Bayesian models of human inductive learning Josh Tenenbaum MIT.

Bayesian models of human inductive learning

Josh TenenbaumMIT

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

“tufa”

Modeling Goals• Explain how and why human learning and reasoning works,

in terms of (approximations to) optimal statistical inference in natural environments.

• Computational-level theories that provide insights into algorithmic- or processing-level questions.

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

• A framework for studying people’s implicit knowledge about the world: how it is structured, used, and acquired.

• A two-way bridge to state-of-the-art AI, machine learning.

1. How does background knowledge guide learning from sparsely observed data? Bayesian inference, with priors based on background knowledge.

2. What form does background knowledge take, across different domains and tasks? Probabilities defined over structured representations: graphs, grammars, rules, logic, relational schemas, theories.

3. How is background knowledge itself learned? Hierarchical Bayesian models, with inference at multiple levels of abstraction.

4. How can background knowledge constrain learning yet maintain flexibility, balancing assimilation and accommodation? Nonparametric models, growing in complexity as the data require.

Explaining inductive learning

Two case studies

• The number game• Property induction

The number game

• Program input: number between 1 and 100

• Program output: “yes” or “no”

The number game

• Learning task:– Observe one or more positive (“yes”) examples.– Judge whether other numbers are “yes” or “no”.

The number game

Examples of“yes” numbers

Generalizationjudgments (N = 20)

60Diffuse similarity

The number game



60

60 80 10 30

Diffuse similarity

Rule: “multiples of 10”

The number game



60

60 80 10 30

60 52 57 55

Diffuse similarity


Focused similarity: numbers near 50-60

The number game



16

16 8 2 64

16 23 19 20

Diffuse similarity

Rule: “powers of 2”

Focused similarity: numbers near 20

Main phenomena to explain:– Generalization can appear either similarity-based (graded) or rule-based (all-or-

none). – Learning from just a few positive examples.

60

60 80 10 30

60 52 57 55

Diffuse similarity


Focused similarity: numbers near 50-60

The number game

Divisions into “rule” and “similarity” subsystems?

• Category learning– Nosofsky, Palmeri et al.: RULEX– Erickson & Kruschke: ATRIUM

• Language processing– Pinker, Marcus et al.: Past tense morphology

• Reasoning– Sloman – Rips– Nisbett, Smith et al.

• H: Hypothesis space of possible concepts:– h1 = {2, 4, 6, 8, 10, 12, …, 96, 98, 100} (“even numbers”)

– h2 = {10, 20, 30, 40, …, 90, 100} (“multiples of 10”)

– h3 = {2, 4, 8, 16, 32, 64} (“powers of 2”)

– h4 = {50, 51, 52, …, 59, 60} (“numbers between 50 and 60”)

– . . .

Bayesian model

Representational interpretations for H:– Candidate rules

– Features for similarity

– “Consequential subsets” (Shepard, 1987)

Three hypothesis subspaces for number concepts

• Mathematical properties (24 hypotheses): – Odd, even, square, cube, prime numbers– Multiples of small integers– Powers of small integers

• Raw magnitude (5050 hypotheses): – All intervals of integers with endpoints between 1 and

100.

• Approximate magnitude (10 hypotheses):– Decades (1-10, 10-20, 20-30, …)

• H: Hypothesis space of possible concepts:– Mathematical properties: even, odd, square, prime, . . . .

– Approximate magnitude: {1-10}, {10-20}, {20-30}, . . . .

– Raw magnitude: all intervals between 1 and 100.

• X = {x1, . . . , xn}: n examples of a concept C.

• Evaluate hypotheses given data:

– p(h) [prior]: domain knowledge, pre-existing biases

– p(X|h) [likelihood]: statistical information in examples.

– p(h|X) [posterior]: degree of belief that h is the true extension of C.

Bayesian model

Hh

hphXp

hphXpXhp

)()|(

)()|()|(

Generalizing to new objects

Given p(h|X), how do we compute , the probability that C applies to some new stimulus y?

x1 x2 x3 x4

h

Background knowledge

X =

)|( XCyp

?Cy

Generalizing to new objects

Hypothesis averaging:

Compute the probability that C applies to some new object y by averaging the predictions of all hypotheses h, weighted by p(h|X):

Hh

XhphCypXCyp )|()|()|(

hy

hy

if 0

if 1

},{

)|(Xyh

Xhp

Likelihood: p(X|h)

• Size principle: Smaller hypotheses receive greater likelihood, and exponentially more so as n increases.

