Graphical models Tom Griffiths UC Berkeley

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Graphical modelsTom GriffithsUC Berkeley
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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)

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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
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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

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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
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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
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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

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Graphical models

• Introduction to graphical models
• definitions
• efficient representation and inference
• explaining away
• Graphical models and cognitive science
• uses of graphical models

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Graphical models

• Introduction to graphical models
• definitions
• efficient representation and inference
• explaining away
• Graphical models and cognitive science
• uses of graphical models

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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

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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
• neural networks (e.g. Boltzmann machines)

X2
X5
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Ising models
X1
X2

• Consist of
• a set of nodes
• a set of edges
• a potential for each clique, multiplied together
  to yield the distribution over variables
• Distribution is specified as

X3
X4
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Ising models
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Boltzmann machines
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
• Distribution is specified as

X2
X5
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Boltzmann machines
True image
Boltzmann
PCA
PCA
Boltzmann
(Hinton Salakhutdinov, 2006)
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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

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Graphical models

• Introduction to graphical models
• definitions
• efficient representation and inference
• explaining away
• Graphical models and cognitive science
• uses of graphical models

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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

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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

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1
4
2
total 8 (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

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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

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Identifying independence
X1 and X3 dependent
X1 and X3 independent
X2
X2
X2
X2
X2
X2
(shaded variables are observed)
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Identifying independence

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

X4 and X2 are independent
X3
X4
X3
X4
X1
X2
X1
X2
X4 and X2 are independent
X4 and X2 are dependent
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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

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Computing with Bayes nets
X3
X4
X1
X2
P(x1, x2, x3, x4) P(x1x3, x4)P(x2x3)P(x3)P(x4)
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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

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Logic and probability

• Bayesian networks are equivalent to a
  probabilistic propositional logic
• Associate variables with atomic propositions
• Bayes net specifies a distribution over possible
  worlds, probability of a proposition is a sum
  over worlds
• More efficient than simply enumerating worlds
• Developing similarly efficient schemes for
  working with other probabilistic logics is a
  major topic of current research

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Graphical models

• Introduction to graphical models
• definitions
• efficient representation and inference
• explaining away
• Graphical models and cognitive science
• uses of graphical models

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Identifying independence
X1 and X3 dependent
X1 and X3 independent
X2
X2
X2
X2
X2
X2
(shaded variables are observed)
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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

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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
• Support kinds of inference that are problematic
  for other cognitive models explaining away
• hard to capture in a production system
• more natural than with spreading activation

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Graphical models

• Introduction to graphical models
• definitions
• efficient representation and inference
• explaining away
• Graphical models and cognitive science
• uses of graphical models

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Uses of graphical models

• Understanding existing cognitive models
• e.g., neural network models

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Sigmoid belief networks
y

• We can view multilayer perceptrons as Bayes nets
  with specific probabilities
• (e.g., Neal, 1992)
• Makes it possible to use Bayesian tools with
  existing neural network models
• (e.g., Mackay, 1992)

z1
z2
x1
x2
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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)

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The holism of confirmation

• If everything we know is one big probability
  distribution, then discovering one small fact
  requires changing all of our beliefs
• Used by Fodor (2001) as an argument against the
  possibility of inductive logic
• Bayes nets everything can be connected to
  everything, but inference can still be efficient

vs.
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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,

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Graphical models and coinflipping
q
d1
d2
d3
d4
d1
d2
d3
d4
d1
d2
d3
d4
Hidden Markov model si Fair coin, Trick
coin
Fair coin P(H) 0.5
P(H) q
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A hierarchical Bayesian model
physical knowledge
Coins
q Beta(FH,FT)
FH,FT
...
Coin 1
Coin 2
Coin 200
q200
q1
q2
d1 d2 d3 d4
d1 d2 d3 d4
d1 d2 d3 d4
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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,
• Modeling human causal reasoning
• more on Friday!

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