Inductive inference in perception and cognition by yaosaigeng

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```									        Bayesian models of human
learning and inference
(http://web.mit.edu/cocosci/Talks/nips06-tutorial.ppt)

Josh Tenenbaum
MIT
Department of Brain and Cognitive Sciences
Computer Science and AI Lab (CSAIL)
Thanks to Tom Griffiths, Charles Kemp, Vikash Mansinghka
The probabilistic revolution in AI
• Principled and effective solutions for
inductive inference from ambiguous data:
–   Vision
–   Robotics
–   Machine learning
–   Expert systems / reasoning
–   Natural language processing

• Standard view: no necessary connection to
how the human brain solves these problems.
Probabilistic inference in
human cognition?
• “People aren‟t Bayesian”
– Kahneman and Tversky (1970‟s-present): “heuristics and
biases” research program. 2002 Nobel Prize in
Economics.
– Slovic, Fischhoff, and Lichtenstein (1976): “It appears
that people lack the correct programs for many important
the opportunity to evolve an intellect capable of dealing
conceptually with uncertainty.”
– Stephen Jay Gould (1992): “Our minds are not built (for
whatever reason) to work by the rules of probability.”
The probability of breast cancer is 1% for a woman at 40 who
participates in a routine screening. If a woman has breast cancer,
the probability is 80% that she will have a positive mammography.
If a woman does not have breast cancer, the probability is 9.6%
that she will also have a positive mammography.

A woman in this age group had a positive mammography in a
routine screening. What is the probability that she actually has
breast cancer?
A. greater than 90%
B. between 70% and 90%           95 out of 100 doctors
C. between 50% and 70%
D. between 30% and 50%
E. between 10% and 30%                                “Base rate
F. less than 10%                 Correct answer        neglect”
Availability biases in
probability judgment
• How likely is that a randomly chosen word
– ends in “g”?
– ends in “ing”?

• When buying a car, how much do you
weigh your friend‟s experience relative to
consumer satisfaction surveys?
Probabilistic inference in
human cognition?
• “People aren‟t Bayesian”
– Kahneman and Tversky (1970‟s-present): “heuristics
and biases” research program. 2002 Nobel Prize in
Economics.
• Psychology is often drawn towards the
mind‟s errors and apparent irrationalities.
• But the computationally interesting question
remains: How does mind work so well?
Bayesian models of cognition
Visual perception [Weiss, Simoncelli, Adelson, Richards, Freeman, Feldman,
Kersten, Knill, Maloney, Olshausen, Jacobs, Pouget, ...]
Language acquisition and processing [Brent, de Marken, Niyogi, Klein,
Manning, Jurafsky, Keller, Levy, Hale, Johnson, Griffiths, Perfors, Tenenbaum, …]
Motor learning and motor control [Ghahramani, Jordan, Wolpert, Kording,
Associative learning [Dayan, Daw, Kakade, Courville, Touretzky, Kruschke, …]
Memory [Anderson, Schooler, Shiffrin, Steyvers, Griffiths, McClelland, …]
Attention [Mozer, Huber, Torralba, Oliva, Geisler, Yu, Itti, Baldi, …]
Categorization and concept learning [Anderson, Nosfosky, Rehder, Navarro,
Griffiths, Feldman, Tenenbaum, Rosseel, Goodman, Kemp, Mansinghka, …]
Reasoning [Chater, Oaksford, Sloman, McKenzie, Heit, Tenenbaum, Kemp, …]
Causal inference [Waldmann, Sloman, Steyvers, Griffiths, Tenenbaum, Yuille, …]
Decision making and theory of mind [Lee, Stankiewicz, Rao, Baker,
Goodman, Tenenbaum, …]
Learning concepts from examples

• Word learning

“horse”          “horse”   “horse”
Learning concepts from examples
“tufa”
“tufa”

“tufa”
Everyday inductive leaps
How can people learn so much about the
world . . .
–   Kinds of objects and their properties
–   The meanings of words, phrases, and sentences
–   Cause-effect relations
–   The beliefs, goals and plans of other people
–   Social structures, conventions, and rules

. . . from such limited evidence?
Contributions of Bayesian models
• Principled quantitative models of human behavior,
with broad coverage and a minimum of free
• Explain how and why human learning and
reasoning works, in terms of (approximations to)
optimal statistical inference in natural
environments.
• A framework for studying people‟s implicit
knowledge about the structure of the world: how it
is structured, used, and acquired.
• A two-way bridge to state-of-the-art AI and
machine learning.
Marr‟s Three Levels of Analysis
• Computation:
“What is the goal of the computation, why is it
appropriate, and what is the logic of the
strategy by which it can be carried out?”

