Part II: Graphical models

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 0,1 0,1 0,1
4
Joint distribution
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
1010 1011 1100 1101 1110 1111

• 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 0,1 0,1 0,1
6
Statistical independence

• Two random variables X1 and X2 are independent if
  P(x1x2) P(x1)
• e.g. coinflips P(x1Hx2H) P(x1H) 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 models
X3
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
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Directed graphical models
X3
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
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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

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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)
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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

1
1
4
2
total 7 (vs 15)
P(x1, x2, x3, x4) P(x1x3, x4)P(x2x3)P(x3)P(x4)
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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

X3
X4
X1
X2
P(x1, x2, x3, x4) P(x1x3, x4)P(x2x3)P(x3)P(x4)
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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 nets
P(x1, x2, x3, x4) P(x1x3, x4)P(x2x3)P(x3)P(x4)
23
Computing with Bayes nets
sum over 8 values
P(x1, x2, x3, x4) P(x1x3, x4)P(x2x3)P(x3)P(x4)
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Computing with Bayes nets
P(x1, x2, x3, x4) P(x1x3, x4)P(x2x3)P(x3)P(x4)
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Computing with Bayes nets
sum over 4 values
P(x1, x2, x3, x4) P(x1x3, x4)P(x2x3)P(x3)P(x4)
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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

• Assume grass will be wet if and only if it
  rained last night, or if the sprinklers were left
  on

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Explaining away
Compute probability it rained last night, given
that the grass is wet
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Explaining away
Compute probability it rained last night, given
that the grass is wet
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Explaining away
Compute probability it rained last night, given
that the grass is wet
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Explaining away
Compute probability it rained last night, given
that the grass is wet
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Explaining away
Compute probability it rained last night, given
that the grass is wet
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Explaining away
Compute probability it rained last night, given
that the grass is wet and sprinklers were left
on
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Explaining away
Compute probability it rained last night, given
that the grass is wet and sprinklers were left
on
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Explaining away
Discounting to prior probability.
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Contrast w/ production system
Rain
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

38
Contrast w/ spreading activation
Rain
Sprinkler
Grass 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!

• Excitatory links Rain Wet, Sprinkler
  Wet

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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

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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

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Contrast w/ spreading activation
(McClelland Rumelhart, 1981)
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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

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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)

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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

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

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Interventions

• Four random variables
• X1 coin toss produces heads
• X2 pencil levitates
• X3 friend has psychic powers
• X4 friend has two-headed coin

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Learning causal graphical models

• Strength how strong is a relationship?
• Structure does a relationship exist?

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Causal structure vs. causal strength

• Strength how strong is a relationship?

B
B
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Causal structure vs. causal strength

• Strength how strong is a relationship?
• requires defining nature of relationship

B
B
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Parameterization

• Structures h1 h0
  
• Parameterization

C
B
C
B
E
E
C
B
h1 P(E 1 C, B)
h0 P(E 1 C, B)
0 0 1 0 0 1 1 1
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Parameterization

• Structures h1 h0
  
• Parameterization

C
B
C
B
E
E
C
B
h1 P(E 1 C, B)
h0 P(E 1 C, B)
0 0 1 0 0 1 1 1
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Parameterization

• Structures h1 h0
  
• Parameterization

C
B
C
B
E
E
C
B
h1 P(E 1 C, B)
h0 P(E 1 C, B)
0 0 1 0 0 1 1 1
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Parameter estimation

• Maximum likelihood estimation
• maximize ?i P(bi,ci,ei w0, w1)
• Bayesian methods as in Part I

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Causal structure vs. causal strength

• Structure does a relationship exist?

