Revealing inductive biases with Bayesian models

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Revealing inductive biases with Bayesian models

• Tom Griffiths
• UC Berkeley

with Mike Kalish, Brian Christian, and Steve
Lewandowsky
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Inductive problems
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Generalization requires induction

• Generalization predicting the properties of
  an entity from observed properties of others

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What makes a good inductive learner?

• Hypothesis 1 more representational power
• more hypotheses, more complexity
• spirit of many accounts of learning and
  development

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Some hypothesis spaces

• Linear functions
• Quadratic functions
• 8th degree polynomials

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Minimizing squared error
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Minimizing squared error
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Minimizing squared error
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Minimizing squared error
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Measuring prediction error
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What makes a good inductive learner?

• Hypothesis 1 more representational power
• more hypotheses, more complexity
• spirit of many accounts of learning and
  development
• Hypothesis 2 good inductive biases
• constraints on hypotheses that match the
  environment

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Outline

• The bias-variance tradeoff
• Bayesian inference and inductive biases
• Revealing inductive biases
• Conclusions

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Outline

• The bias-variance tradeoff
• Bayesian inference and inductive biases
• Revealing inductive biases
• Conclusions

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A simple schema for induction

• Data D are n pairs (x,y) generated from function
  f
• Hypothesis space of functions, y g(x)
• Error is E (y - g(x))2
• Pick function g that minimizes error on D
• Measure prediction error, averaging over x and y

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Bias and variance

• A good learner makes (f(x) - g(x))2 small
• g is chosen on the basis of the data D
• Evaluate learners by the average of (f(x) -
  g(x))2 over data D generated from f

(Geman, Bienenstock, Doursat, 1992)
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Making things more intuitive

• The next few slides were generated by
• choosing a true function f(x)
• generating a number of datasets D from p(x,y)
  defined by uniform p(x), p(yx) f(x) plus noise
• finding the function g(x) in the hypothesis space
  that minimized the error on D
• Comparing average of g(x) to f(x) reveals bias
• Spread of g(x) around average is the variance

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Linear functions (n 10)
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Linear functions (n 10)
pink is g(x) for each dataset red is average
g(x) black is f(x)
y
x
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Quadratic functions (n 10)
pink is g(x) for each dataset red is average
g(x) black is f(x)
y
x
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8-th degree polynomials (n 10)
pink is g(x) for each dataset red is average
g(x) black is f(x)
y
x
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Bias and variance
(for our (quadratic) f(x), with n 10) Linear
functions high bias, medium variance Quadratic
functions low bias, low variance 8-th order
polynomials low bias, super-high variance
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In general

• Larger hypothesis spaces result in higher
  variance, but low bias across several f(x)
• The bias-variance tradeoff
• if we want a learner that has low bias on a range
  of problems, we pay a price in variance
• This is mainly an issue when n is small
• the regime of much of human learning

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Quadratic functions (n 100)
pink is g(x) for each dataset red is average
g(x) black is f(x)
y
x
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8-th degree polynomials (n 100)
pink is g(x) for each dataset red is average
g(x) black is f(x)
y
x
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The moral

• General-purpose learning mechanisms do not work
  well with small amounts of data
• more representational power isnt always better
• To make good predictions from small amounts of
  data, you need a bias that matches the problem
• these biases are the key to successful induction,
  and characterize the nature of an inductive
  learner
• So how can we identify human inductive biases?

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Outline

• The bias-variance tradeoff
• Bayesian inference and inductive biases
• Revealing inductive biases
• Conclusions

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Bayesian inference

• Rational procedure for updating beliefs
• Foundation of many learning algorithms
• Lets us make the inductive biases of learners
  precise

Reverend Thomas Bayes
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Bayes theorem
h hypothesis d data
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Priors and biases

• Priors indicate the kind of world a learner
  expects to encounter, guiding their conclusions
• In our function learning example
• likelihood gives probability to data that
  decrease with sum squared errors (i.e. a
  Gaussian)
• priors are uniform over all functions in
  hypothesis spaces of different kinds of
  polynomials
• having more functions corresponds to a belief in
  a more complex world

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Outline

• The bias-variance tradeoff
• Bayesian inference and inductive biases
• Revealing inductive biases
• Conclusions

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Two ways of using Bayesian models

• Specify models that make different assumptions
  about priors, and compare their fit to human data
• (Anderson Schooler, 1991
• Oaksford Chater, 1994
• Griffiths Tenenbaum, 2006)
• Design experiments explicitly intended to reveal
  the priors of Bayesian learners

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Iterated learning(Kirby, 2001)
What are the consequences of learners learning
from other learners?
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Objects of iterated learning

• Knowledge communicated across generations through
  provision of data by learners
• Examples
• religious concepts
• social norms
• myths and legends
• causal theories
• language

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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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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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Analyzing iterated learning
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Iterated Bayesian learning
PL(hd)
PL(hd)
PP(dh)
PP(dh)
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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

• If iterated learning converges to the prior, it
  might provide a tool for determining the
  inductive biases of human learners
• We can test this by reproducing iterated learning
  in the lab, with stimuli for which human biases
  are well understood

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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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Outline

• The bias-variance tradeoff
• Bayesian inference and inductive biases
• Revealing inductive biases
• Conclusions

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Conclusions

• Solving inductive problems and forming good
  generalizations requires good inductive biases
• Bayesian inference provides a way to make
  assumptions about the biases of learners explicit
• Two ways to identify human inductive biases
• compare Bayesian models assuming different priors
• design tasks to extract biases from Bayesian
  learners
• Iterated learning provides a lens for magnifying
  the inductive biases of learners
• small effects for individuals are big effects for
  groups

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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,
1961)
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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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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Discovering the biases of models
Generic neural network
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Discovering the biases of models
EXAM (Delosh, Busemeyer, McDaniel, 1997)
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Discovering the biases of models
POLE (Kalish, Lewandowsky, Kruschke, 2004)
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