The document summarizes Noah Goodman's talk on using mathematical principles to understand thought. It discusses how thought is productive through compositional representations and probabilistic inference. Thought combines basic elements in infinite combinations, like words into sentences. Thinking involves probabilistic inference over mental representations to explain observations and plan actions. The talk suggests thought may operate via a probabilistic language of thought based on probabilistic lambda calculus.

1. ngoodman@
stanford.edu
(Reverse)
Engineering Intelligence
Noah D. Goodman
Stanford University
H+ Summit,
June 12, 2010

2. What is thought?

3. What is thought?
• How are thoughts structured?

4. What is thought?
• How are thoughts structured?
• How does this structure support
flexible, successful thinking?

5. What is thought?
• How are thoughts structured?
• How does this structure support
flexible, successful thinking?
What mathematical principles can help us understand
thought?

6. What is thought?
• How are thoughts structured?
• How does this structure support
flexible, successful thinking?
e ngi ne e r
What mathematical principles can help us understand
thought?

7. Composition and probability

8. Composition and probability
Thought is productive:
“the infinite use of
finite means”

9. Composition and probability
Thought is productive:
“the infinite use of
finite means”
..a big green
bear who loves
chocolate..

10. Composition and probability
Thought is productive:
“the infinite use of
finite means”
..a big green
bear who loves
chocolate..

11. Composition and probability
Thought is productive:
“the infinite use of
finite means”
p=mv

12. Composition and probability
Thought is productive:
“the infinite use of
finite means”
p=mv

13. Composition and probability
Thought is productive:
“the infinite use of
finite means”
p=mv
Compositional
representations

14. Composition and probability
Thought is productive:
“the infinite use of
finite means”
Compositional
representations

15. Composition and probability
Thought is productive:
“the infinite use of
finite means”
Compositional
representations

16. Composition and probability
Thought is productive:
“the infinite use of
finite means”
Compositional
representations

17. Composition and probability
Thought is productive:
“the infinite use of
finite means”
Compositional
representations

18. Composition and probability
Thought is productive:
“the infinite use of
finite means”
Compositional
representations

19. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Compositional
representations

20. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Compositional
representations

21. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Why did he yell at me?
Compositional
representations

22. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Why did he yell at me?
He wanted to hurt me.
He thought I was a telemarketer.
Compositional
representations

23. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Why did he yell at me?
Belief Desire
Action
He wanted to hurt me.
He thought I was a telemarketer.
Compositional
representations

24. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Why did he yell at me?
Belief Desire
Action
He wanted to hurt me.
He thought I was a telemarketer.
Compositional Probabilistic
representations inference

25. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Compositional Probabilistic
representations inference

26. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Compositional Probabilistic
representations inference

27. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
a+b+c =
Compositional Probabilistic
representations inference

28. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
a+b+c =
0 1 2 3
Compositional Probabilistic
representations inference

29. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
a+b+c =
0 1 2 3
P (H|d) ∝ P (d|H)P (H)
Compositional Probabilistic
representations inference

30. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Compositional Probabilistic
representations inference

31. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
Compositional Probabilistic
representations inference

32. Composition and probability
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
∀x King(x) =⇒ M an(x)
∀y M an(y) ⇐⇒ ¬W oman(y)
Compositional Probabilistic
representations inference

33. Composition and probability
Probabilistic language of
thought hypothesis
Thought is productive: Thought is useful
“the infinite use of in an uncertain
finite means” world
∀x King(x) =⇒ M an(x)
∀y M an(y) ⇐⇒ ¬W oman(y)
Compositional Probabilistic
representations inference

34. A probabilistic language

35. A probabilistic language
Lambda calculus:

36. A probabilistic language
Lambda calculus:
(define double
(λ (x) (+ x x)))

37. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))

38. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))

39. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12

40. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12
Probabilistic lambda calculus:
Goodman, Mansinghka, Roy, Bonawitz, Tenenabum (2008)

41. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12
Probabilistic lambda calculus:
(define a (flip 0.3))
(define b (flip 0.3))
(define c (flip 0.3))
(+ a b c)
Goodman, Mansinghka, Roy, Bonawitz, Tenenabum (2008)

42. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12
Probabilistic lambda calculus:
(define a (flip 0.3)) => 1
(define b (flip 0.3)) => 0
(define c (flip 0.3)) => 1
(+ a b c) => 2
Goodman, Mansinghka, Roy, Bonawitz, Tenenabum (2008)

43. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12
Probabilistic lambda calculus:
(define a (flip 0.3)) => 1 0
(define b (flip 0.3)) => 0 0
(define c (flip 0.3)) => 1 0
(+ a b c) => 2 0
Goodman, Mansinghka, Roy, Bonawitz, Tenenabum (2008)

44. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12
Probabilistic lambda calculus:
(define a (flip 0.3)) => 1 0 0
(define b (flip 0.3)) => 0 0 0
(define c (flip 0.3)) => 1 0 1
(+ a b c) => 2 0 1
Goodman, Mansinghka, Roy, Bonawitz, Tenenabum (2008)

45. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12
Probabilistic lambda calculus:
(define a (flip 0.3)) => 1 0 0
(define b (flip 0.3)) => 0 0 0
(define c (flip 0.3)) => 1 0 1
(+ a b c) => 2 0 1 ..
Goodman, Mansinghka, Roy, Bonawitz, Tenenabum (2008)

46. A probabilistic language
Lambda calculus:
(define double
(double 3) => 6
(λ (x) (+ x x)))
(define repeat
(λ (f) (λ (x) (f (f x)))))
((repeat double) 3) => 12
Probabilistic lambda calculus:
probability / frequency
(define a (flip 0.3)) => 1 0 0
(define b (flip 0.3)) => 0 0 0
(define c (flip 0.3)) => 1 0 1
(+ a b c) => 2 0 1 ..
0 1 2 3
Goodman, Mansinghka, Roy, Bonawitz, Tenenabum (2008)

47. Hypothesis
• The probabilistic language of thought
hypothesis:
Mental representations are functions
in a probabilistic lambda calculus.
• Thoughts are built compositionally (like molecules).
• Thinking is probabilistic inference.
http://projects.csail.mit.edu/church

48. Bob’s box
Goodman, Baker, Tenenbaum (2009; in prep.)

49. Bob’s box
• Bob has a box with two
buttons and a light.
A B
Goodman, Baker, Tenenbaum (2009; in prep.)

50. Bob’s box
• Bob has a box with two
buttons and a light.
A B
• He presses both buttons,
and the light comes on.
Goodman, Baker, Tenenbaum (2009; in prep.)

51. Bob’s box
• Bob has a box with two
buttons and a light.
A B
• He presses both buttons,
and the light comes on.
• How does the
box work? A A A A A
B B B B B
C C C C C
A alone B alone A or B A and B Nothing
causes C. causes C. cause C. causes C. causes C.
Goodman, Baker, Tenenbaum (2009; in prep.)

52. Human judgements
Social
50
*
40 Social condition
Mean Bets ($)
30
Physical
50
20 Physical condition
40 ns
30
10
20
0
A B AorB A&B none 10
N=15 0
A B AorB A&B none
A alone B alone A or B A and B Nothing
causes C. causes C. cause C. causes C. causes C.

53. Purely causal learning
Causal!only model
0.5
(query
Causal-only
(define world-cs (cs-prior)) 0.4
(define action (uniform))
Probability
0.3
(define outcome (world-cs
init-state 0.2
action))
0.1
world-cs
(and (press-A action) 0
A B AorB A&B none
A or B
A&B
B only
none
A only
(press-B action) Cause of C
(light-on outcome)))
No conclusion is possible.
The evidence is confounded.

