Ping Pong in Church
Productive use of concepts in human probabilistic inference
To
b
i
University
College
London
Tobias
Gerstenberg
What does a computational model need in order
to capture the abstract knowledge people use for
everyday reasoning?
Ping Pong
strength
match
Connectionist
Network
Logic
A
A B
B
Bayesian
Network
A
vs.
2
How do people make inferences from complex
patterns of evidence across diverse situations?
1
B
C
D
winner
being lazy
A
B
Stanford
University
Model
(mh-query 1000 100 ;Monte Carlo Inference
;CONCEPTS
(define
define personstrength (mem
mem (lambda
lambda (person) (gaussian 10 3))))
(define
define lazy (mem
mem (lambda
lambda (person game) (flip
flip 0.1))))
(define
define (teamstrength
teamstrength team game)
(sum
sum (map
map (lambda
lambda (person)
(if
if (lazy person game)
(/ (personstrength person) 2)
(personstrength person)))
team)))
(define
define (winner
winner team1 team2 game)
(if
if (< (teamstrength team1 game)
(teamstrength team2 game))
'team2 'team1))
;QUERY
(personstrength 'A)
;EVIDENCE
(and
and
(= 'team1 (winner '(TG) '(NG) 1))
(= 'team1 (winner '(NG) '(AS) 2))
(= 'team1 (winner '(NG) '(BL) 3))
(lazy '(NG) 1) ;additional evidence, used in Experiment 2
)
)
Thinking is compositional
and productive.
team
Noah
Goodman
Singles
personstrength
normally distributed
persistent property
lazy
p(lazy) = 10%
not persistent
teamstrength
individual strengths
combine additively
winner
team with greater
strength wins
Doubles
C
C
A B
A
B
B
B C
E
D
B
C
A
A C
Thinking = probabilistic
inference over compositionally
structured representations
data
1.5
model
1
1
Overall Correlation
1.5
r = 0.989
RMSE = 0.161
0
−0.5
0.5
−1
−1.5
−1.5
A
>
B
A
>
B *
A
>
B *
A
>
B
A
>
B *
B
>
C
B
<
C
A
>
C*
A
>
B
B
>
D
B
<
D
A
>
D
confounded
* = lazy in experiment 2
weak
indirect
strong
indirect
14
12
−1
−0.5
0
0.5
Model Prediction
1
diverse
data
strength
0.5
confounded
8
strong
indirect
weak
indirect
diverse
irrelevant
6
• people can reason flexibly
based on different patterns
and sources of evidence
2
1.5
after
1
0
Individual Correlation
4
0
Doubles
model
1.5
10
# participants
0
before
0.5
Empirical Means
strength
1.5
Commentator
4
Singles
strength
3
0.75
0.8
0.85
0.9
0.95
Pearson correlation
1
• compositional concepts
model
support productive extensions over novel situations
and objects
1
• only a handful of concepts required to predict inferences
in fourty different situations
0.5
A
B
>
C
D
A
B
>
E
F
A
B
>
E
F
A
B
>
E
F
A
B
>
E
F
A
B
>
C
D
A
B
>
E
F
A
C
>
E
G
B
C
<
E
F
B
C
>
E
F
A
C
>
G
H
A
C
>
B
D
A
B
>
G
H
A
D
>
E
H
B
D
<
E
F
B
D
>
E
F
A
D
>
I
J
A
D
>
B
C
confounded confounded
partner
opponent
strong
indirect
weak
indirect
diverse
round
robin
• understanding people’s intuitive
theories through models based on
probabilistic programs
Gerstenberg, T. G., Goodman, N. D., Lagnado, D. A., & Tenenbaum, J. B. (2012). Noisy
Newtons: Unifying process and dependency accounts of causal attribution. Cognitive
Science Proceedings
Goodman, N. D., Mansinghka, V. K., Roy, D., Bonawitz, K., & Tenenbaum, J. B. (2008).
ah
Church: A language for generative models. Uncertainty in Artificial Intelligence
No
Goodman, N. D., & Tenenbaum, J. B. (in prep). The probabilistic language of thought.