Ping Pong in ChurchProductive use of concepts in human probabilistic inference

Tobias Gerstenberg

Noah Goodmanvs.

(mh-query 1000 100 ;Monte Carlo Inference ;CONCEPTS (definedefine personstrengthpersonstrength (memmem (lambdalambda (person) (gaussian 10 3)))) (definedefine lazylazy (memmem (lambdalambda (person game) (flipflip 0.1)))) (definedefine (teamstrengthteamstrength team game) (sumsum (mapmap (lambdalambda (person) (ifif (lazy person game) (/ (personstrength person) 2) (personstrength person))) team))) (definedefine (winnerwinner team1 team2 game)

(ifif (< (teamstrength team1 game) (teamstrength team2 game)) 'team2 'team1)) ;QUERY (personstrength 'A) ;EVIDENCE (andand (= 'team1 (winner '(TG) '(NG) 1)) (= 'team1 (winner '(NG) '(AS) 2)) (= 'team1 (winner '(NG) '(BL) 3)) (lazy '(NG) 1) ;additional evidence, used in Experiment 2 ))

How do people make inferences from complexpatterns of evidence across diverse situations?

What does a computational model need in order to capture the abstract knowledge people use for everyday reasoning?

strength

matchteamwinner

being lazy

1

>

>

>

strong indirect

A

B

B

B

C

D

>

<

<

weakindirect

A

B

B

B

C

D

>

>

>

diverse

A

A

A

B

C

D

>

>

>

confounded

A

A

A

B

B

B

0

0.5

1

1.5

stre

ngth

data model

*

* *

*

* = lazy in experiment 2

Doubles

Thinking = probabilistic inference over compositionally structured representations

Thinking is compositional and productive.

A B C

A BC

AB

C

CB AB

AB

C

D

AA BB

A B

C

D

Ping Pong Logic ConnectionistNetwork

BayesianNetwork

Model

E

3

Singles Doubles

CommentatorSingles

2

0.75 0.8 0.85 0.9 0.95 10

2

4

6

8

10

12

14

# pa

rtic

ipan

ts

Pearson correlation

Individual Correlation

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Model Prediction

Empi

rical

Mea

ns

r = 0.989RMSE = 0.161

Overall Correlation

>

>

>

B

B

B

C

E

G

A

A

A

D

F

H

>

>

>

B

C

D

E

E

E

A

A

A

F

G

H

>

<

<

B

C

D

E

E

E

A

B

B

F

F

F

>

>

>

B

C

D

E

E

E

A

B

B

F

F

F

>

>

>

B

C

D

E

G

I

A

A

A

F

H

J

>

>

>

B

C

D

C

B

B

A

A

A

D

D

C

0.5

1

1.5

stre

ngth

data model

confoundedpartner

confoundedopponent

strongindirect

weakindirect

diverse roundrobin

4

personstrengthnormally distributedpersistent propertylazyp(lazy) = 10%not persistent

teamstrengthindividual strengthscombine additively

winnerteam with greaterstrength wins

0

0.5

1

1.5

stre

ngth

strong indirect

weakindirect

diverseconfounded irrelevant

• people can reason flexibly based on di�erent patterns and sources of evidence

• compositional concepts support productive exten-sions over novel situations and objects

• only a handful of concepts required to predict inferences in fourty di�erent situations

• understanding people’s intuitive theories through models based on probabilistic programs

before

after

model

University CollegeLondon

StanfordUniversity

Tobi

Noah

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). Church: A language for generative models. Uncertainty in Arti�cial Intelligence

Goodman, N. D., & Tenenbaum, J. B. (in prep). The probabilistic language of thought.