Bayesian models of human inductive learning Josh Tenenbaum MIT

Bayesian models of human inductive learning

Josh TenenbaumMIT

Department of Brain and Cognitive SciencesComputer Science and AI Lab (CSAIL)

Charles KempPat Shafto

Lauren SchmidtChris Baker

Collaborators

Funding: US NSF, AFOSR, ONR, DARPA, NTT Communication Sciences Laboratories, Schlumberger, Eli Lilly & Co., James S. McDonnell Foundation

Vikash Mansinghka Tom Griffiths

Takeshi Yamada Naonori Ueda

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.

Bayesian models of cognitionVisual 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, Kawato, Doya, Todorov, Shadmehr, …]

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, …]

Everyday inductive leaps

How can people learn so much about the world from such limited evidence?– Learning concepts from examples

“horse” “horse” “horse”

Learning concepts from examples

“tufa”

“tufa”

“tufa”

Everyday inductive leaps

How can people learn so much about the world from such limited evidence?– 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

The solution

Strong prior knowledge (inductive bias).

What is the relation between y and x?




The solutionStrong prior knowledge (inductive bias).

– How does background knowledge guide learning from sparsely observed data?

– What form does the knowledge take, across different domains and tasks?

– How is that knowledge itself learned?

Our goal: Computational models that answer these questions, with strong quantitative fits to human behavioral data and a bridge to state-of-the-art AI and machine learning.

1. How does background knowledge guide learning from sparsely observed data?

Bayesian inference:

2. What form does background knowledge take, across different domains and tasks?

Probabilities defined over structured representations: graphs, grammars, predicate logic, schemas, theories.

3. How is background knowledge itself acquired, constraining learning while maintaining flexibility?

Hierarchical probabilistic models, with inference at multiple levels of abstraction. Nonparametric models in which complexity grows automatically as the data require.

The approach: from statistics to intelligence

Hhii

i

hPhdPhPhdPdhP

)()|()()|()|(

Basics of Bayesian inference

• Bayes’ rule:• An example

– Data: John is coughing – Some hypotheses:

1. John has a cold2. John has lung cancer3. 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

Hhii

i

hPhdPhPhdPdhP

)()|()()|()|(

• How likely is the conclusion, given the premises?

“Similarity”, “Typicality”,

“Diversity”

Gorillas have T9 hormones.Seals have T9 hormones.Squirrels have T9 hormones.

Horses have T9 hormones.Gorillas have T9 hormones.Chimps have T9 hormones.Monkeys have T9 hormones.Baboons have T9 hormones.

Horses have T9 hormones.

Gorillas have T9 hormones.Seals have T9 hormones.Squirrels have T9 hormones.

Flies have T9 hormones.

Property induction

The computational problem

?

?????

??

Features New property

?

HorseCow

ChimpGorillaMouse

SquirrelDolphin

SealRhino

Elephant

85 features for 50 animals (Osherson et al.): e.g., for Elephant: ‘gray’, ‘hairless’, ‘toughskin’, ‘big’, ‘bulbous’, ‘longleg’, ‘tail’, ‘chewteeth’, ‘tusks’, ‘smelly’, ‘walks’, ‘slow’, ‘strong’, ‘muscle’, ‘fourlegs’,…

“Transfer Learning”, “Semi-Supervised Learning”

???????

?

HorseCow

ChimpGorillaMouse

SquirrelDolphin

SealRhino

Elephant

... ...

Horses have T9 hormonesRhinos have T9 hormones

Cows have T9 hormones

X

Y

}

Xh

YXh

hP

hPXYP

with consistent

, with consistent

)(

)()|(

Prior P(h)

Hypotheses h

???????

?

HorseCow

ChimpGorillaMouse

SquirrelDolphin

SealRhino

Elephant

... ...

Horses have T9 hormonesRhinos have T9 hormones

Cows have T9 hormones

}

Prediction P(Y | X) Hypotheses h

Prior P(h)

X

Y

Xh

YXh

hP

hPXYP

with consistent

, with consistent

)(

)()|(

F: form

S: structure

D: data

Tree with species at leaf nodes

mouse

squirrel

chimp

gorilla

mousesquirrel

chimpgorilla

F1

F2 F3 F4 Has

T9

horm

ones

??