• Algorithm:
Cognitive psychology

• Implementation:
Neurobiology
• The human mind is not a universal Bayesian engine.
• But, the mind does appear adapted to solve
important real-world inference problems in
approximately Bayesian ways, e.g.
– Predicting everyday events
– Causal learning and reasoning
– Learning concepts from examples
children solve these problems mostly
unconsciously, effortlessly, and successfully.
Technical themes
• Inference in probabilistic models
– Role of priors, explaining away.
• Learning in graphical models
– Parameter learning, structure learning.
• Bayesian model averaging
– Being Bayesian over network structures.
• Bayesian Occam‟s razor
– Trade off model complexity against data fit.
Technical themes
• Structured probabilistic models
– Grammars, first-order logic, relational schemas.
• Hierarchical Bayesian models
– Acquire abstract knowledge, supports transfer.
• Nonparametric Bayes
– Flexible models that grow in complexity as new
data warrant.
• Tractable approximate inference
– Markov chain Monte Carlo (MCMC),
Sequential Monte Carlo (particle filtering).
Outline
• Predicting everyday events
• Causal learning and reasoning
• Learning concepts from examples
Outline
• Predicting everyday events
• Causal learning and reasoning
• Learning concepts from examples
Basics of Bayesian inference
P(d | h) P(h)
• Bayes‟ rule: P(h | d ) 
• An example
 P(d | hi ) P(hi )
hi H

– Data: John is coughing
– Some hypotheses:
1. John has a cold
2. John has lung cancer
3. John has a stomach flu
– Likelihood P(d|h) favors 1 and 2 over 3
– Prior probability P(h) favors 1 and 3 over 2
– Posterior probability P(h|d) favors 1 over 2 and 3
Bayesian inference in perception and
sensorimotor integration

(Weiss, Simoncelli & Adelson 2002)   (Kording & Wolpert 2004)
Memory retrieval as Bayesian inference
(Anderson & Schooler, 1991)
Power law of        Additive effects                         Spacing effects
forgetting:         of practice & delay:                     in forgetting:
Log memory strength

Mean # recalled
Log delay            Log delay                            Retention interval
(hours)             (seconds)                                 (days)
Memory retrieval as Bayesian inference
(Anderson & Schooler, 1991)

For each item in
memory, estimate
the probability that it
will be useful in the
present context.

Use priors based on
the statistics of
natural information
sources.
Memory retrieval as Bayesian inference
(Anderson & Schooler, 1991)
Power law of        Additive effects                       Spacing effects
forgetting:         of practice & delay:                   in forgetting:
Log need odds

Log need odds
Log # days since      Log # days since                     Log # days since
last occurrence       last occurrence                      last occurrence

[New York Times data; c.f. email sources, child-directed speech]
Everyday prediction problems
(Griffiths & Tenenbaum, 2006)

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?
Making predictions
• You encounter a phenomenon that has
existed for tpast units of time. How long will
it continue into the future? (i.e. what‟s
ttotal?)

• We could replace “time” with any other
quantity that ranges from 0 to some
unknown upper limit.
Bayesian inference

P(ttotal|tpast)  P(tpast|ttotal) P(ttotal)
posterior         likelihood     prior
probability
Bayesian inference

P(ttotal|tpast)  P(tpast|ttotal) P(ttotal)
posterior           likelihood       prior
probability

      1/ttotal        1/ttotal
Assume            “Uninformative”
random            prior
sample
(0 < tpast < ttotal)       (e.g., Jeffreys,
Jaynes)
Bayesian inference

P(ttotal|tpast)  1/ttotal           1/ttotal
posterior           Random         “Uninformative”
probability          sampling        prior

P(ttotal|tpast)

ttotal
tpast
Bayesian inference

P(ttotal|tpast)  1/ttotal            1/ttotal
posterior            Random         “Uninformative”
probability           sampling        prior

P(ttotal|tpast)

ttotal
tpast
Best guess for ttotal: t such that P(ttotal > t|tpast) = 0.5:
Bayesian inference

P(ttotal|tpast)  1/ttotal            1/ttotal
posterior            Random         “Uninformative”
probability           sampling        prior

P(ttotal|tpast)

ttotal
tpast
Yields Gott‟s Rule: P(ttotal > t|tpast) = 0.5 when t = 2tpast
i.e., best guess for ttotal = 2tpast .
Evaluating Gott‟s Rule
to date. How much money will it make in total?
– “\$156 million” seems reasonable.
• 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.

e.g., wealth,            e.g., height,
contacts                 lifespan
[Gott: P(ttotal) ttotal-1 ]
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 move has been running for 55 minutes
–   A U.S. congressman has served for 11 years
–   A cake has been in the oven for 34 minutes
• Use 5 values of tpast for each.
• People predict ttotal .
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?
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?
Summary: prediction
• Predictions about the extent or magnitude of
• Contrast with Bayesian inference in perception,
motor control, memory: no “universal priors” here.
• Predictions depend rationally on priors that are
appropriately calibrated for different domains.
– Form of the prior (e.g., power-law or exponential)
– Specific distribution given that form (parameters)
– Non-parametric distribution when necessary.

