1. Part II: Graphical models

2. Challenges of probabilistic models
Specifying well-defined probabilistic models with many variables is hard (for modelers)
Representing probability distributions over those variables is hard (for computers/learners)
Computing quantities using those distributions is hard (for computers/learners)

3. Representing structured distributions
Four random variables:
X1 coin toss produces heads
X2 pencil levitates
X3 friend has psychic powers
X4 friend has two-headed coin
Domain
{0,1}

4. 0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Joint distribution
Requires 15 numbers to specify probability of all values x1,x2,x3,x4
N binary variables, 2N-1 numbers
Similar cost when computing conditional probabilities

5. How can we use fewer numbers?
Four random variables:
X1 coin toss produces heads
X2 coin toss produces heads
X3 coin toss produces heads
X4 coin toss produces heads
Domain
{0,1}

6. Statistical independence
Two random variables X1 and X2 are independent if P(x1|x2) = P(x1)
e.g. coinflips: P(x1=H|x2=H) = P(x1=H) = 0.5
Independence makes it easier to represent and work with probability distributions
We can exploit the product rule:
If x1, x2, x3, and x4 are all independent…

7. Expressing independence
Statistical independence is the key to efficient probabilistic representation and computation
This has led to the development of languages for indicating dependencies among variables
Some of the most popular languages are based on “graphical models”

8. Part II: Graphical models
Introduction to graphical models
representation and inference
Causal graphical models
causality
learning about causal relationships
Graphical models and cognitive science
uses of graphical models
an example: causal induction

9. Part II: Graphical models
Introduction to graphical models
representation and inference
Causal graphical models
causality
learning about causal relationships
Graphical models and cognitive science
uses of graphical models
an example: causal induction

10. Graphical models
Express the probabilistic dependency structure among a set of variables (Pearl, 1988)
Consist of
a set of nodes, corresponding to variables
a set of edges, indicating dependency
a set of functions defined on the graph that specify a probability distribution

11. Undirected graphical modelsX3 X4 X1
Consist of
a set of nodes
a set of edges
a potential for each clique, multiplied together to yield the distribution over variables
Examples
statistical physics: Ising model, spinglasses
early neural networks (e.g. Boltzmann machines) X2 X5

12. Directed graphical modelsX3 X4 X1
Consist of
a set of nodes
a set of edges
a conditional probability distribution for each node, conditioned on its parents, multiplied together to yield the distribution over variables
Constrained to directed acyclic graphs (DAGs)
Called Bayesian networks or Bayes nets X2 X5

13. Bayesian networks and Bayes
Two different problems
Bayesian statistics is a method of inference
Bayesian networks are a form of representation
There is no necessary connection
many users of Bayesian networks rely upon frequentist statistical methods
many Bayesian inferences cannot be easily represented using Bayesian networks

14. Properties of Bayesian networks
Efficient representation and inference
exploiting dependency structure makes it easier to represent and compute with probabilities
Explaining away
pattern of probabilistic reasoning characteristic of Bayesian networks, especially early use in AI

15. Properties of Bayesian networks
Efficient representation and inference
exploiting dependency structure makes it easier to represent and compute with probabilities
Explaining away
pattern of probabilistic reasoning characteristic of Bayesian networks, especially early use in AI

16. Efficient representation and inference
Four random variables:
X1 coin toss produces heads
X2 pencil levitates
X3 friend has psychic powers
X4 friend has two-headed coin X1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)

17. The Markov assumption
Every node is conditionally independent of its non-descendants, given its parents
where Pa(Xi) is the set of parents of Xi
(via the product rule)

18. Efficient representation and inference
Four random variables:
X1 coin toss produces heads
X2 pencil levitates
X3 friend has psychic powers
X4 friend has two-headed coin X1 X2 X3 X4 1
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4) 1 4 2
total = 7 (vs 15)
P(x1, x2, x3, x4) = P(x1|x3, x4)P(x2|x3)P(x3)P(x4)

19. Reading a Bayesian network
The structure of a Bayes net can be read as the generative process behind a distribution
Gives the joint probability distribution over variables obtained by sampling each variable conditioned on its parents

20. Reading a Bayesian network
Four random variables:
X1 coin toss produces heads
X2 pencil levitates
X3 friend has psychic powers
X4 friend has two-headed coin
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4) X4 X3 X1 X2
P(x1, x2, x3, x4) = P(x1|x3, x4)P(x2|x3)P(x3)P(x4)

