Inference in Bayesian Networks

1
Inference in Bayesian Networks
MINDLab. Seminars on Reasoning and Planning under
Uncertainty

• Ugur Kuter
• MIND Lab.
• 8400 Baltimore Avenue, Ste. 200
• College Park, Maryland, 20742
• Web Site for the seminars http//www.cs.umd.edu/u
  sers/ukuter/uncertainty/

2
From Last Week

• A Bayesian Network is a DAG
• Nodes represent the events
• Arcs represent the causal influences between the
  linked events
• The strength (i.e. the quantification) of a
  causal link is defined directly by the
  conditional probabilities on the linked events
• Conditional Independence in Bayesian Networks
• The event a d-separates the event b from c, if
  
• along every undirected link between b and c,
  there is an event w that satisfies the following
• if w does not have converging arrows then it is
  equal to a
• if w has converging arrows then a is not equal to
  w or any of ws descendants

3
Inference over Bayesian Networks

• Inference ? computing/updating our belief in some
  designated query events, given the values for
  some evidence
• Given that I know a and c occurred in the world,
  what is the probability that e and b will
  occur/has occurred?
• Exact vs. Approximate Inference
• Predictive vs. Diagnostic Inference
• Different inference algorithms for different
  structures of the network models
• Singly-Connected Networks
• Multiply-Connected Networks

a
b
c
d
e
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Exact Inference in Singly-Connected Networks

• Singly-connected networks
• there is at most one path between each pair of
  events
• e.g, chains, trees, poly-trees (a.k.a., forests)
• Note no loops
• Predictive inference is done via the chain rule
  
• of the probability theory
• P(a,b, , f) P(a) P(b a) . P(f a, b, ,
  e)
• Diagnostic inference is done via the chain rule
  and the Bayes rule
• P(a b) P(b a) P(a) / P(b)

a
b
b
c
f
e
d
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Exact Inference over Chains

• Two-event chains
• Prediction Given an evidence on , what is
  the probability of ?
• The value of the conditional probability P( b a
  )
• Diagnosis Given an evidence on , what is
  the probability of ?
• The value of the conditional probability P( a b
  )
• Use Bayes Rule to compute this conditional
  probability
• P(a b) P(b a) P(a) / P(b)

a
b
a
b
b
a
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Exact Inference over Chains (contd)

• Multiple-Event Chains
• Prediction Given an evidence on , what is
  the probability of ?
• P( c a ) ?Btrue,falseP(c B, a)
  ?Btrue,falseP(c B) P(B a)
• Diagnosis Given an evidence on , what is
  the probability of ?
• Use Bayes Rule to compute this conditional
  probability
• P(a c) P(c a) P(a) / P(c)

a
b
c
a
c
c
a
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Exact Inference over Trees and Poly-Trees

• Pearls Message Passing Techniques
• Based on successive local computations
• of beliefs for each event
• Inference via propagating messages
• between the events
• Two types of messages associated with each event
  x
• ?-messages x receives the message ?(x) from its
  parents
• ?-messages x receives the message ?(x) from its
  children

a
b
?
?
?
?
c
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Message Passing

• Three specific parameters for computing
• the belief of an event, say c
• Conditional Probability Table for c
• i.e., P(c parents(c))
• Predictive Support
• the probability of the parents of c,
• given all the evidence connected to a and b,
• except via c
• ?(c) P(c parents(c)) P(parents(c) all
  evidence)
• Diagnostic Support
• The probability of all of the evidence connected
  to the children of c, except via c
• ?(c) P(all evidence except via c c)

a
b
?(c)
c
f
?(c)
e
d
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Message Passing over Trees
a
?(c)

• Consider the event c
• Our belief in the occurrence of c is
• P(c all evidence) ? P(all evidence via
  children of c c)
• P(c all evidence via parents of c)
• In other words, our belief in c is computed by
• ? ?(c) ?(c)

?(a)
b
c
?(c)
e
d
f
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Computing ?(c) and ?(c)

• Computing the ?-Message
• ?(c) 1, if we have evidence on c
• ?(c) 0, if we do not have evidence on c
• ?(c) ?c(D) ?c(E) ?c(F), otherwise
• Computing the ? -Message
• ?(c) ? P( c A) ?c(A)

a
?(c)
b
c
?(c)
e
d
f
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Computing Individual ?-Messages