?

…

P(structure | form)

P(data | structure)

P(form)

Hierarchical Bayesian Framework

Smooth: P(h) high

P(D|S): How the structure constrains the data of experience

• Define a stochastic process over structure S that generates candidate property extensions h.– Intuition: properties should vary smoothly over structure.

Not smooth: P(h) low

S

y

Gaussian Process (~ random walk, diffusion)

Threshold


h

[Zhu, Lafferty & Ghahramani 2003]

S

y

Gaussian Process (~ random walk, diffusion)

Threshold


[Zhu, Lafferty & Ghahramani 2003]

h

Species 1Species 2Species 3Species 4Species 5Species 6Species 7Species 8Species 9Species 10

Structure S

Data D

Features85 features for 50 animals (Osherson et al.): e.g., for Elephant: ‘gray’, ‘hairless’, ‘toughskin’, ‘big’, ‘bulbous’, ‘longleg’, ‘tail’, ‘chewteeth’, ‘tusks’, ‘smelly’, ‘walks’, ‘slow’, ‘strong’, ‘muscle’, ‘fourlegs’,…

[c.f., Lawrence, 2004; Smola & Kondor 2003]

Species 1Species 2Species 3Species 4Species 5Species 6Species 7Species 8Species 9Species 10

Features New property

Structure S

Data D ?

?????

??

85 features for 50 animals (Osherson et al.): e.g., for Elephant: ‘gray’, ‘hairless’, ‘toughskin’, ‘big’, ‘bulbous’, ‘longleg’, ‘tail’, ‘chewteeth’, ‘tusks’, ‘smelly’, ‘walks’, ‘slow’, ‘strong’, ‘muscle’, ‘fourlegs’,…

Gorillas have property P.Mice have property P.Seals have property P.

All mammals have property P.

Cows have property P.Elephants have property P.

Horses have property P.

Tree

2D

Testing different priors

Correctbias

Wrongbias

Too weakbias

Too strongbias

Inductive bias

Learning about spatial properties Geographic inference task: “Given that a certain kind of

native American artifact has been found in sites near city X, how likely is the same artifact to be found near city Y?”

Tree

2D

Hierarchical Bayesian Framework

F: form

S: structure

D: data mousesquirrel

chimpgorilla

F1

F2 F3 F4

Tree

mouse

squirrel

chimp

gorilla

mousesquirrel

chimpgorilla

SpaceChain

chimp

gorilla

squirrel

mouse

Discovering structural forms

Ostrich

Robin

Crocod

ile

Snake

Bat

Orangu

tan

Turtle

Ostrich Robin Crocodile Snake Bat OrangutanTurtle

Ostrich

Robin

Crocod

ile

Snake

Bat

Orangu

tan

Turtle

Angel

GodRock

Plant

Ostrich Robin Crocodile Snake Bat OrangutanTurtle

Discovering structural forms

Linnaeus

“Great chain of being”

• Scientific discoveries

• Children’s cognitive development– Hierarchical structure of category labels– Clique structure of social groups– Cyclical structure of seasons or days of the week– Transitive structure for value

People can discover structural forms

Tree structure for biological species

Periodic structure for chemical elements

(1579) (1837)

Systema Naturae

Kingdom Animalia Phylum Chordata Class Mammalia Order Primates Family Hominidae Genus Homo Species Homo sapiens

(1735)

“great chain of being”

Typical structure learning algorithms assume a fixed structural form

Flat Clusters

K-MeansMixture modelsCompetitive learning

Line

Guttman scalingIdeal point models

Tree

Hierarchical clusteringBayesian phylogenetics

Circle

Circumplex models

Euclidean Space

MDSPCAFactor Analysis

Grid

Self-Organizing MapGenerative topographic

mapping

The ultimate goal

“Universal Structure Learner”

K-MeansHierarchical clusteringFactor AnalysisGuttman scalingCircumplex modelsSelf-Organizing maps···

Data Representation

A “universal grammar” for structural forms

Form FormProcess Process

F: form

S: structure

D: data mousesquirrel

chimpgorilla

F1

F2 F3 F4

Favors simplicity

Favors smoothness[Zhu et al., 2003]

Tree

mouse

squirrel

chimp

gorilla

ClustersLinear

chimp

gorilla

squirrel

mouse

mouse

squirrel

chimp

gorilla

Development of structural forms as more data are observed

“blessing of abstraction”

Summary so far

Bayesian inference over hierarchies of structured representations provides a framework to understand core questions of human cognition:– What is the content and form of

human knowledge, at multiple levels of abstraction?

