Bayesian models of human
inference
Josh Tenenbaum
MIT
The Bayesian 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 in AI: no necessary connection to
how the human brain solves these problems.
– Heuristics & Biases program in the background (“We
know people aren’t Bayesian, but…”).
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,
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, …]
How to meet up with mainstream JDM
research (i.e., heuristics & biases)?
1. How to reconcile apparently contradictory
messages of H&B and Bayesian models?
Are people Bayesian or aren’t they? When are they,
when aren’t they, and why?
2. How to integrate the H&B and Bayesian
research approaches?
When are people Bayesian, and why?
• Low level hypothesis (Shiffrin, Maloney, etc.)
– People are Bayesian in low-level input or output processes
that have a long evolutionary history shared with other
species, e.g. vision, motor control, memory retrieval.
When are people Bayesian, and why?
• Low level hypothesis (Shiffrin, Maloney, etc.)
• Information format hypothesis (Gigerenzer)
– Higher-level cognition
can be Bayesian when
information is
presented in formats
that we have evolved
to process, and that
support simple
heuristic algorithms,
e.g., base-rate neglect
disappears with
“natural frequencies”.
Explicit
probabilities
Natural
frequencies
When are people Bayesian, and why?
• Low level hypothesis (Shiffrin, Maloney, etc.)
• Information format hypothesis (Gigerenzer)
• Core capacities hypothesis
– Bayes can illuminate distinctively human cognitive
capacities for inductive inference – learning words and
concepts, projecting properties of objects, causal inference,
or action understanding: problems we solve effortlessly,
unconsciously, and successfully in natural contexts, which
a five-year-old solves better than any animal or computer.
Figure 13: Procedure used in Sobel et al. (2002), Experi
One-Cause Condition
When are people Bayesian, and why?
• Low level hypothesis (Shiffrin, Maloney, etc.)
Figure 13: Procedure used in Sobel et al. (2002), Experiment 2
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• Core capacities hypothesis
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When are people Bayesian, and why?
• Low level hypothesis (Shiffrin, Maloney, etc.)
• Information format hypothesis (Gigerenzer)
• Core capacities hypothesis
Word learning
Hypothesis
space
Data
(Tenenbaum & Xu)
When are people Bayesian, and why?
• Low level hypothesis (Shiffrin, Maloney, etc.)
• Information format hypothesis (Gigerenzer)
• Core capacities hypothesis
– Bayes can illuminate distinctively human cognitive
capacities for inductive inference – learning words and
concepts, projecting properties of objects, causal inference,
or action understanding: problems we solve effortlessly,
unconsciously, and successfully in natural contexts, which
a five-year-old solves better than any animal or computer.
– The mind is not good at explicit Bayesian reasoning about
verbally or symbolically presented statistics, unless core
capacities can be engaged.
When are people Bayesian, and why?
Statistical version of
Diagnosis problem
Causal version of
Diagnosis problem
Correct Base-rate
neglect
• Low level hypothesis (Shiffrin, Maloney, etc.)
• Information format hypothesis (Gigerenzer)
• Core competence hypothesis
(Krynski & Tenenbaum)
How to meet up with mainstream JDM
research (i.e., heuristics & biases)?
1. How to reconcile apparently contradictory
messages of H&B and Bayesian models?
Are people Bayesian or aren’t they? When are they,
when aren’t they, and why?
2. How to integrate the H&B and Bayesian
research approaches?
Reverse engineering
• Goal is to reverse-engineer human inference.
– A computational understanding of how the mind
works and why it works it does.
• Even for core inferential capacities, we are
likely to observe behavior that deviates from
any ideal Bayesian analysis.
• These deviations are likely to be informative
about how the mind works.
Analogy to visual illusions
(Adelson)
(Shepard)
• Highlight the problems the visual system is designed to
solve: inferring world structure from images, not judging
properties of the images themselves.
• Reveal the implicit visual system’s implicit assumptions
about the physical world and the processes of image
formation that are needed to solve these problems.
How do we interpret deviations from a
Bayesian analysis?
• H&B: People aren’t Bayesian, but use some other
means of inference.
–
–
–
–
Base-rate neglect: representativeness heuristic
Recency bias: availability heuristic
Order of evidence effects: anchoring and adjustment
…
• Not so compelling as reverse engineering.
– What engineer would want to design a system based on
“representativeness”, without knowing how it is computed,
why it is computed that way, what problem it attempts to
solve, when it works, or how its accuracy and efficiency
compares to some ideal computation or other heuristics.
How do we interpret deviations from a
Bayesian analysis?
Multiple levels of analysis (Marr)
• Computational theory
– What is the goal of the computation – the outputs and
available inputs? What is the logic by which the inference
can be performed? What constraints (prior knowledge) do
people assume to make the solution well-posed?
• Representation and algorithm
– How is the information represented? How is the computation
carried out algorithmically, approximating the ideal
computational theory with realistic time & space resources?
• Hardware implementation
How do we interpret deviations from a
Bayesian analysis?
Multiple levels of analysis (Marr)
• Computational theory
– What is the goal of the computation – the outputs andBayes
available inputs? What is the logic by which the inference
can be performed? What constraints (prior knowledge) do
people assume to make the solution well-posed?