• In the absence of concrete experience, priors may
be generated by qualitative background knowledge.
Outline
• Predicting everyday events
• Causal learning and reasoning
• Learning concepts from examples
Bayesian networks
Nodes: variables
Four random variables:
X1   coughing
Each node has a conditional
X2   high body temperature
probability distribution
X3   flu
X4   lung cancer
Data: observations of X1, ..., X4

P(x4) X4                   X3 P(x3)

P(x1|x3, x4)     X1             X2 P(x2|x3)
Causal Bayesian networks
Nodes: variables
Four random variables:
X1   coughing
Each node has a conditional
X2   high body temperature
probability distribution
X3   flu
X4   lung cancer
Data: observations of and
interventions on X1, ..., X4

P(x4) X4                X3 P(x3)

P(x1|x3, x4)   X1             X2 P(x2|x3)
(Pearl; Glymour
& Cooper)
Inference in causal graphical models
• Explaining away or “discounting” in                 A       B

social reasoning (Kelley; Morris & Larrick)
C

• “Screening off” in intuitive causal reasoning
(Waldmann, Rehder & Burnett, Blok & Sloman, Gopnik & Sobel)
B
A     B      C                         P(c|b) vs. P(c|b, a)
P(c|b, not a)
A        C
– Better in chains than common-cause structures; common-cause
better if mechanisms clearly independent
• Understanding and predicting the effects of
interventions (Sloman & Lagnado; Gopnik & Schulz)
Learning graphical models
• Structure learning: what causes what?
P(x4) X4            X3 P(x3)

P(x1|x3, x4) X1          X2 P(x2|x3)

• Parameter learning: how do causes work?
P(x4) X4            X3 P(x3)

P(x1|x3, x4) X1          X2 P(x2|x3)
Bayesian learning of causal structure
Data d                                     Causal hypotheses h
d1  X 1  1, X 2  1, X 3  1, X 4  1       X4            X3                  X4        X3
d 2  X 1  1, X 2  0, X 3  0, X 4  1
d 3  X 1  0, X 2  1, X 3  0, X 4  1           X1                X2              X1        X2
d 4  X 1  1, X 2  0, X 3  1, X 4  1

1. What is the most likely                              P(d | hi )P(hi )
network h given                            P(hi | d) 
observed data d ?                                       P(d | h j )P(h j )
j
2. How likely
is there to be a
P( X 4  X 2 | d )     P( X
h j H
4     X 2 | h j ) P(h j | d )
link X4     X2 ?                                        (Bayesian model averaging)
Bayesian Occam‟s Razor

M1

p(D = d | M )
(MacKay, 2003;
Ghahramani
tutorials)
M2

All possible data sets d

For any model M,                    p(D  d | M )  1
all d D
Law of “conservation of belief”: A model that can predict many
possible data sets must assign each of them low probability.
Learning causation from contingencies
C present C absent
(c+)      (c-)
e.g., “Does injecting
E present   (e+)       a        c
this chemical cause
mice to express a
E absent (e-)          b        d       certain gene?”

Subjects judge the extent C to which causes E
(rate on a scale from 0 to 100)
Two models of causal judgment

• Delta-P (Jenkins & Ward, 1965):
P  P(e  | c  )  P(e  | c  )

• Power PC (Cheng, 1997):
P
Power 
p
1  P (e  | c  )
Judging the probability that C                  E
(Buehner & Cheng, 1997; 2003)
P        0.00          0.25       0.50    0.75 1.00

People

P

Power

• Independent effects of both P and causal power.
• At P=0, judgments decrease with base rate.
(“frequency illusion”)
Learning causal strength
(parameter learning)
Assume this causal structure:
B        C
w0       w1
E

P and causal power are maximum likelihood
estimates of the strength parameter w1, under
different parameterizations for P(E|B,C):
linear  P, Noisy-OR  causal power
Learning causal structure
(Griffiths & Tenenbaum, 2005)
• Hypotheses:           h1:     B        C        h0:   B          C
w0       w1             w0
E                        E

P (d | h1 ) likelihood ratio
• Bayesian causal support: log             (Bayes factor)
P (d | h0 ) gives evidence
P(d | h0 )   0 P(d | w0 ) p(w0 | h0 ) dw0
1                                     in favor of h1

 
1    1
P(d | h1 )              P(d | w0 ,w1) p(w0 ,w1 | h1) dw0 dw1
0     0

noisy-OR
(assume uniform parameter priors, but see Yuille et al., Danks et al.)
Buehner and Cheng (1997)

People

P (r = 0.89)

Power (r = 0.88)

Support (r = 0.97)
Implicit background theory
• Injections may or may not cause gene expression,
but gene expression does not cause injections.
– No hypotheses with E    C
• Other naturally occurring processes may also
cause gene expression.
– All hypotheses include an always-present background
cause B    C
• Causes are generative, probabilistically sufficient
and independent, i.e. each cause independently
produces the effect in some proportion of cases.
– Noisy-OR parameterization
Sensitivity analysis

People

Support (Noisy-OR)

2

Support (generic parameterization)
Generativity is essential
P(e+|c+)      8/8      6/8      4/8      2/8      0/8
P(e+|c-)      8/8      6/8      4/8      2/8      0/8
100                        Support
50
0

• Predictions result from “ceiling effect”
– ceiling effects only matter if you believe a cause
increases the probability of an effect
Blicket detector
Both objects activate     Object A does not       Chi
the detector        activate the detector   each
by itself        Then
they
maket

(Sobel, Gopnik, and colleagues)
Procedure used in Sobel et al. (2002), Experiment 2
Backward Blocking Condition
e Condition