54. Explaining actions
Beliefs: Desires:
A B
C
Decision
Rational action:
Actions: (define decide
(λ (state causal-model utility)
(query
(define action (action-prior))
action
(flip (utility
(causal-model
state action))))))

55. Causal learning models Causal!only model
0.5
Causal-only model
Causal-only
Causal-only
(define world-cs (cs-prior)) 0.4
(define action (uniform)) model
Probability
(define outcome (world-cs 0.3
init-state 0.2
action))
0.1
0
A B AorB A&B none
A or B
A&B
B only
none
A only
Cause of C
(define world-cs (cs-prior))
(define utility (uniform))
Social & causal
(define cs-belief world-cs) Knowledgeable
(define action (decide
init-state
agent assumption
cs-belief Rational
utility))
(define outcome (world-cs agent assumption
init-state
action))

56. Causal learning models Causal!only model
0.5
Causal-only model
Causal-only
Causal-only
(define world-cs (cs-prior)) 0.4
(define action (uniform)) model
Probability
(define outcome (world-cs 0.3
init-state 0.2
action))
0.1
0
A B AorB A&B none
A or B
A&B
B only
none
A only
Cause of C
(define world-cs (cs-prior))
(define utility (uniform))
Social & causal
(define cs-belief world-cs)
(define action (decide
init-state
cs-belief
utility))
(define outcome (world-cs
init-state
action))

57. Causal learning models Causal!only model
0.5
Causal-only model
Causal-only
(define world-cs (cs-prior)) 0.4
(define action (uniform))
Probability
(define outcome (world-cs 0.3
init-state 0.2
action))
0.1
0
A B AorB A&B none
A or B
A&B
B only
none
A only
Cause of C
(define world-cs (cs-prior))
(define utility (uniform))
Social & causal
(define cs-belief world-cs)
(define action (decide
init-state
cs-belief
utility))
(define outcome (world-cs
init-state
action))

58. Causal learning models Causal!only model
0.5
Causal-only model
Causal-only
(define world-cs (cs-prior)) 0.4
(define action (uniform))
Probability
(define outcome (world-cs 0.3
init-state 0.2
action))
0.1
0
A B AorB A&B none
A or B
A&B
B only
none
A only
Cause of C
(define world-cs (cs-prior))
(define utility (uniform))
Social & causal
Social!causal model
0.5
(define cs-belief world-cs) Social + causal model
(define action (decide 0.4
init-state
Posterior probability
Probability
0.3
cs-belief
utility)) 0.2
(define outcome (world-cs
init-state 0.1
action)) 0
A B AorB A&B none

59. Scalar implicature
Some of the plants
have sprouted
(Plants usually sprout.) Goodman, et al (in prep)

60. Scalar implicature
Desires:
-informative
Beliefs -parsimonious
Actions:
“...”
Some of the plants
have sprouted
(Plants usually sprout.) Goodman, et al (in prep)

61. Scalar implicature
Desires: Model:
-informative
Beliefs -parsimonious
Plausibility (Z-score)
2
1
0
-1
Actions:
-2
“...” 0:5 1:5 2:5 3:5 4:5 5:5
Number sprouted
Some of the plants
have sprouted
(Plants usually sprout.) Goodman, et al (in prep)

62. Scalar implicature
Desires: Model:
-informative
Beliefs -parsimonious
Plausibility (Z-score)
2
1
0
-1
Actions:
-2
“...” 0:5 1:5 2:5 3:5 4:5 5:5
Number sprouted
Some of the plants
have sprouted
(Plants usually sprout.) Goodman, et al (in prep)

63. Scalar implicature
Desires: Model: Partial
-informative Full knowledge knowledge
Beliefs -parsimonious
Plausibility (Z-score)
2
1
0
-1
Actions:
-2
“...” 0:5 1:5 2:5 3:5 4:5 5:5 0:5 1:5 2:5 3:5 4:5 5:5
Number sprouted
Some of the plants
have sprouted
(Plants usually sprout.) Goodman, et al (in prep)

64. Scalar implicature
Desires: Model: Partial
-informative Full knowledge knowledge
Beliefs -parsimonious
Plausibility (Z-score)
2
1
0
-1
Actions:
-2
“...” 0:5 1:5 2:5 3:5 4:5 5:5 0:5 1:5 2:5 3:5 4:5 5:5
Number sprouted
Human:
Some of the plants
have sprouted
(Plants usually sprout.) Goodman, et al (in prep)

65. Summary
• The probabilistic language of thought
combines composition and probability.
• We can explain complex, flexible human
thinking...
• And engineer flexible computer
intelligence.