– How does abstract domain knowledge guide learning of new concepts?

– How is abstract domain knowledge learned? What must be built in?

F: form

S: structure

D: data

mouse

squirrel

chimp

gorilla

mousesquirrel

chimpgorilla

F1

F2 F3 F4

Other questions• How can we learn domain structures if we do not already

know in advance which features are relevant? • How can we discover richer models of a domain, with

multiple ways of structuring objects? • How can we learn models for more complex domains, with

not just a single object-property matrix but multiple different types of objects, their properties and relations to each other?

• How do these ideas & tools apply to other aspects of cognition, beyond categorizing and predicting the properties of objects?

Raw data matrix:

A single way of structuring a domain rarely describes all its features…

Conventional clustering (CRP mixture):

A single way of structuring a domain rarely describes all its features…

Learning multiple structures to explain different feature subsets

(Shafto et al.; Shafto, Mansinghka, Tenenbaum, Yamada & Ueda, 2007)

System 1 System 2 System 3CrossCat:

Discovering structure in relational data

391

135

117

142

106

1248

15

3 9 1 13 5 11 7 14 2 10 6 12 4 8 15

3 9 113 511

7 14 210 6

12 4 8 15

123456789

101112131415

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Input Output

pers

on

TalksTo(person,person)

person

O

z

Infinite Relational Model (IRM)(Kemp, Tenenbaum, Griffiths, Yamada & Ueda, AAAI 06)

3 9 113 511

7 14 210 6

12 4 8 15

0.90.1 0.1

0.1 0.1 0.9

0.9 0.1 0.1

391

135

117

142

106

1248

15

3 9 1 13 5 11 7 14 2 10 6 12 4 8 15

conc

ept

concept

predicate

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 a medical ontology

e.g., Diseases affect Organisms

Chemicals interact with Chemicals

Chemicals cause Diseases

International relations circa 1965 (Rummel)– 14 countries: UK, USA, USSR, China, ….– 54 binary relations representing interactions between countries: exports

to( USA, UK ), protests( USA, USSR ), …. – 90 (dynamic) country features: purges, protests, unemployment,

communists, # languages, assassinations, ….


patients

conditions

has(patient,condition)

Learning causal models

Bayesian network

Observed events

Classes = {R, D, S}Laws = {R D, D S}

( : possible causal link)Classes = {R, D, S}Laws = {S D}

Classes = {C}Laws = {C C}

Abstract causal theories

patients

conditions


Classes = {R, D, S}Laws = {R D, D S}

R: working in factory, smoking, stress, high fat diet, …D: flu, bronchitis, lung cancer, heart disease, …S: headache, fever, coughing, chest pain, …

Abstract theory

Observed events

Bayesian network

Learning causal theories

patients

conditions


Causal graphical model

Classes z

Laws

1 2 3 40.30.0 0.01

0.0 0.0 0.25

0.0 0.0 0.0

5 6 7 8

9 1011 12

‘B’ ‘D’

‘S’

‘B’ ‘D’ ‘S’‘B’

‘D’

‘S’

Abstract theory

Observed events

Bayesian network

IRM

1

2

3

4

5

6

7

8

9

10

11

12

True structure of Bayesian network N:

edge (N)

class (Z)

edge (N)

1 2 3 4 5 6

7 8 9 10 11 12 13 14 15 16

# of samples: 20 80 1000

Data D

Network N

Data D

Network N

AbstractTheory

1 2 3 4 5 6…

7 8 9 10 11 12 1314 15 16…

…

0.40.0

0.0 0.0…

…

(Mansinghka, Kemp, Tenenbaum, Griffiths UAI 06)

c1 c2

c1

c2

c1

c2

Classes Z

“blessing of abstraction”