Both objects activate   Object Aactivates the     Chi

See this? It‟s a                       Let‟s put this one                          Oooh, it‟s a
s activate          Object A does not          Children are asked if   the detector         detector by itself     each
tector             activate the detector      each is a blicket                                                    Then
they
by itself           Then
blicket machine.                       on the machine.
makehe machine go
t                                       blicket!
Blickets
Blocking Condition make it go.

s activate        Object Aactivates the        Children are asked if
tector              detector by itself        each is a blicket
Then
Both objects activate                Object A does not                Children are a

“Backwards blocking”
the detector                   activate the detector            each is a blicke
by itself                 Then
makehe machin
t
el et al. (2002), Experiment 2
Figure 13: Procedure used in Sobel et al. (2002), Experiment 2
(Sobel, Tenenbaum & Gopnik, 2004)
Backward Blocking Condition
One-Cause Condition

t A does not                 A                B
Children are asked if objects activate
Both                    AB Trial
Both objects activate
Object A does not
the detector
A Trial
Object Aactivates the
detector by itself
Children are a
each is a blicke
e the detector           each is a blicket      the detector                  activate the detector            each is a blicket
by itself                Then
by itself                 Then
makehe machine go
t                                                                                                          makehe machin
t
makehe machine go
t

–   Initially: Nothing on detector – detector silent (A=0, B=0, E=0)
Backward Blocking Condition
–   Trial 1: A B on detector – detector active (A=1, B=1, E=1)
–   Trial 2: A on detector – detector active (A=1, B=0, E=1)
–   4-year-olds judge if each object is a blicket                         A B
Aactivates the
A: a blicketasked if     (100% say yes)
Children are Both objects activate                            ? ?
Object Aactivates the  Children are asked if
tor by itself           each is a blicket
B: probablytonot detector
Then
make
the
say itself
a blicket (34% detector byyes)      each is a blicket
Then E
the machine go                                                                     makehe machine go
t

(cf. “explaining away in weight space”, Dayan & Kakade)
Possible hypotheses?
A       B   A       B   A       B   A       B   A       B   A       B   A       B   A       B

E           E           E           E           E           E           E           E

A       B   A       B   A       B   A       B   A       B   A       B   A       B   A       B

E           E           E           E           E           E           E           E

A       B   A       B   A       B   A       B   A       B   A       B   A       B   A       B

E           E           E           E           E           E           E           E
Bayesian causal learning
With a uniform prior on hypotheses,
generic parameterization:              Probability of
being a blicket:
A         B
0.32      0.32

0.34     0.34
A stronger hypothesis space
• Links can only exist from blocks to detectors.
• Blocks are blickets with prior probability q.
• Blickets always activate detectors, detectors never activate on their
own (i.e., deterministic OR parameterization, no hidden causes).

P(h00) = (1 – q)2     P(h01) = (1 – q) q   P(h10) = q(1 – q)   P(h11) = q2

A       B     A       B            A       B          A       B

E             E                    E                  E
P(E=1 | A=0, B=0):       0             0                    0                  0
P(E=1 | A=1, B=0):       0             0                    1                  1
P(E=1 | A=0, B=1):       0             1                    0                  1
P(E=1 | A=1, B=1):       0             1                    1                  1
Manipulating prior probability
(Tenenbaum, Sobel, Griffiths, & Gopnik)
Figure 13: Procedure used in Sobel et al. (2002), Experiment 2
One-Cause Condition

Both objects activate                        Object A does not                       Children are ask
the detector                           activate the detector                   each is a blicket
by itself                        Then
Figure 13: Procedure used in Sobel et al. (2002), Experiment 2                                                                                                         makehe machine
t
ed in Sobel et al. (2002), Experiment 2
et al. (2002), Experiment 2
One-Cause Condition                  Figure 13: Procedure used in Sobel et al. (2002), Experiment 2
Backward Blocking Condition
One-Cause Condition

Both objects activate                         Object A does not                          Children are asked if
A does not the detector not
Object A does                 Initial    activate theare asked if
detector
Children
byaitself
AB Trial  each is a blicket
Both objects activate
Then
A Trial
Object Aactivates the                     Children are ask
activate the detector
he detector                     each is a blicket each is blicket Both objects activate the detector           Object A does not detector by itself                    eachif a blicket
makehe machine go
t                                                                      Then
y itself        by itself        Then              Then
they are asked to they are asked to the detector                              activate the detector                   each is a blicketthey are asked to
makehe machine go the machine go
t             make                                                               by itself                                         makehe machine
Then
Learning more complex structures
• Tenenbaum et al., Griffiths & Sobel: detectors with
more than two objects and noisy mechanisms
• Steyvers et al., Sobel & Kushnir: active learning
with interventions (c.f. Tong & Koller, Murphy)

from interventions on
continuous dynamical
systems
Inferring hidden causes
Common unobserved cause

4x           2x           2x
Independent unobserved causes

1x           2x           2x           2x          2x
One observed cause
The “stick ball”
2x           4x                 machine
(Kushnir, Schulz, Gopnik, & Danks, 2003)
Bayesian learning with unknown
number of hidden variables

(Griffiths
et al 2006)
Common unobserved cause

Independent unobserved causes

One observed cause

a = 0.3
w = 0.8

r = 0.94
Inferring latent causes in
classical conditioning
(Courville, Daw, Gordon, Touretzky 2003)
e.g.,
A noise
X tone
B click
US shock

Training:
A US
A X
B US
Test:
X
X B
Inferring latent causes in
perceptual learning
(Orban, Fiser, Aslin, Lengyel 2006)

Learning to recognize objects and segment scenes:
Inferring latent causes in
sensory integration
(Kording et al. 2006, NIPS 06)
Coincidences
(Griffiths & Tenenbaum, in press)

• The birthday problem
– How many people do you need to have in the
room before the probability exceeds 50% that
two of them have the same birthday?    23.

