), x(t+1)) = 1
p(x)
Outline
Markov chain Monte Carlo
Sampling from the prior
Sampling from category distributions
Iterated learning(Kirby, 2001)
What are the consequences of learners learning from other learners?
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Analyzing iterated learning
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PL(h|d): probability of inferring hypothesis h from data d
PP(d|h): probability of generating data d from hypothesis h
PL(h|d)
PP(d|h)
PL(h|d)
PP(d|h)
Iterated Bayesian learning
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PL(h|d)
PP(d|h)
PL(h|d)
PP(d|h)
€
PL (h | d) =PP (d | h)P(h)
PP (d | ′ h )P( ′ h )′ h ∈H
∑
Assume learners sample from their posterior distribution:
Analyzing iterated learning
d0 h1 d1 h2PL(h|d) PP(d|h) PL(h|d)
d2 h3PP(d|h) PL(h|d)
d PP(d|h)PL(h|d)h1 h2d PP(d|h)PL(h|d)
h3
A Markov chain on hypotheses
d0 d1h PL(h|d) PP(d|h)d2h PL(h|d) PP(d|h) h PL(h|d) PP(d|h)
A Markov chain on data
Stationary distributions
• Markov chain on h converges to the prior, P(h)
• Markov chain on d converges to the “prior predictive distribution”
€
P(d) = P(d | h)h
∑ P(h)
(Griffiths & Kalish, 2005)
Explaining convergence to the prior
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PL(h|d)
PP(d|h)
PL(h|d)
PP(d|h)
• 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)
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
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”
General strategy
• Use well-studied and simple stimuli for which people’s inductive biases are known– function learning– concept learning– color words
• Examine dynamics of iterated learning– convergence to state reflecting biases– predictable path to convergence
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
data hypotheses
(Kalish, Griffiths, & Lewandowsky, in press)
Function learning experiments
Stimulus
Response
Slider
Feedback
Examine iterated learning with different initial data
1 2 3 4 5 6 7 8 9
IterationInitialdata
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
• 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
Statistics and cultural evolution
Outline
Markov chain Monte Carlo
Sampling from the prior
Sampling from category distributions
Categories are central to cognition
Sampling from categories
Frog distribution
P(x|c)
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 samplesWhich is the frog? Which is the frog? Which is the frog?
Trial 1 Trial 2 Trial 3
Verifying the method
Training
Subjects were shown schematic fish of different sizes and trained on whether they came from the ocean
(uniform) or a fish farm (Gaussian)
Between-subject conditions
Choice task
Subjects judged which of the two fish came from the fish farm (Gaussian) distribution
Examples of subject MCMC chains
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
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 plane from LDA)
Mean animals by subject
giraffe
horse
cat
dog
S1 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
Relative volume of categoriesMinimum Enclosing Hypercube
Giraffe Horse Cat Dog
0.00004 0.00006 0.00003 0.00002
Convex hull content divided by enclosing hypercube content
Convex Hull
Discrimination method(Olman & Kersten, 2004)
Parameter space for discrimination
Restricted so that most random draws were animal-like
MCMC and discrimination means
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, …
A different approach…
Instead of asking whether people are rational, use assumption of rationality to investigate cognition
If we can predict people’s responses, we can design experiments that measure psychological variables
Randomized algorithms Psychological experiments
€
PL (h | d)∝PP (d | h)P(h)
PP (d | ′ h )P( ′ h )′ h ∈H
∑
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
r
r = 1 r = 2 r =
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)
From sampling to maximizing
Implications for linguistic universals
• When learners sample from P(h|d), 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
Iterated concept learning
• Each learner sees examples from a species
• Identifies species of four amoebae
• Iterated learning is run within-subjects
data hypotheses
(Griffiths, Christian, & Kalish, in press)
Two positive examples
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data (d)
hypotheses (h)
Bayesian model(Tenenbaum, 1999; Tenenbaum & Griffiths, 2001)
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P(h | d) =P(d | h)P(h)
P(d | ′ h )P( ′ h )′ h ∈H
∑d: 2 amoebaeh: set of 4 amoebae
€
P(d | h) =1/ h
m
0
⎧ ⎨ ⎩
d ∈ h
otherwise
m: # of amoebae in the set d (= 2)|h|: # of amoebae in the set h (= 4)
€
P(h | d) =P(h)
P( ′ h )h '|d ∈h'
∑Posterior is renormalized prior
What is the prior?
Classes of concepts(Shepard, Hovland, & Jenkins, 1958)
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
shape
size
color
Experiment design (for each subject)Class 1Class 2Class 3Class 4Class 5Class 6Class 1Class 2Class 3Class 4Class 5Class 6
6 iterated learning chains
6 independent
learning “chains”
Estimating the prior
data (d)hy
poth
eses
(h)
Estimating the prior
Class 1Class 2
Class 3
Class 4
Class 5
Class 6
0.8610.087
0.009
0.002
0.013
0.028
Prior
r = 0.952
Bayesian modelHuman subjects
Two positive examples(n = 20)
Prob
abil
ity
Iteration
Prob
abil
ity
Iteration
Human learners Bayesian model
Two positive examples(n = 20)
Prob
abil
ity
Bayesian model
Human learners
Three positive examples
data (d)
hypotheses (h)
Three positive examples(n = 20)
Prob
abil
ity
Iteration
Prob
abil
ity
Iteration
Human learners Bayesian model
Three positive examples(n = 20)
Bayesian model
Human learners
Classification objects
Parameter space for discrimination
Restricted so that most random draws were animal-like
MCMC and discrimination means
Problems with classification objects
Category 1
Category 2
Category 1
Category 2
Problems with classification objectsMinimum Enclosing Hypercube
Giraffe Horse Cat Dog
0.00004 0.00006 0.00003 0.00002
Convex hull content divided by enclosing hypercube content
Convex Hull
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(x|c)
Category drift
• For fragile categories, the MCMC procedure could influence the category representation
• Interleaved training and test blocks in the training experiments