The flexibility of a nonparametric prior

edge (N)

class (Z)

edge (N)

12

3

4567

8

9

1011 12

# of samples: 40 100 1000

True structure of Bayesian network N:

Data D

Network N

Data D

Network N

AbstractTheory 1 2 3 4

5 6 7 89 10 11 12

…

0.1

c1

c1

c1

Classes Z

……

…

VerbVPNPVPVP

VNPRelRelClauseRelClauseNounAdjDetNP

VPNPS

][][][

Phrase structure

Utterance

Speech signal

Grammar

“Universal Grammar” Hierarchical phrase structure grammars (e.g., CFG, HPSG, TAG)

P(phrase structure | grammar)

P(utterance | phrase structure)

P(speech | utterance)

(c.f. Chater and Manning, 2006)

P(grammar | UG)

(Han & Zhu, 2006; c.f.,Zhu, Yuanhao & Yuille NIPS 06 )

Vision as probabilistic parsing

Goal-directed action (production and comprehension)

(Wolpert, Doya and Kawato, 2003)

Goal inference as inverseprobabilistic planning

(Baker, Tenenbaum & Saxe)

Constraints Goals

Actions

Rational planning(PO)MDP

model predictions

hum

an

judg

men

ts

Conclusions• The big questions: How does the mind build rich models of the world from sparse

data? What is the form and function of abstract knowledge, and how can abstractions be learned? – These questions are central in vision, language, categorization, causal reasoning, planning, social

understanding… perhaps all of cognition?

• Some powerful tools for making progress on these questions:– 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

• New ways to think about development of cognitive systems.– Domain-specific representations can be learned by domain-general mechanisms.– Structure symbolic knowledge can support and even be acquired via statistical learning.– Powerful abstractions can be learned “from the top down”, together with or prior to learning more

concrete knowledge.

Extra slides

Summary

Structure

Data

mouse

squirrel

chimp

gorilla

mousesquirrel

chimpgorilla

F1

F2 F3 F4

Abstractknowledge

Modeling human inductive learning as Bayesian inference over hierarchies of flexibly structured representations.

Classes of variables: B, D, SCausal laws: B D, D S

“dax”

“zav”

“fep”

“zav”

“zav”

“zav”“dax”

“dax”

“dax” “fep”

“fep”

“fep”

Shape varies across categories but not within categories.

Texture, color, size vary within categories.

Word learning Property induction Causal learning

Conclusions• Learning algorithms for discovering domain

structure, given feature or relational data. • Broader themes

– Combining structured representations with statistical inference yields powerful knowledge discovery tools.

– Hierarchical Bayesian modeling allows us to learn domain structure at multiple levels of abstraction.

– Nonparametric Bayesian formulations allow the complexity of representations to be determined automatically and on the fly, growing as the data require.

Beyond similarity-based induction• Reasoning based

on dimensional thresholds: (Smith et al., 1993)

• Reasoning based on causal relations: (Medin et al., 2004; Coley & Shafto, 2003)

Poodles can bite through wire.

German shepherds can bite through wire.

Dobermans can bite through wire.


Salmon carry E. Spirus bacteria.

Grizzly bears carry E. Spirus bacteria.



Different sources for priors

Chimps have T9 hormones.

Gorillas have T9 hormones.





Taxonomic similarity

Jaw strength

Food web relations

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 C

Class AClass D

Class E

Class G

Class F

Class BClass C

Class A

Class D

Class E

Class G

Class F

Class B

Class AClass BClass CClass DClass EClass FClass G

. . . . . . . . .

Class C

Class GClass FClass EClass D

Class BClass A

Hypotheses

Reasoning with two property types

Bio

logi

cal

prop

erty

Dis

ease

prop

erty

Tree Web

Kelp Human

Dolphin

Sand shark

Mako sharkTunaHerring

Kelp

HumanDolphin

Sand sharkMako sharkTuna

Herring

(Shafto, Kemp, Bonawitz, Coley & Tenenbaum)

“Given that X has property P, how likely is it that Y does?”