• The bombing of London
How much of
a coincidence?
P (d | latent)
Bayesian coincidence factor: log
P (d | random)

Chance:                Latent common cause:
C

x   x   x   x   x           x   x   x      x     x

August

Alternative hypotheses:
proximity in date, matching
days of the month, matching
month, ....
How much of
a coincidence?
P (d | latent)
Bayesian coincidence factor: log
P (d | random)

Chance:                   Latent common cause:
C

x   x   x   x   x         x   x   x   x    x

uniform
uniform                       +
regularity
Summary: causal inference & learning
• Human causal induction can be explained using
core principles of graphical models.
– Bayesian inference (explaining away, screening off)
– Bayesian structure learning (Occam‟s razor,
model averaging)
– Active learning with interventions
– Identifying latent causes
Summary: causal inference & learning
• Crucial constraints on hypothesis spaces come
from abstract prior knowledge, or “intuitive
theories”.
– What are the variables?
– How can they be connected?
– How are their effects parameterized?

• Big open questions…
– How can these theories be described formally?
– How can these theories be learned?
Hierarchical Bayesian framework
Abstract
Principles

Structure

Data

(Griffiths, Tenenbaum, Kemp et al.)
A theory for blickets
(c.f. PRMs, BLOG, FOPL)
Learning with a
uniform prior on
network structures:

True network
attributes (1-12)
Sample 75 observations…
patients   observed data
Learning a block-          0.0 0.8 0.01
z     1 2
3 4          5 6

structured prior on        0.0 0.0 0.75   h                          7 8

9 10
0.0 0.0 0.0
network structures:                                           11 12

(Mansinghka et al. 2006)

True network
attributes (1-12)
Sample 75 observations…
patients       observed data
The “blessing of abstraction”
True structure of                      1    2    3    4    5    6

graphical model G:
7   8    9   10   11   12   13   14   15   16

# of samples: 20                  80                  1000
Graph G
edge
(G)
Data D

Abstract theory Z                                                                edge
(G)
Graph G
class
Data D                                                                            (z)
Outline
• Predicting everyday events
• Causal learning and reasoning
• Learning concepts from examples
“tufa”

“tufa”

“tufa”

Learning from just one or a few examples, and mostly
unlabeled examples (“semi-supervised learning”).
Simple model of concept learning
“This is a blicket.”
“Can you show me the
other blickets?”
Simple model of concept learning
“This is a blicket.”

Other blickets.
Simple model of concept learning
“This is a blicket.”

Other blickets.

Learning from just one positive example is possible if:
– Assume concepts refer to clusters in the world.
– Observe enough unlabeled data to identify clear clusters.
(c.f. Learning with mixture models and EM, Ghahramani &
Jordan, 1994; Nigam et al. 2000)
Concept learning with mixture
models in cognitive science
• Fried & Holyoak (1984)
– Modeled unsupervised and
semi-supervised categorization
as EM in a Gaussian mixture.

• Anderson (1990)
– Modeled unsupervised and semi-supervised
categorization as greedy sequential search in an
infinite (Chinese restaurant process) mixture.
Infinite (CRP) mixture models
• Construct from k-component mixtures by integrating
out mixing weights, collapsing equivalent partitions,
and taking the limit as k  .
• Does not require that we commit to a fixed – or even
finite – number of classes.
• Effective number of classes can grow with number
of data points, balancing complexity with data fit.
• Computationally much simpler than applying
Bayesian Occam‟s razor or cross-validation.
• Easy to learn with standard Monte Carlo
approximations (MCMC, particle filtering),
hopefully avoiding local minima.
High school lunch room analogy
Sampling from
the CRP:

“punks”

“preppies”

“jocks”

“nerds”
Assign to       Group with
Gibbs sampler (Neal):      larger groups   similar objects

“punks”

“preppies”

“jocks”

“nerds”
A typical cognitive experiment
F1   F2   F3   F4   Label

Training stimuli:   1    1    1    1      1
1    0    1    0      1
0    1    0    1      1
0    0    0    0      0
0    1    0    0      0
1    0    1    1      0

Test stimuli:   0    1    1    1     ?
1    1    0    1     ?
1    1    1    0     ?
1    0    0    0     ?
0    0    1    0     ?
0    0    0    1     ?
Anderson (1990), “Rational
model of categorization”:
Greedy sequential search
in an infinite mixture
model.

