Generating Bayesian Networks from Probability Logic knowledge Bases

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Generating Bayesian Networks from Probability
Logic knowledge Bases

• Peter Haddawy

Kim Kangil
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Contents

• Introduction
• Representation Language
• Bayesian Knowledge Bases
• Network Generation Algorithm
• Conclusion

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Introduction

• Problem of Bayesian network
• Propositional model
• Static representation
• Approach
• Use Knowledge base of probability sentences

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Representation Language

• Network fragment
• g consequent
• f antecedent
• x is a subset of Random variable X

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Contd

• Rule

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Contd

• Definition for each conditional probability
  sentence

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Contd

• Constraints for isomorphic mapping to Bayesian
  network

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Bayesian Knowledge base

• We have to define some concepts for representing
  the conditional probability completely
• Definition 1 Path, directed edge, converging
  arrows, direct predecessor, direct successor,
  root, leaf

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Contd

• Definition 2 d-seperation
• Condition about ground terms to define Z
• W has converging arrows and none of W or its
  descendents are in Z, or
• W does not have converging arrows and W is in Z

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Contd

• Definition 3 Bayesian knowledge base

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Contd

• Lemma
• A ground term in a Bayesian knowledge base is
  independent of all ground terms which are not its
  successors, given its direct predecessors
• 4 case for proof this
• The path is a direct link from G to F
• The path is a direct link from F to G
• The path must pass through one of Gs direct
  predecessors
• The path must pass through one of Gs direct
  successors

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• Theorem 1
• A Bayesian knowledge base is a complete
  specification of a joint probability distribution
  over the ground terms contained in any non-empty
  set of ground instances of its rules in which
  every ground term is the consequent of some rule
  instance.

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Network generation algorithm

• Backward chaining
• P(QE)
• Theorem 2
• The equality of probability between Bayesian
  Knowledge base and network

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Examples
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Conclusion

• From Bayesian Knowledge base, we can get a
  network completely matched to Bayesian network.
• We dont need to consider some variables not used
  in a Query and evidence.