Summary so far• A framework for modeling human inductive

reasoning as rational statistical inference over structured knowledge representations– Qualitatively different priors are appropriate for different

domains of property induction. – In each domain, a prior that matches the world’s structure

fits people’s judgments well, and better than alternative priors.

– A language for representing different theories: graph structure defined over objects + probabilistic model for the distribution of properties over that graph.

• Remaining question: How can we learn appropriate theories for different domains?

Principles

Structure

Data

Whole-object principleShape biasTaxonomic principleContrast principleBasic-level bias

Learning word meanings

“tufa” “tufa”

“tufa”

Word learningBayesian inference over tree-structured hypothesis space:

(Xu & Tenenbaum; Schmidt & Tenenbaum)

Causal learning with prior knowledge(Griffiths, Sobel, Tenenbaum & Gopnik)

AB Trial A TrialInitial

“Backwards blocking” paradigm:

Learning grounded causal models(Goodman, Mansinghka & Tenenbaum)

A child learns that petting the cat leads to purring, while pounding leads to growling. But how to learn these symbolic event concepts over which causal links are defined?

a

b

c

a b c a b c a b c

The big picture• 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– Understanding other people’s actions, plans, thoughts, goals– Language comprehension and production– Scene understanding

• 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

Overhypotheses• Syntax: Universal Grammar• Phonology Faithfulness constraints

Markedness constraints• Word Learning Shape bias

Principle of contrastWhole object bias

• Folk physics Objects are unified, bounded and persistent bodies

• Predicability M-constraint• Folk biology Taxonomic principle ... ...

(Spelke)

(Markman)

(Keil)

(Atran)

(Chomsky)(Prince, Smolensky)

Beyond similarity-based induction• Inference based on

dimensional thresholds: (Smith et al., 1993)

• Inference based on causal relations: (Medin et al., 2004; Coley & Shafto, 2003)









Property type “has T9 hormones” “can bite through wire” “carry E. Spirus bacteria”

Form of background knowledge taxonomic tree directed chain directed network + diffusion process + drift process + noisy transmission

Class C

Class AClass D

Class E

Class G

Class F

Class BClass C

Class A

Class D

Class E

Class G

Class F

Class B

Class AClass BClass CClass DClass EClass FClass G

. . . . . . . . .

Class C

Class GClass FClass EClass D

Class BClass A

Hypotheses

Beyond similarity-based induction

Bio

logi

cal

prop

erty

Dis

ease

prop

erty

Tree Web

Kelp Human

Dolphin

Sand shark

Mako sharkTunaHerring

Kelp

HumanDolphin

Sand sharkMako sharkTuna

Herring

(Shafto, Kemp, Bonawitz, Coley & Tenenbaum)

“Given that X has property P, how likely is it that Y does?”

Node-replacement graph grammars

Production(Line) Derivation





Model fitting• Evaluate each form in parallel• For each form, heuristic search over structures

based on greedy growth from a one-node seed:

Synthetic 2D data

Flat Line Ring Tree Grid

Flat Line Ring Tree Grid

log posterior probabilities

Model Selection results:Data:Continuous features drawn from a Gaussian field over these points.

Flat Line Ring Tree Grid ScoresTrue

Clustering models for relational data

• Social networks: block models

Does person x respect person y?

Does prisoner xlike prisoner y?

Learning a hierarchical ontology

More abstract relational structure

Edges

Classgraph

Graphtype

DominanceCliques Ring Tree hierarchy

Conclusions• Computational tools for studying core questions of human learning (and

building more human-like ML?)– What is the content and form of human knowledge, at multiple levels of abstraction?– How does abstract domain knowledge guide new learning? – How can abstract domain knowledge itself be learned? – How can inductive biases be so strong yet so flexible?

• Go beyond the traditional dichotomies of cog sci (and AI). – Instead of “nature vs. nurture”: Powerful abstractions can be learned “from the top

down”, together with or prior to learning more concrete knowledge.– Instead of “domain-general” vs. “domain-specific”: Domain-general learning

mechanisms acquire domain-specific knowledge representations? – Instead of “statistics” vs. “structure”: How can structured symbolic representations

be acquired by statistical learning?

Documents

Bayesian models of human inductive learning Josh Tenenbaum MIT