Sanborn, Griffiths, Navarro
(2006), “More rational
model of categorization”:
Particle filter with a
small # of particles
Towards more natural concepts
CrossCat: Discovering multiple structures
that capture different subsets of features
(Shafto, Kemp, Mansinghka, Gordon & Tenenbaum, 2006)
Infinite relational models
(Kemp, Tenenbaum, Griffiths, Yamada & Ueda, AAAI 06)

(c.f. Xu,
Tresp, et al.
SRL 06)
concept

predicate

concept

Biomedical predicate data from UMLS (McCrae et al.):
– 134 concepts: enzyme, hormone, organ, disease, cell function ...
– 49 predicates: affects(hormone, organ), complicates(enzyme, cell
function), treats(drug, disease), diagnoses(procedure, disease) …
Infinite relational models
(Kemp, Tenenbaum, Griffiths, Yamada & Ueda, AAAI 06)

e.g., Diseases affect   Chemicals interact   Chemicals cause
Organisms              with Chemicals       Diseases
Learning from very few examples
“tufa”

“tufa”
• Word learning
“tufa”

• Property induction
Cows have T9 hormones.          Cows have T9 hormones.
Seals have T9 hormones.         Sheep have T9 hormones.
Squirrels have T9 hormones.     Goats have T9 hormones.

All mammals have T9 hormones.   All mammals have T9 hormones.
The computational problem
(c.f., semi-supervised learning)
?

Horse
Cow                                                              ?
Chimp                                                              ?
Gorilla                                                            ?
Mouse                                                              ?
Squirrel                                                            ?
Dolphin                                                             ?
Seal                                                             ?
Rhino
?
Elephant
Features                              New property

(85 features from Osherson et al., e.g., for Elephant: „gray‟, „hairless‟,
„toughskin‟, „big‟, „bulbous‟, „longleg‟, „tail‟, „chewteeth‟, „tusks‟,
„smelly‟, „walks‟, „slow‟, „strong‟, „muscle‟, „quadrapedal‟,…)
Many sources of priors
Chimps have T9 hormones.                  Taxonomic
Gorillas have T9 hormones.                similarity

Poodles can bite through wire.            Jaw strength
Dobermans can bite through wire.

Salmon carry E. Spirus bacteria.
Food web
Grizzly bears carry E. Spirus bacteria.   relations
Hierarchical Bayesian Framework
(Kemp & Tenenbaum)
P(form)

F: form                        Tree

mouse
P(structure | form)
squirrel

S: structure                chimp

gorilla

hormones
P(data | structure)

Has T9
F1
F2
F3
F4
mouse
D: data                   squirrel         …
?
?
chimp
gorilla              ?
P(D|S): How the structure constrains the
data of experience
• Define a stochastic process over structure S that
generates hypotheses h.
– For generic properties, prior should favor hypotheses that
vary smoothly over structure.
– Many properties of biological species were actually
generated by such a process (i.e., mutation + selection).

Smooth: P(h) high                      Not smooth: P(h) low
P(D|S): How the structure constrains the
data of experience
S
Gaussian Process
(~ random walk,
diffusion)
~1
   (S )
[Zhu, Ghahramani
& Lafferty 2003]   y

Threshold

h
Structure S

Data D             Species 1
Species 2
Species 3
Species 4
Species 5
Species 6
Species 7
Species 8
Species 9
Species 10
Features
(85 features from Osherson et al., e.g., for Elephant: „gray‟, „hairless‟,
„toughskin‟, „big‟, „bulbous‟, „longleg‟, „tail‟, „chewteeth‟, „tusks‟,
„smelly‟, „walks‟, „slow‟, „strong‟, „muscle‟, „quadrapedal‟,…)
Structure S

Data D        Species 1
Species 2                                                           ?
Species 3                                                           ?
Species 4                                                           ?
Species 5                                                           ?
Species 6                                                           ?
Species 7                                                           ?
Species 8                                                           ?
Species 9
Species 10                                                          ?
Features                           New property
(85 features from Osherson et al., e.g., for Elephant: „gray‟, „hairless‟,
„toughskin‟, „big‟, „bulbous‟, „longleg‟, „tail‟, „chewteeth‟, „tusks‟,
„smelly‟, „walks‟, „slow‟, „strong‟, „muscle‟, „quadrapedal‟,…)
Cows have property P.
Elephants have property P.

Horses have property P.

Tree
2D
Gorillas have property P.
Mice have property P.
Seals have property P.

