Bayesian models of inductive learning and reasoning

Josh TenenbaumMIT

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

Charles Kemp

Pat ShaftoVikash Mansinghka Amy Perfors Lauren Schmidt

Chris Baker Noah Goodman

Collaborators

Tom Griffiths

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

Prior knowledge (inductive bias).

The solution

Prior knowledge (inductive bias).– How does background knowledge guide learning

from sparsely observed data? – What form does background knowledge take,

across different domains and tasks?– How is background knowledge itself acquired?

The challenge: Can we answer these questions in precise computational terms?

Modeling goals

• Principled quantitative models of human inductive inferences, with broad coverage and a minimum of free parameters and ad hoc assumptions.

• An understanding of how and why human learning and reasoning works, as a species of rational (approximately optimal) statistical inference given the structure of natural environments.

• A two-way bridge to artificial intelligence and machine learning.

Bayesian inference

• Bayes’ rule:

• An example– 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

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i

hPhdP

hPhdPdhP

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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? Hierarchical probabilistic models, with inference at multiple levels of abstraction.

Flexible nonparametric models in which complexity grows with the data.

The Bayesian modeling toolkit

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A case study: learning about objects and their properties

“Property induction”, “category-based induction” (Rips, 1975; Osherson, Smith et al., 1990)

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.

“Similarity”, “Typicality”,

“Diversity”

• 20 subjects rated the strength of 45 arguments:

X1 have property P. (e.g., Cows have T4 hormones.)

X2 have property P.

X3 have property P.

All mammals have property P. [General argument]

• 20 subjects rated the strength of 36 arguments:X1 have property P.

X2 have property P.

Horses have property P. [Specific argument]

Experiments on property induction(Osherson, Smith, Wilkie, Lopez, Shafir, 1990)

?

?????

??

Features New property

?

HorseCow

ChimpGorillaMouse

SquirrelDolphin

SealRhino

Elephant

85 features for 50 animals (Osherson & Wilkie feature rating task). e.g., for Elephant: ‘gray’, ‘hairless’, ‘toughskin’, ‘big’, ‘bulbous’,

‘longleg’, ‘tail’, ‘chewteeth’, ‘tusks’, ‘smelly’, ‘walks’, ‘slow’, ‘strong’, ‘muscle’, ‘fourlegs’,…

Property induction as acomputational problem

Model

Dat

aSimilarity-based models

Each “ ” represents one argument:X1 have property P.X2 have property P.X3 have property P.

All mammals have property P.

.

Beyond similarity in 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.

German shepherds can bite through wire.

Salmon carry E. Spirus bacteria.

Grizzly bears carry E. Spirus bacteria.

Grizzly bears carry E. Spirus bacteria.

Salmon carry E. Spirus bacteria.

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? Hierarchical probabilistic models, with inference at multiple levels of abstraction.

Flexible nonparametric models in which complexity grows with the data.

The Bayesian modeling toolkit

Hhii

i

hPhdP

hPhdPdhP

)()|(

)()|()|(

F: form

S: structure

D: data

Tree with species at leaf nodes

mouse

squirrel

chimp

gorilla

mousesquirrel

chimpgorilla

F1

F2

F3

F4

Ha

s T

9h

orm

on

es

??

?

…

P(structure | form)

P(data | structure)

P(form)

Model overview

F: form

S: structure

D: data

Tree with species at leaf nodes

mouse

squirrel

chimp

gorilla

mousesquirrel

chimpgorilla

F1

F2

F3

F4

Ha

s T

9h

orm

on

es

??

?

…

Model overview

???????

?

HorseCow

ChimpGorillaMouse

SquirrelDolphin

SealRhino

Elephant

... ...

Horses have T9 hormonesRhinos have T9 hormones

Cows have T9 hormones

X

Y

}

Xh

YXh

hP

hP

XYP

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

hP

XYP

with consistent

, with consistent

)(

)(

)|(

HorseCow

ChimpGorillaMouse

SquirrelDolphin

SealRhino

Elephant

... ...

Prior P(h)

Why not just enumerate all logically possible hypothesesalong with their relative prior probabilities?

Where does the prior come from?

Knowledge-based priors

Chimps have T9 hormones.

Gorillas have T9 hormones.

Poodles can bite through wire.

Dobermans can bite through wire.

Salmon carry E. Spirus bacteria.

Grizzly bears carry E. Spirus bacteria.

Taxonomic similarity

Jaw strength

Food web relations

F: form

S: structure

D: data

Tree with species at leaf nodes

mouse

squirrel

chimp

gorilla

mousesquirrel

chimpgorilla

F1

F2

F3

F4

Ha

s T

9h

orm

on

es

??

?

…

Model overview

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

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

h

ji

T

ij

ji yyd

yySyp

,

2

2

1)(

4

1exp)|(

dij = length of the edge between i and j

(= if i and j are not connected)

A Gaussian prior ~ N(0, ), with (Zhu, Lafferty & Ghahramani, 2003)

).(~ 1 S

)()|( yyhp

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

Structure S

Data D

Features

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’,…

Modeling feature covariance based on distance in graph(Zhu et al., 2003; c.f. Sattath & Tversky, 1977)

Modeling feature covariance based on distance in two-dimensional space(Lawrence, 2004; Smola & Kondor 2003; c.f. Shepard, 1987)

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.

Tre

e

2D

Testing different priors

Correctbias

Wrongbias

Too weakbias

Too strongbias

Inductive bias

x

Spatially varying 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?”

Tre

e

2D

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 A

Class 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 G

Class F

Class E

Class D

Class B

Class A

Hypotheses

Bio

logi

cal

prop

erty

Dis

ease

prop

erty

Tree Web

“Given that A has property P, how likely is it that B does?”

Kelp

Human

Dolphin

Sand shark

Mako shark

Tuna

Herring

Kelp Human

Dolphin

Sand shark

Mako sharkTunaHerring

e.g., P = “has X cells”

e.g., P = “has X disease”

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 structures for different domains?

Model overview

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

Croco

dile

Snake

Bat

Orangu

tan

Turtle

Ostrich Robin Crocodile Snake Bat OrangutanTurtle

Ostrich

Robin

Croco

dile

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

Node-replacement graph grammars

Production(Line) Derivation

Production(Line) Derivation

Node-replacement graph grammars

Production(Line) Derivation

Node-replacement graph grammars

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

GridLinear

chimp

gorilla

squirrel

mouse

mouse squirrel

chimp gorilla

x

Learning algorithm• Evaluate each form in parallel• For each form, heuristic search over structures

based on greedy growth from a one-node seed:

anim

als

features

cases

judg

es

objects

obje

cts

similarities

Primate troop Bush administration Prison inmates Kula islands “x beats y” “x told y” “x likes y” “x trades with y”

Dominance hierarchy Tree Cliques Ring

Structural forms from relational data

Development of structural forms as more data are observed

“blessing of abstraction”

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.

Conclusion 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

– How can domain-general learning mechanisms acquire domain-

specific representations? How can probabilistic inference work together with symbolic, flexibly structured representations?

Principles

Structure

Data

Whole-object principleShape biasTaxonomic principleContrast principleBasic-level bias

Learning word meanings

Causal learning and reasoning

Principles

Structure

Data

(Griffiths, Tenenbaum, et al.)

VerbVP

NPVPVP

VNPRelRelClause

RelClauseNounAdjDetNP

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)

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.– Language comprehension and production– Scene understanding– Causal inference– 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