21. Reading a Bayesian network
The structure of a Bayes net can be read as the generative process behind a distribution
Gives the joint probability distribution over variables obtained by sampling each variable conditioned on its parents
Simple rules for determining whether two variables are dependent or independent
Independence makes inference more efficient

22. Computing with Bayes netsX1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)
P(x1, x2, x3, x4) = P(x1|x3, x4)P(x2|x3)P(x3)P(x4)

23. Computing with Bayes nets
sum over 8 values X1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)
P(x1, x2, x3, x4) = P(x1|x3, x4)P(x2|x3)P(x3)P(x4)

24. Computing with Bayes netsX1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)
P(x1, x2, x3, x4) = P(x1|x3, x4)P(x2|x3)P(x3)P(x4)

25. Computing with Bayes nets
sum over 4 values X1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)
P(x1, x2, x3, x4) = P(x1|x3, x4)P(x2|x3)P(x3)P(x4)

26. Computing with Bayes nets
Inference algorithms for Bayesian networks exploit dependency structure
Message-passing algorithms
“belief propagation” passes simple messages between nodes, exact for tree-structured networks
More general inference algorithms
exact: “junction-tree”
approximate: Monte Carlo schemes (see Part IV)

27. Properties of Bayesian networks
Efficient representation and inference
exploiting dependency structure makes it easier to represent and compute with probabilities
Explaining away
pattern of probabilistic reasoning characteristic of Bayesian networks, especially early use in AI

28. Explaining away
Rain
Sprinkler
Grass Wet
Assume grass will be wet if and only if it rained last night, or if the sprinklers were left on:

29. Explaining away Rain Sprinkler Grass Wet
Compute probability it rained last night, given that the grass is wet:

30. Explaining away Rain Sprinkler Grass Wet
Compute probability it rained last night, given that the grass is wet:

31. Explaining away Rain Sprinkler Grass Wet
Compute probability it rained last night, given that the grass is wet:

32. Explaining away Rain Sprinkler Grass Wet
Compute probability it rained last night, given that the grass is wet:

33. Explaining away Rain Sprinkler Grass Wet
Compute probability it rained last night, given that the grass is wet:
Between 1 and P(s)

34. Explaining away Rain Sprinkler Grass Wet
Compute probability it rained last night, given that the grass is wet and sprinklers were left on:
Both terms = 1

35. Explaining away Rain Sprinkler Grass Wet
Compute probability it rained last night, given that the grass is wet and sprinklers were left on:

36. Explaining away Rain Sprinkler Grass Wet “Discounting” to
prior probability.

37. Contrast w/ production system
Rain
Sprinkler
Grass Wet
Formulate IF-THEN rules:
IF Rain THEN Wet
IF Wet THEN Rain
Rules do not distinguish directions of inference
Requires combinatorial explosion of rules
IF Wet AND NOT Sprinkler
THEN Rain

38. Contrast w/ spreading activation
Rain
Sprinkler
Grass Wet
Excitatory links: Rain Wet, Sprinkler Wet
Observing rain, Wet becomes more active.
Observing grass wet, Rain and Sprinkler become more active
Observing grass wet and sprinkler, Rain cannot become less active. No explaining away!

39. Contrast w/ spreading activation
Rain
Sprinkler
Grass Wet
Excitatory links: Rain Wet, Sprinkler Wet
Inhibitory link: Rain Sprinkler
Observing grass wet, Rain and Sprinkler become more active
Observing grass wet and sprinkler, Rain becomes less active: explaining away

40. Contrast w/ spreading activation
Rain
Burst pipe
Sprinkler
Grass Wet
Each new variable requires more inhibitory connections
Not modular
whether a connection exists depends on what others exist
big holism problem
combinatorial explosion

41. Contrast w/ spreading activation
(McClelland & Rumelhart, 1981)

42. Graphical models Capture dependency structure in distributions
Provide an efficient means of representing and reasoning with probabilities
Allow kinds of inference that are problematic for other representations: explaining away
hard to capture in a production system
more natural than with spreading activation

43. Part II: Graphical models
Introduction to graphical models
representation and inference
Causal graphical models
causality
learning about causal relationships
Graphical models and cognitive science
uses of graphical models
an example: causal induction

44. Causal graphical models
Graphical models represent statistical dependencies among variables (ie. correlations)
can answer questions about observations
Causal graphical models represent causal dependencies among variables (Pearl, 2000)
express underlying causal structure
can answer questions about both observations and interventions (actions upon a variable)

45. Bayesian networks Nodes: variables Links: dependency
Each node has a conditional probability distribution
Data: observations of x1, ..., x4
Four random variables:
X1 coin toss produces heads
X2 pencil levitates
X3 friend has psychic powers
X4 friend has two-headed coin X1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)

