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Bayesian models of human 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
Lab members
Tom Griffiths*
Funding: AFOSR Cognition and Decision Program, AFOSR MURI, DARPA IPTO, NSF, HSARPA, NTT Communication Sciences Laboratories, James S. McDonnell Foundation
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
Modeling Goals• Principled quantitative models of human behavior, with
broad coverage and a minimum of free parameters and ad hoc assumptions.
• 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.
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 approach: from statistics to intelligence
Hhii
i
hPhdP
hPhdPdhP
)()|(
)()|()|(
Basics of 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
Hhii
i
hPhdP
hPhdPdhP
)()|(
)()|()|(
• You read about a movie that has made $60 million 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 meet a US congressman who has served for 11 years. How long will he serve in total?
• You encounter a phenomenon or event with an unknown extent or duration, ttotal, at a random time or value of t <ttotal. What is the total extent or duration ttotal?
Everyday prediction problems(Griffiths & Tenenbaum, 2006)
Bayesian analysis
p(ttotal|t) p(t|ttotal) p(ttotal)
1/ttotal p(ttotal)
Assume randomsample
(for 0 < t < ttotal
else = 0)
Form of p(ttotal)? e.g., uninformative (Jeffreys) prior 1/ttotal
Priors P(ttotal) based on empirically measured durations or magnitudes for many real-world events in each class:
Median human judgments of the total duration or magnitude ttotal of events in each class, given that they are first observed at a duration or magnitude t, versus Bayesian predictions (median of P(ttotal|t)).
“tufa” “tufa”
“tufa”
Concept learningBayesian inference over tree-structured hypothesis space:
(Xu & Tenenbaum; Schmidt & Tenenbaum)
Some questions• How confident are we that a tree-structured model is the best
way to characterize this learning task?
• How do people construct an appropriate tree-structured hypothesis space?
• What other kinds of structured probabilistic models may be needed to explain other inductive leaps that people make, and how do people acquire these different structured models?
• Are there general unifying principles that explain our capacity to learn and reason with structured probabilistic models across different domains?
• Property induction
“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.
How can people generalize new concepts from just a few examples?
The computational problem(c.f., semi-supervised learning)
?
?????
??
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’,…
Model predictions
Human judgmentsof argument strength
Similarity-based models
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.
All mammals have property P.
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.
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.
Different sources for 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
??
?
…
P(structure | form)
P(data | structure)
P(form)
Bac
kgro
und
know
ledg
eHierarchical Bayesian Framework
The value of structural form knowledge: inductive bias
F: form
S: structure
D: data
Tree with species at leaf nodes
Hierarchical Bayesian Framework
mouse
squirrel
chimp
gorilla
mousesquirrel
chimpgorilla
F1
F2
F3
F4
Ha
s T
9h
orm
on
es
??
?
…
Property induction
Smooth: P(h) high
P(D|S): How the structure constrains the data of experience
• Define a stochastic process over structure S that generates hypotheses h.– Intuitively, properties should vary smoothly over structure.
Not smooth: P(h) low
S
y
Gaussian Process (~ random walk, diffusion)
Threshold
P(D|S): How the structure constrains the data of experience
[Zhu, Ghahramani & Lafferty 2003]
h
S
y
Gaussian Process (~ random walk, diffusion)
Threshold
P(D|S): How the structure constrains the data of experience
[Zhu, Lafferty & Ghahramani 2003]
h
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’,…
[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.
Tre
e
2D
Reasoning about spatially varying properties
“Native American artifacts” task
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
Reasoning with two property types
Bio
logi
cal
prop
erty
Dis
ease
prop
erty
Tree Web
Kelp Human
Dolphin
Sand shark
Mako sharkTunaHerring
Kelp
Human
Dolphin
Sand shark
Mako shark
Tuna
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?
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
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
F: form
S: structure
D: data
Hierarchical Bayesian Framework
Favors simplicity
Favors smoothness[Zhu et al., 2003]
mousesquirrel
chimpgorilla
F1
F2
F3
F4
mouse
squirrel
chimp
gorilla
Model fitting
• Evaluate each form in parallel• For each form, heuristic search over structures
based on greedy growth from a one-node seed:
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
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 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?