• Follows from assumption of randomly sampled examples + law of “conservation of belief”:

• Captures the intuition of a “representative” sample.

hxx

n

nhhXp

,,if

1)size(

1)|(

hxi any if 0

1)|( all

MdDpDd

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

Illustrating the size principle

h1 h2

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100


h1 h2

Data slightly more of a coincidence under h1

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100


h1 h2

Data much more of a coincidence under h1

Prior: p(h)

• Choice of hypothesis space embodies a strong prior: effectively, p(h) ~ 0 for many logically possible but conceptually unnatural hypotheses.

• Prevents overfitting by highly specific but unnatural hypotheses, e.g. “multiples of 10 except 50 and 70”.

Prior: p(h)



e.g., X = {60 80 10 30}:

0001.010

1)10 of multiples|(

4

Xp

00024.08

1)70 50,except 10 of multiples|(

4

Xp

The “ugly duckling” theorem

Hypotheses

How would we generalize without any inductive bias – without constraints on the hypothesis space, informative priors or likelihoods?

1234

Objects


1234

)|}3{( hXp

Hypotheses

Objects

1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0


})3{|4( XCp

})3{|2( XCp

1234

Hypotheses

Objects

)|}3{( hXp 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0

1 + 1 + 1 + 1 = 4/8

})3{|1( XCp 1 1 + 1 1 = 4/8

1 1 + 1 1 = 4/8


1234

Hypotheses

Objects

Without any inductive bias – constraints on hypotheses, informative priors or likelihoods – no meaningful generalization!

})1,3{|4( XCp

)|}1,3{( hXp 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0

1 + 1 = 2/4

})1,3{|2( XCp 1 1 + = 2/4

Posterior:

• X = {60, 80, 10, 30}

• Why prefer “multiples of 10” over “even numbers”? p(X|h).

• Why prefer “multiples of 10” over “multiples of 10 except 50 and 20”? p(h).

• Why does a good generalization need both high prior and high likelihood? p(h|X) ~ p(X|h) p(h)

Hh

hphXp

hphXpXhp

)()|(

)()|()|(

Prior: p(h)



• p(h) encodes relative weights of alternative theories:

H1: Math properties (24)

• even numbers• powers of two• multiples of three ….

H2: Raw magnitude (5050)

• 10-15• 20-32• 37-54 ….

H3: Approx. magnitude (10)

• 10-20• 20-30• 30-40 ….

H: Total hypothesis spacep(H1) = 1/5

p(H2) = 3/5p(H3) = 1/5

p(h) = p(H1) / 24 p(h) = p(H2) / 5050 p(h) = p(H3) / 10

+ Examples Human generalization

60

60 80 10 30

60 52 57 55

Bayesian Model

16

16 8 2 64

16 23 19 20

Examples: 16

Examples: 16 8 2 64

Examples: 16 23 19 20

Summary of the Bayesian model

• How do the statistics of the examples interact with prior knowledge to guide generalization?

• Why does generalization appear rule-based or similarity-based?

priorlikelihoodposterior

principle size averaging hypothesis

broad p(h|X): similarity gradient narrow p(h|X): all-or-none rule

Summary of the Bayesian model

• How do the statistics of the examples interact with prior knowledge to guide generalization?

• Why does generalization appear rule-based or similarity-based?

priorlikelihoodposterior

principle size averaging hypothesis

broad p(h|X): Many h of similar size, or very few examples (i.e. 1)narrow p(h|X): One h much smaller

Alternative models

• Neural networks

• Hypothesis ranking and elimination

• Similarity to exemplars

Time?

Alternative models• Neural networks

even multiple of 10

power of 2

multiple of 3

80

10

30

60



even multiple of 10

power of 2

multiple of 3

80

10

30

60

Hypothesis ranking: 1 2 3 4 ….

….

Model (r = 0.80)Data



• Similarity to exemplars– Average similarity:

60

60 80 10 30

60 52 57 55

),(sim||

1)|( j

Xx

xyX

XCyp

j

Alternative models

• Neural networks


• Similarity to exemplars– Flexible similarity? Bayes.