46. Causal Bayesian networks
Nodes: variables
Links: causality
Each node has a conditional probability distribution
Data: observations of and interventions on x1, ..., x4
Four random variables:
X1 coin toss produces heads
X2 pencil levitates
X3 friend has psychic powers
X4 friend has two-headed coin X1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)

47. Interventions X
Cut all incoming links for the node that we intervene on
Compute probabilities with “mutilated” Bayes net
Four random variables:
X1 coin toss produces heads
X2 pencil levitates
X3 friend has psychic powers
X4 friend has two-headed coin
hold down pencil X1 X2 X3 X4
P(x4)
P(x3)
P(x2|x3)
P(x1|x3, x4)

48. Learning causal graphical models
Strength: how strong is a relationship?
Structure: does a relationship exist? E B C

49. Causal structure vs. causal strength
Strength: how strong is a relationship? B E B C E B C B

50. Causal structure vs. causal strength
Strength: how strong is a relationship?
requires defining nature of relationship B E B C w0 w1 E B C w0 B

51. Parameterization Generic Structures: h1 = h0 = Parameterization: C B B
h1: P(E = 1 | C, B)
h0: P(E = 1| C, B)

52. Parameterization Linear Structures: h1 = h0 = Parameterization: C B Bw0 w1
w0, w1: strength parameters for B, C E E w1 w0
w1+ w0
Linear C B
h1: P(E = 1 | C, B)
h0: P(E = 1| C, B)

53. Parameterization “Noisy-OR” Structures: h1 = h0 = Parameterization: CB C B C w0 w1
w0, w1: strength parameters for B, C E E w1 w0
w1+ w0 – w1 w0
“Noisy-OR” C B
h1: P(E = 1 | C, B)
h0: P(E = 1| C, B)

54. maximize i P(bi,ci,ei; w0, w1)
Parameter estimation
Maximum likelihood estimation:
maximize i P(bi,ci,ei; w0, w1)
Bayesian methods: as in Part I

55. Causal structure vs. causal strength
Structure: does a relationship exist? B E B C E B C B

56. Approaches to structure learning
Constraint-based:
dependency from statistical tests (eg. 2)
deduce structure from dependencies B B C E
(Pearl, 2000; Spirtes et al., 1993)

57. Approaches to structure learning
Constraint-based:
dependency from statistical tests (eg. 2)
deduce structure from dependencies B B C E
(Pearl, 2000; Spirtes et al., 1993)

58. Approaches to structure learning
Constraint-based:
dependency from statistical tests (eg. 2)
deduce structure from dependencies B B C E
(Pearl, 2000; Spirtes et al., 1993)

59. Approaches to structure learning
Constraint-based:
dependency from statistical tests (eg. 2)
deduce structure from dependencies B B C E
(Pearl, 2000; Spirtes et al., 1993)
Attempts to reduce inductive problem to deductive problem

60. Approaches to structure learning
Constraint-based:
dependency from statistical tests (eg. 2)
deduce structure from dependencies B B C E
(Pearl, 2000; Spirtes et al., 1993)
Bayesian:
compute posterior
probability of structures,
given observed data B C B C E E
P(h1|data)
P(h0|data)
P(h|data)  P(data|h) P(h)
(Heckerman, 1998; Friedman, 1999)

61. Bayesian Occam’s Razor
h0 (no relationship)
P(d | h )
h1 (relationship)
All possible data sets d
For any model h,

62. Causal graphical models
Extend graphical models to deal with interventions as well as observations
Respecting the direction of causality results in efficient representation and inference
Two steps in learning causal models
strength: parameter estimation
structure: structure learning

63. Part II: Graphical models
Introduction to graphical models
representation and inference
Causal graphical models
causality
learning about causal relationships
Graphical models and cognitive science
uses of graphical models
an example: causal induction

64. Uses of graphical models
Understanding existing cognitive models
e.g., neural network models
Representation and reasoning
a way to address holism in induction (c.f. Fodor)
Defining generative models
mixture models, language models (see Part IV)
Modeling human causal reasoning

65. Human causal reasoning
How do people reason about interventions?
(Gopnik, Glymour, Sobel, Schulz, Kushnir & Danks, 2004; Lagnado & Sloman, 2004; Sloman & Lagnado, 2005; Steyvers, Tenenbaum, Wagenmakers & Blum, 2003)
How do people learn about causal relationships?
parameter estimation (Shanks, 1995; Cheng, 1997)
constraint-based models (Glymour, 2001)
Bayesian structure learning
(Steyvers et al., 2003; Griffiths & Tenenbaum, 2005)