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
Principles
Structure
Data
Whole-object principleShape biasTaxonomic principleContrast principleBasic-level bias
Learning word meanings
AbstractPrinciples
Structure
Data
(Griffiths, Tenenbaum, Kemp et al.)
Learning causal relations
First-order probabilistic theories for causal inference
True structure of graphical model G:
edge (G)
class (z)
edge (G)
1 2 3 4 5 6
7 8 9 10 11 12 13 14 15 16
# of samples: 20 80 1000
Data D
Graph G
Data D
Graph G
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
Goal-directed action (production and comprehension)
(Wolpert et al., 2003)
Goal inference as inverseprobabilistic planning
(Baker, Tenenbaum & Saxe)
Constraints Goals
Actions
Rational planning(PO)MDP
model predictions
hum
an
judg
men
ts
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
A raw data matrix:
The chicken-and-egg problem of structure learning and feature selection
Conventional clustering (CRP mixture):
The chicken-and-egg problem of structure learning and feature selection
Learning multiple structures to explain different feature subsets
(Shafto, Kemp, Mansinghka, Gordon & Tenenbaum, 2006)
System 1 System 2 System 3CrossCat:
The “nonparametric safety-net”
edge (G)
class (z)
edge (G)
12
3
4567
8
9
1011 12
# of samples: 40 100 1000
Data D
Graph G
Data D
Graph G
Abstract theory Z
True structure of graphical model G:
Bayesian prediction
P(ttotal|tpast)
ttotal
What is the best guess for ttotal? Compute t such that P(ttotal > t|tpast) = 0.5:
P(ttotal|tpast) 1/ttotal P(tpast)
posterior probability
Randomsampling
Domain-dependent prior
We compared the medianof the Bayesian posteriorwith the median of subjects’judgments… but what about the distribution of subjects’ judgments?
• 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.
Sources of individual differences
Individual differences in prediction
P(ttotal|tpast)
ttotal
Quantile of Bayesian posterior distribution
Pro
port
ion
of ju
dgm
ents
bel
ow p
redi
cted
val
ue
Individual differences in prediction
Average over all prediction tasks:• movie run times• movie grosses• poem lengths• life spans• terms in congress• cake baking times
P(ttotal|tpast)
ttotal
Individual differences in concept learning
• 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.
Why probability matching?
Markov chain Monte Carlo
(Metropolis-Hastings algorithm)
Bayesian inference in perception and sensorimotor integration
(Weiss, Simoncelli & Adelson 2002) (Kording & Wolpert 2004)
• You read about a movie that has made $60 million 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 meet a US congressman who has served for 11 years. How long will he serve in total?
• You encounter a phenomenon or event with an unknown extent or duration, ttotal, at a random time or value of t <ttotal. What is the total extent or duration ttotal?
Everyday prediction problems(Griffiths & Tenenbaum, 2006)
Bayesian analysis
p(ttotal|t) p(t|ttotal) p(ttotal)
1/ttotal p(ttotal)
Assume randomsample
(for 0 < t < ttotal
else = 0)
Form of p(ttotal)? e.g., uninformative (Jeffreys) prior 1/ttotal
Priors P(ttotal) based on empirically measured durations or magnitudes for many real-world events in each class:
Median human judgments of the total duration or magnitude ttotal of events in each class, given that they are first observed at a duration or magnitude t, versus Bayesian predictions (median of P(ttotal|t)).
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 typicalpharaoh reign in ancientegypt?
Summary: prediction• Predictions about the extent or magnitude of everyday events
follow Bayesian principles.
• 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.
Learning concepts from examples
Cows have T9 hormones.Sheep have T9 hormones.Goats have T9 hormones.
All mammals have T9 hormones.
Cows have T9 hormones.Seals have T9 hormones.Squirrels have T9 hormones.
All mammals have T9 hormones.
• Property induction
• Word learning
“tufa”
“tufa”
“tufa”
Clustering models for relational data
• Social networks: block models
Does person x respect person y?
Does prisoner xlike prisoner y?
conc
ept
concept
predicate
Learning systems of concepts with infinite relational models
(Kemp, Tenenbaum, Griffiths, Yamada & Ueda, AAAI 06)
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
Clustering arbitrary relational systems
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, ….
Learning a hierarchical ontology