• Representation and algorithm
– How is the information represented? How is the computation
carried out algorithmically, approximating the ideal
computational theory with realistic time & space resources?
• Hardware implementation
Different philosophies
• H&B
– One canonical Bayesian analysis of any given task, and we
know what it is.
– Ideal Bayesian solution can be computed.
– The question “Are people Bayesian?” is empirically
meaningful on any given task.
• Bayes+Marr
– Many possible Bayesian analyses of any given task, and we
need to discover which best characterize cognition.
– Ideal Bayesian solution can only be approximately computed.
– The question “Are people Bayesian?” is not an empirical one,
at least not for an individual task. Bayes is a frameworklevel assumption, like distributed representations in
connectionism or condition-action rules in ACT-R.
How do we interpret deviations from a
Bayesian analysis?
Multiple levels of analysis (Marr)
• Computational theory
– What is the goal of the computation – the outputs and
available inputs? What is the logic by which the inference
can be performed? What constraints (prior knowledge) do
people assume to make the solution well-posed?
• Representation and algorithm
– How is the information represented? How is the computation
carried out algorithmically, approximating the ideal
computational theory with realistic time & space resources?
• Hardware implementation
The centrality of causal inference
• In visual perception:
– Judge P(scene|image features) rather than P(image
features|scene) or P(image features|other image features).
• Coin–flipping: Which sequence is more likely to come
from flipping a fair coin, HHTHT or HHHHH?
• Coincidences: How likely that 2 people in a random
party of 25 have the same birthday? 3 in a party of 10?
(Griffiths & Tenenbaum)
Judgments of randomness:
Judgments of coincidence:
P(data | random)
P(random | data) 
P(data | regular )
Rational measure of
evidential support:
P(data | h1 )
P(data | h0 )
P(data | regular )
P(regular | data) 
P(data | random)
(Griffiths & Tenenbaum)
How do we interpret deviations from a
Bayesian analysis?
Multiple levels of analysis (Marr)
• Computational theory
– What is the goal of the computation – the outputs and
available inputs? What is the logic by which the inference
can be performed? What constraints (prior knowledge) do
people assume to make the solution well-posed?
• Representation and algorithm
– How is the information represented? How is the computation
carried out algorithmically, approximating the ideal
computational theory with realistic time & space resources?
• Hardware implementation
Assuming the world is simple
• In visual
perception:
Figure 13: Procedure used in Sobel et al. (2002), Experimen
– “Slow and smooth”
prior on visual motion
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Recognizing the world is complex
• In visual
perception:
– Need uncertainty
about coherence
ratio and velocity
of coherent motion.
(Lu & Yuille)
• Property induction:
– Properties should
be distributed
stochastically over
tree structure, not
just focused on
single branches.
Gorillas have T9 cells.
Seals have T9 cells.
Horses have T9 cells.
Bayes:
single
branch
prior
r = 0.50
(Kemp & Tenenbaum)
Recognizing the world is complex
• In visual
perception:
– Need uncertainty
about coherence
ratio and velocity
of coherent motion.
(Lu & Yuille)
• Property induction:
– Properties should
be distributed
stochastically over
tree structure, not
just focused on
single branches.
Gorillas have T9 cells.
Seals have T9 cells.
Horses have T9 cells.
Bayes:
“mutation”
prior
r = 0.92
(Kemp & Tenenbaum)
“has T9
hormones”
“can bite
through wire”
“is found near
Minneapolis”
“carry E.
Spirus
bacteria”
(Kemp & Tenenbaum)
How do we interpret deviations from a
Bayesian analysis?
Multiple levels of analysis (Marr)
• Computational theory
– What is the goal of the computation – the outputs and
available inputs? What is the logic by which the inference
can be performed? What constraints (prior knowledge) do
people assume to make the solution well-posed?
• Representation and algorithm
– How is the information represented? How is the computation
carried out algorithmically, approximating the ideal
computational theory with realistic time & space resources?
• Hardware implementation
Sampling-based approximate inference
• In visual
perception:
– Temporal dynamics
of bi-stability due to
fast sampling-based
approximation of a
bimodal posterior (Schrater & Sundareswara).
• Order effects in category learning
– Particle filter (sequential Monte Carlo), an online
approximate inference algorithm assuming stationarity.
• Probability matching in classification decisions
– Sampling-based approximations with guarantees of near
optimal generalization performance.
(Griffiths et al., Goodman et al.)
Conclusions
• “Are people Bayesian?”, “When are they Bayesian?”
– Maybe not the most interesting questions in the long run….
• What is the best way to reverse engineer cognition at
multiple levels of analysis? Assuming core inductive
capacities are approximately Bayesian at the
computational-theory level offers several benefits:
–
–
–
–
Explanatory power: why does cognition work?
Fewer degrees of freedom in modeling
A bridge to state-of-the-art AI and machine learning
Tools to study the big questions: What are the goals of
cognition? What does the mind know about the world? How
is that knowledge represented? What are the processing
mechanisms and why do they work as they do?
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)
Latent common cause:
Chance:
C
x
x
x
x
x
uniform
x
x
x
x
x
uniform
+
regularity