66. Causation from contingencies
C present
(c+)
C absent
(c-) a
E present (e+) c d
E absent (e-) b
“Does C cause E?”
(rate on a scale from 0 to 100)

67. Two models of causal judgment
Delta-P (Jenkins & Ward, 1965):
Power PC (Cheng, 1997):
Power

68. Buehner and Cheng (1997)
0.00
0.25
0.50
0.75
1.00 DP
People DP
Power

69. Buehner and Cheng (1997) Constant P, changing judgments People DP
Power
Constant P, changing judgments

70. Buehner and Cheng (1997) Constant causal power, changing judgments
People DP
Power
Constant causal power, changing judgments

71. Buehner and Cheng (1997)
People DP
Power
P = 0, changing judgments

72. Causal structure vs. causal strength
Strength: how strong is a relationship?
Structure: does a relationship exist? B E B C w0 w1 E B C w0 B

73. Causal strength Assume structure:
DP and causal power are maximum likelihood estimates of the strength parameter w1, under different parameterizations for P(E|B,C):
linear  DP, Noisy-OR  causal power B E B C w0 w1

74. likelihood ratio (Bayes factor) gives evidence in favor of h1
Causal structure
Hypotheses: h1 = h0 =
Bayesian causal inference:
support = B E B C B E B C
P(d|h1)
likelihood ratio (Bayes factor) gives evidence in favor of h1
P(d|h0)

75. Buehner and Cheng (1997) People DP (r = 0.89) Power (r = 0.88)
Support (r = 0.97)

76. The importance of parameterization
Noisy-OR incorporates mechanism assumptions:
generativity: causes increase probability of effects
each cause is sufficient to produce the effect
causes act via independent mechanisms
(Cheng, 1997)
Consider other models:
statistical dependence: 2 test
generic parameterization (cf. Anderson, 1990)

77. People
Support (Noisy-OR) 2
Support (generic)

78. Generativity is essential
P(e+|c+)
8/8
6/8
4/8
2/8
0/8
P(e+|c-)
100 50
Support
Predictions result from “ceiling effect”
ceiling effects only matter if you believe a cause increases the probability of an effect

79. Blicket detector (Dave Sobel, Alison Gopnik, and colleagues)
Oooh, it’s a
blicket!
Let’s put this one
on the machine.
See this? It’s a
blicket machine.
Blickets make it go.

80. “Backwards blocking” (Sobel, Tenenbaum & Gopnik, 2004)
AB Trial
A Trial
Two objects: A and B
Trial 1: A B on detector – detector active
Trial 2: A on detector – detector active
4-year-olds judge whether each object is a blicket
A: a blicket (100% say yes)
B: probably not a blicket (34% say yes)

81. Possible hypotheses A B E A B A B A B A B A B A B A B A B E E E E E E

82. Bayesian inference Evaluating causal models in light of data:
Inferring a particular causal relation:

83. Probability of being a blicket
Bayesian inference
With a uniform prior on hypotheses, and the generic parameterization
Probability of being a blicket A B
0.32
0.32
0.34
0.34

84. Modeling backwards blocking
Assume…
Links can only exist from blocks to detectors
Blocks are blickets with prior probability q
Blickets always activate detectors, but detectors never activate on their own
deterministic Noisy-OR, with wi = 1 and w0 = 0

85. Modeling backwards blocking
P(h00) = (1 – q)2
P(h01) = (1 – q) q
P(h10) = q(1 – q)
P(h11) = q2 A B A B A B A B E E E E
P(E=1 | A=0, B=0):
P(E=1 | A=1, B=0):
P(E=1 | A=0, B=1):
P(E=1 | A=1, B=1):

86. Modeling backwards blocking
P(h00) = (1 – q)2
P(h01) = (1 – q) q
P(h10) = q(1 – q)
P(h11) = q2 A B A B A B A B E E E E
P(E=1 | A=1, B=1):

87. Modeling backwards blocking
P(h01) = (1 – q) q
P(h10) = q(1 – q)
P(h11) = q2 A B A B A B E E E
P(E=1 | A=1, B=0):
P(E=1 | A=1, B=1):

88. Manipulating prior probability (Tenenbaum, Sobel, Griffiths, & Gopnik, submitted)
Initial
AB Trial
A Trial

89. Summary
Graphical models provide solutions to many of the challenges of probabilistic models
defining structured distributions
representing distributions on many variables
efficiently computing probabilities
Causal graphical models provide tools for defining rational models of human causal reasoning and learning