• Consider the message from the event C to its
  child D
• If any evidence that says c occurs in the world
• then
• ?D(c) 1, ?E(c) 1
• If any evidence that says c does not occur in the
  world
• then
• ?D(c) 0, ?E(c) 0
• Otherwise,
• ?D(c) ? ?(c) ? (?X ? D ?c(X))

a
?(c)
b
c
?(c)
e
d
f
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Computing Individual ?-Messages

• The message from the node D to its parent C
• ?D(c) ?Dtrue,false ?(D) P(D c)

a
?(c)
b
c
?(c)
e
d
f
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Example The Flow of Messages
a
b
c
e
f
e
d
f
Evidence
e
f
Evidence
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The Flow of Messages - Step 1
?-message
a
?-message
Node ready to send a message
b
c
e
f
e
d
f
Evidence
e
f
Evidence
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The Flow of Messages - Step 2
?-message
a
?-message
Node ready to send a message
b
c
e
f
e
d
f
Evidence
e
f
Evidence
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The Flow of Messages - Step 3
?-message
a
?-message
Node ready to send a message
b
c
e
f
e
d
f
Evidence
e
f
Evidence

• And so on, until the network reaches to an
  equilibrium
• Theoretically, if message passing is done
  infinitely often, then the technique computes an
  exact probability for each non-evidence event

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Message Passing in Poly-Trees

• Poly-Trees (i.e., forests)
• There is only one path between every pair of
• events in the network
• Recall that
• Our belief in the occurrence of c is
• P(c all evidence) ? P(all evidence via
  children of c c)
• P(c all evidence via parents of
  c)
• In other words, P(c all evidence) ? ?(c) ?(c)

a
b
?
?
?
?
c
f
e
d
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Computing ?(c) and ? (c)

• These are compound messages
• ?-Message from all evidence through all children
  of c
• ?-Message from all evidence through all parents
  of c
• We compute these messages via its individual
  components
• Computing a ?-Message
• ?(c) 1, if we have evidence on c
• ?(c) 0, if we do not have evidence on c
• ?(c) ?c(D) ?c(E)
• Computing a ?-Message
• ?(c) ? P( c parents(C)) ?X ? parents(C)
  ?X(X)

a
b
?x(a)
?x(b)
c
f
?x(d)
?x(e)
e
d
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Computing the individual ?-messages

• Consider the event c
• If any evidence that says c occurs in the world
• then
• ?D(c) 1, ?E(c) 1
• If any evidence that says c does not occur in the
  world
• then
• ?D(c) 0, ?E(c) 0
• Otherwise,
• ?D(c) ? ? (?X ? D ?c(X))
• ( ? P( c A, B) ?X ? A,B ?c(X) )

a
b
?x(a)
?x(b)
c
f
?x(e)
e
d
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Computing the individual ?-messages

• The message from a node D to its
• parent C is computed as follows
• ?D(c) ?Dtrue,false ?(D)
• ? P( D parents(D) - C)
• ?x ? parents(D) - C ?c(x) )

a
b
?x(a)
?x(b)
c
f
?x(e)
e
d
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Characteristics of Message Passing

• The algorithm uses the locality principle, i.e.,
  it considers information from a events immediate
  parents and children
• If the number of parents is small then the
  message passing quickly converges to an
  equilibrium (or near-equilibrium)
• Otherwise, the algorithm is not feasible
• Since the computation of messages are exponential
  in the number of parents of an event

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Summary

• Exact Inference Algorithms over Singly-Connected
  Networks
• Inference is done based on
• Conditional Probabilities
• Predictive support for the occurrence of an event
• Given the ancestors, how is our belief of the
  event affected?
• Diagnostic support for the occurrence of an
  event
• Given the descendants, how is our belief of the
  event affected?
• Different inference mechanisms for different
  network structures
• Chains Simple application of the Bayes Rule and
  the Chain Rule
• Trees and Poly-Trees Message passing techniques
• Computationally expensive in most cases,
  infeasible in some cases

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Schedule for the Weeks Ahead (Tentative)

• Next week (October 27)
• no seminar
• November 3
• Exact Inference Algorithms over
  Multiply-Connected Networks
• November 10, 17, and 24
• no seminars
• December 1
• Approximate Inference Algorithms