B
B
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Approaches to structure learning

• Constraint-based
• dependency from statistical tests (eg. ?2)
• deduce structure from dependencies

C
B
B
E
(Pearl, 2000 Spirtes et al., 1993)
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Approaches to structure learning

• Constraint-based
• dependency from statistical tests (eg. ?2)
• deduce structure from dependencies

C
B
B
E
(Pearl, 2000 Spirtes et al., 1993)
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Approaches to structure learning

• Constraint-based
• dependency from statistical tests (eg. ?2)
• deduce structure from dependencies

C
B
B
E
(Pearl, 2000 Spirtes et al., 1993)
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Approaches to structure learning

• Constraint-based
• dependency from statistical tests (eg. ?2)
• deduce structure from dependencies

C
B
B
E
(Pearl, 2000 Spirtes et al., 1993)
Attempts to reduce inductive problem to deductive
problem
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Approaches to structure learning

• Constraint-based
• dependency from statistical tests (eg. ?2)
• deduce structure from dependencies

C
B
B
E
(Pearl, 2000 Spirtes et al., 1993)

• Bayesian
• compute posterior
• probability of structures,
• given observed data

C
B
C
B
E
E
P(h1data)
P(h0data)
P(hdata) ? P(datah) P(h)
(Heckerman, 1998 Friedman, 1999)
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Bayesian Occams Razor
h0 (no relationship)
P(d h )
h1 (relationship)
All possible data sets d
For any model h,
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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

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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)

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Causation from contingencies
C present (c)
C absent (c-)
a
c
E present (e)
d
b
E absent (e-)
Does C cause E? (rate on a scale from 0 to 100)
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Two models of causal judgment

• Delta-P (Jenkins Ward, 1965)
• Power PC (Cheng, 1997)

Power
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Buehner and Cheng (1997)
People
DP
Power
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Buehner and Cheng (1997)
People
DP
Power
Constant ?P, changing judgments
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Buehner and Cheng (1997)
People
DP
Power
Constant causal power, changing judgments
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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
B
73
Causal strength

• Assume structure
• DP and causal power are maximum likelihood
  estimates of the strength parameter w1, under
  different parameterizations for P(EB,C)
  
• linear ? DP, Noisy-OR ? causal power

B
74
Causal structure

• Hypotheses h1 h0
• Bayesian causal inference
• support

B
B
P(dh1)
likelihood ratio (Bayes factor) gives evidence in
favor of h1
P(dh0)
75
Buehner and Cheng (1997)
People
DP (r 0.89)
Power (r 0.88)
Support (r 0.97)
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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
0/8 0/8
P(ec)
8/8 8/8
6/8 6/8
4/8 4/8
2/8 2/8
P(ec-)
100 50 0
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)
80
Backwards blocking (Sobel, Tenenbaum Gopnik,
2004)
A Trial
AB 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)

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Possible hypotheses
B
A
E
B
A
B
A
B
A
B
A
B
A
B
A
B
A
B
A
E
E
E
E
E
E
E
E
B
A
B
A
B
A
B
A
B
A
B
A
B
A
B
A
E
E
E
E
E
E
E
E
B
A
B
A
B
A
B
A
B
A
B
A
B
A
B
A
E
E
E
E
E
E
E
E
82
Bayesian inference

• Evaluating causal models in light of data
• Inferring a particular causal relation

83
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(h10) q(1 q)
P(h01) (1 q) q
P(h11) q2
B
A
B
A
B
A
B
A
E
E
E
E
P(E1 A0, B0) 0 0
0
0 P(E1 A1, B0) 0
0 1
1 P(E1 A0, B1) 0
1 0
1 P(E1 A1, B1)
0 1
1 1
86
Modeling backwards blocking
P(h00) (1 q)2
P(h10) q(1 q)
P(h01) (1 q) q
P(h11) q2
B
A
B
A
B
A
B
A
E
E
E
E
P(E1 A1, B1) 0
1 1
1
87
Modeling backwards blocking
P(h10) q(1 q)
P(h01) (1 q) q
P(h11) q2
B
A
B
A
B
A
E
E
E
P(E1 A1, B0) 0
1
1 P(E1 A1, B1)
1 1
1
88
Manipulating prior probability(Tenenbaum, Sobel,
Griffiths, Gopnik, submitted)
A Trial
Initial
AB 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

90
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