Bayesian Networks: Review Representation, independence, and inference a few problems for your enjoym

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Bayesian Networks ReviewRepresentation,
independence, and inference( a few problems for
your enjoyment)

• Stanislav Funiak
• 10-701 Recitation, 3/30/2006

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Conference Submission Network
Done
Beer
in Time
Quality
Sleep1
Sleep2
Comm- ents1
Comm-ents2
Recom-mended
Accepted
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Bayesian Network

• DAG, each node is a variable
• For each var. X, CPD p(X Pa X)
• represents the distribution as

A,B,D,T,Q,S1,S2,C1,C2,R
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Smaller Network
C2t
Qt
C2f
Qf
p(S1f, Qt, C1f, C2t, Rt, At)
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Independence Relations

• where the force lies
• example independencies

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Factorization gt Independence relations

• We have seen
• starting from p factorizes according to G
• show p satisfies some independence relations
• Which independence assumptions in G?

p factorizes according to G
p satisfies indep. relations in G
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Independence relations encoded in G

• 1. Local Markov Assumptions
• X indep. NonDescendants(X) Pa X

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Independence relations encoded in G

• 2. absence of active trails
• Variables X indep of variables Ygiven Z if no
  active trail betweenX and Y given Z

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Active trail Review

• A path X1 X2 Xk is an active trail
  when variables OµX1,,Xn are observed if for
  each consecutive triplet in the trail
• Xi-1?Xi?Xi1, and Xi is not observed (Xi?O)
• Xi-1?Xi?Xi1, and Xi is not observed (Xi?O)
• Xi-1?Xi?Xi1, and Xi is not observed (Xi?O)
• Xi-1?Xi?Xi1, and Xi is observed (Xi2O), or one
  of its descendents

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Independence relations encoded in G

• 2. absence of active trails
• Variables X indep of variables Ygiven Z if no
  active trail betweenX and Y given Z

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Independence relations encoded in G

• 1. Local Markov Assumptions
• 2. absence of active trails (d-separation)

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Independence relations gt Factorization

• Before
• How about

p factorizes according to G
p satisfies indep. relations in G
p factorizes according to G
p satisfies indep. relations in G
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Independence relations gt Factorization

• Suppose
• Prove that p factorizes according to G

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Queries

• Marginal probability
• Most probable explanation (MPE)
• Active data collection

Variable elimination
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Conditioning on evidence
Q
S1
C1
C2
R
C2t
Qt
A
C2f
Qf
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Marginal probability query

• Lets compute

Q
S1
C1
C2
R
A
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Most probable explanation

• Now, lets compute

Q
S1
C1
C2
R
A
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Do we need this algorithm?

• Couldnt we just take argmax of marginals?

Take
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What you need to know

• Representation
• Independence relations
• local Markov assumption
• active trails / d-separation
• independence relations gt factorization
• Variable elimination
• marginal queries
• argmax queries