All mammals have property P.
varying properties
Property type
“has T9 hormones”           “can bite through wire”       “carry E. Spirus bacteria”

Theory Structure
taxonomic tree                directed chain              directed network
+ diffusion process           + drift process              + noisy transmission
Class D                                       Class D
Class A
Class B               Class A                 Class A
Class F                                       Class E
Class C
Class D               Class C           Class C         Class B
Class E                                                             Class G
Class E
Class F
Class B
Class G
Class F
Hypotheses                         Class G

Class A
Class B
Class C
Class D               ...                     ...                         ...
Class E
Class F
Class G
Herring

Tuna
Mako shark
Sand shark

Dolphin
Human

Kelp

Sand shark

Kelp   Herring          Tuna         Mako shark   Human

Dolphin
Hierarchical Bayesian Framework

F: form        Chain                  Tree      Space

chimp             mouse       mouse
squirrel
squirrel
S: structure   gorilla

squirrel              chimp                gorilla
chimp
mouse             gorilla

F1
F2
F3
F4
D: data                    mouse
squirrel
chimp
gorilla
Discovering structural forms

Ostrich Robin Crocodile Snake Turtle Bat   Orangutan
Discovering structural forms
“Great chain of being”

Linnaeus

Ostrich Robin Crocodile Snake Turtle Bat   Orangutan
People can discover structural forms
• Scientists
– Tree structure for living kinds (Linnaeus)
– Periodic structure for chemical elements
(Mendeleev)

• Children
–   Hierarchical structure of category labels
–   Clique structure of social groups
–   Cyclical structure of seasons or days of the week
–   Transitive structure for value
Typical structure learning algorithms
assume a fixed structural form

Flat Clusters             Line                     Circle
K-Means                   Guttman scaling          Circumplex models
Mixture models            Ideal point models
Competitive learning

Tree                      Grid                     Euclidean Space
Hierarchical clustering   Self-Organizing Map      MDS
Bayesian phylogenetics    Generative topographic   PCA
mapping                  Factor Analysis
Hierarchical Bayesian Framework
F: form
Favors simplicity
mouse

squirrel
S: structure                   chimp

gorilla
Favors smoothness
[Zhu et al., 2003]

F1
F2
F3
F4
D: data                   mouse
squirrel
chimp
gorilla
Structural forms as graph grammars
Form     Process     Form     Process
Development of structural forms
as more data are observed
Beyond “Nativism” versus “Empiricism”
• “Nativism”: Explicit knowledge of structural
forms for core domains is innate.
– Atran (1998): The tendency to group living kinds into
hierarchies reflects an “innately determined cognitive
structure”.
– Chomsky (1980): “The belief that various systems of mind are
organized along quite different principles leads to the natural
conclusion that these systems are intrinsically determined, not
simply the result of common mechanisms of learning or
growth.”
• “Empiricism”: General-purpose learning systems
without explicit knowledge of structural form.
– Connectionist networks (e.g., Rogers and McClelland, 2004).
– Traditional structure learning in probabilistic graphical models.
Summary: concept learning
Models based on                     F: form
Bayesian inference over
hierarchies of structured                               mouse

squirrel
representations.                    S: structure        chimp
– How does abstract domain                              gorilla
knowledge guide learning of

F1
F2
F3
F4
new concepts?
mouse
– How can this knowledge be        D: data         squirrel
chimp
represented, and how might it                     gorilla
be learned?
– How can probabilistic inference work together with
flexibly structured representations to model complex,
real-world learning and reasoning?
Contributions of Bayesian models
• Principled quantitative models of human behavior,
with broad coverage and a minimum of free
• Explain how and why human learning and
reasoning works, in terms of (approximations to)
optimal statistical inference in natural
environments.
• A framework for studying people‟s implicit
knowledge about the structure of the world: how it
is structured, used, and acquired.
• A two-way bridge to state-of-the-art AI and
machine learning.
Looking forward
• What we need to understand: the mind‟s ability to build
rich models of the world from sparse data.
–   Learning about objects, categories, and their properties.
–   Causal inference
–   Language comprehension and production
–   Scene understanding
–   Understanding other people‟s actions, plans, thoughts, goals
• What do we need to understand these abilities?
–   Bayesian inference in probabilistic generative models
–   Hierarchical models, with inference at all levels of abstraction
–   Structured representations: graphs, grammars, logic
–   Flexible representations, growing in response to observed data
Learning word meanings
Whole-object principle
Abstract      Shape bias
Taxonomic principle
Principles    Contrast principle
Basic-level bias

Structure

Data

(Tenenbaum & Xu)
Causal learning and reasoning
Abstract
Principles

Structure

Data

(Griffiths, Tenenbaum, Kemp et al.)
“Universal Grammar”                       Hierarchical phrase structure
grammars (e.g., CFG, HPSG, TAG)
P(grammar | UG)
S  NP VP
Grammar
NP  Det [ Adj] Noun [ RelClause]
RelClause  [ Rel] NP V
VP  VP NP
P(phrase structure | grammar)
VP  Verb

Phrase structure

P(utterance | phrase structure)

Utterance
P(speech | utterance)
Speech signal
(c.f. Chater and Manning, 2006)
Vision as probabilistic parsing

(Han & Zhu, 2006; c.f.,
Zhu, Yuanhao
& Yuille NIPS 06 )
Goal-directed action
(production and comprehension)

