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.