Distance in psychological space

Probability of generalization

The universal law of generalization

Explaining the universal law(Tenenbaum & Griffiths, 2001)

Bayesian generalization when the hypotheses correspond to convex regions in a low-dimensional metric space (e.g., intervals in one dimension), with an isotropic prior.

)|( xCyp x

y

xy1

y2

y3

• Assume a gradient of typicality • Examples sampled in proportion to their typicality:

• Size of hypotheses now

horse camel

Asymmetric generalization

Asymmetric generalization

Symmetry may depend heavily on the context:– Healthy levels of hormone (left) versus healthy levels

of toxin (right)

– Predicting the durations or magnitudes of events.

“tufa” “tufa”

“tufa”

Modeling word learningBayesian inference over tree-structured hypothesis space:

(Xu & Tenenbaum; Schmidt & Tenenbaum)

Taking stock• A model of high-level, knowledge-driven inductive

reasoning that makes strong quantitative predictions with minimal free parameters. (r2 > 0.9 for mean judgments on 180 generalization stimuli, with 3

free numerical parameters)

• Explains qualitatively different patterns of generalization (rules, similarity) as the output of a single general-purpose rational inference engine.

• Differently structured hypothesis spaces account for different kinds of generalization behavior seen in different domains and contexts.

What’s missing: How do we choose a good prior?

• Can we describe formally how these priors are generated by abstract knowledge or theories?

• Can we move from ‘weak rational analysis’ to ‘strong rational analysis’ in inductive learning?– “Weak”: behavior consistent with some reasonable prior.– “Strong”: behavior consistent with the “correct” prior given the structure

of the world (c.f., ideal observer analyses in vision).

• Can we explain how people learn these rich priors? • Can we work with more flexible priors, not just restricted to a

small subset of all logically possible concepts? – Would like to be able to learn any concept, even complex and unnatural

ones, given enough data (a non-dogmatic prior).

• 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

hP

XYP

with consistent

, with consistent

)(

)(

)|(

Prior P(h)

Hypotheses h

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)

Hierarchical Bayesian Framework(Kemp & Tenenbaum)

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

Features


[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 ?

?????

??


Probability of generalization:

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

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?”

Tre

e

2D

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?

Hierarchical Bayesian Framework

F: form

S: structure

D: data mousesquirrel

chimpgorilla

F1

F2

F3

F4

Tree

mouse

squirrel

chimp

gorilla

ClustersLinear

chimp

gorilla

squirrel

mouse

mouse

squirrel

chimp

gorilla

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

Hypothesis space of structural forms

Order Chain RingPartition

Hierarchy Tree Grid Cylinder

Development of structural forms as more data are observed

The “blessing of abstraction”• Often quicker to learn at higher levels of abstraction.

– Quicker to learn that you have a biased coin than to learn its precise bias, or to learn that you have a second-order polynomial than to learn the precise coefficients.

– Quicker to learn that shape matters most for labeling object categories than to learn the labels for most categories.

– Quicker to learn that a domain is tree-structured than to learn the precise tree that best

characterizes it.

• Explanation in hierarchical Bayesian models: – At higher levels, hypothesis space gets smaller and simpler, and draw support (albeit

indirectly) from a broader sample of data.

– Total hypothesis space gets bigger when we add levels of abstraction, but the effective number of degrees of freedom only decreases, because higher levels specify constraints on lower levels.

– Hence the overall learning problem becomes easier.

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.

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

building more human-like machine learning)– What is the structure of 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 provide strong constraints yet be flexible?

• A different way to think about the development of cognition.– Powerful abstractions can be learned “from the top down”, together with or prior to

learning more concrete knowledge.

• Go beyond the traditional “either-or” dichotomies: – How can probabilistic inference over symbolic hypotheses span the range of “rule-based”

to “similarity-based” generalization?

– How can domain-general learning mechanisms acquire domain-specific representations?

– How can structured symbolic representations be acquired by statistical learning?

Bayesian models of human inductive learning Josh Tenenbaum MIT.

Documents

number game slide

diffuse similarity slide

inductive learning slide

number game examples

diffuse similarity rule

number game learning

focused similarity

numbers h