(Wolpert et al., 2003)
Open directions and challenges
• Effective methods for learning structured knowledge
– How to balance expressiveness/learnability tradeoff?
• More precise relation to psychological processes
– To what extent do mental processes implement boundedly
rational methods of approximate inference?
• Relation to neural computation
– How to implement structured representations in brains?
• Modeling individual subjects and single trials
– Is there a rational basis for probability matching?
• Understanding failure cases
– Are these simply “not Bayesian”, or are people using a
different model? How do we avoid circularity?
• Special issue of Trends in Cognitive
Sciences (TiCS), July 2006 (Vol. 10, no. 7),
on “Probabilistic models of cognition”.
• Tom Griffiths‟ reading list, a/k/a
http://bayesiancognition.com
• Summer school on probabilistic models of
cognition, July 2007, Institute for Pure and
Applied Mathematics (IPAM) at UCLA.
Extra slides
Bayesian prediction
P(ttotal|tpast)     1/ttotal        P(tpast)
posterior         Random        Domain-dependent
probability        sampling       prior

What is the best guess for ttotal?
Compute t such that P(ttotal > t|tpast) = 0.5:

P(ttotal|tpast)                       We compared the median
of the Bayesian posterior
with the median of subjects‟
the distribution of subjects‟
judgments?
Sources of individual differences
• Individuals‟ judgments could by noisy.
• Individuals‟ judgments could be optimal,
but with different priors.
– e.g., each individual has seen only a sparse
sample of the relevant population of events.
• Individuals‟ inferences about the posterior
could be optimal, but their judgments could
be based on probability (or utility) matching
rather than maximizing.
Individual differences in prediction
Proportion of judgments below predicted value

P(ttotal|tpast)

ttotal

Quantile of Bayesian posterior distribution
Individual differences in prediction

P(ttotal|tpast)

ttotal

Average over all
•   movie run times
•   movie grosses
•   poem lengths
•   life spans
•   terms in congress
•   cake baking times
Individual differences in concept learning
Why probability matching?
• Optimal behavior under some
(evolutionarily natural) circumstances.
–   Optimal betting theory, portfolio theory
–   Optimal foraging theory
–   Competitive games
–   Dynamic tasks (changing probabilities or utilities)

• Side-effect of algorithms for approximating
complex Bayesian computations.
– Markov chain Monte Carlo (MCMC): instead of integrating over
complex hypothesis spaces, construct a sample of high-probability
hypotheses.
– Judgments from individual (independent) samples can on average
be almost as good as using the full posterior distribution.
Markov chain Monte Carlo

(Metropolis-Hastings algorithm)
The puzzle of coincidences
Discoveries of hidden causal structure are
often driven by noticing coincidences. . .
• Science
– Halley‟s comet (1705)
(Halley, 1705)
(Halley, 1705)
The puzzle of coincidences
Discoveries of hidden causal structure are
often driven by noticing coincidences. . .
• Science
– Halley‟s comet (1705)
– John Snow and the cause of cholera (1854)
Rational analysis of cognition
• Often can show that apparently irrational behavior
is actually rational.

Which cards do you have to turn over to test this rule?
“If there is an A on one side, then there is a 2 on the other side”
Rational analysis of cognition
• Often can show that apparently irrational behavior
is actually rational.
• Oaksford & Chater‟s rational analysis:
– Optimal data selection based
on maximizing expected
information gain.
– Test the rule “If p, then q”
against the null hypothesis
that p and q are independent.
– Assuming p and q are rare
predicts people‟s choices:
Integrating multiple forms of reasoning
(Kemp, Shafto, Berke & Tenenbaum NIPS 06)

2) Causal relations                 … Parameters of causal
between features                    relations vary smoothly
over the category hierarchy.
1) Taxonomic
relations
between
categories

T9 hormones cause elevated heart rates.
Elevated heart rates cause faster metabolisms.
Mice have T9 hormones.

…?
Integrating multiple forms of reasoning
Infinite relational models
(Kemp, Tenenbaum, Griffiths, Yamada & Ueda, AAAI 06)

(c.f. Xu,
Tresp, et al.
SRL 06)
concept

predicate

concept

Biomedical predicate data from UMLS (McCrae et al.):
– 134 concepts: enzyme, hormone, organ, disease, cell function ...
– 49 predicates: affects(hormone, organ), complicates(enzyme, cell
function), treats(drug, disease), diagnoses(procedure, disease) …
Learning relational theories

e.g., Diseases affect   Chemicals interact   Chemicals cause
Organisms              with Chemicals       Diseases
Learning annotated hierarchies
from relational data
(Roy, Kemp, Mansinghka, Tenenbaum NIPS 06)
Learning abstract relational structures
Dominance
hierarchy             Tree             Cliques           Ring

Primate troop   Bush administration   Prison inmates     Kula islands
“x beats y”        “x told y”          “x likes y”    “x trades with y”
Bayesian inference in
neural networks

(Rao, in press)
The big problem of intelligence
• The development of intuitive theories in
childhood.
– Psychology: How do we learn to understand
others‟ actions in terms of beliefs, desires,
plans, intentions, values, morals?
– Biology: How do we learn that people, dogs,
bees, worms, trees, flowers, grass, coral, moss
are alive, but chairs, cars, tricycles, computers,
the sun, Roomba, robots, clocks, rocks are not?
The big problem of intelligence
• Common sense reasoning.
Consider a man named Boris.
– Is the mother of Boris‟s father his grandmother?
– Is the mother of Boris‟s sister his mother?
– Is the son of Boris‟s sister his son?
(Note: Boris and his family were
stranded on a desert island when he
was a young boy.)

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