Probabilistic Horn abduction and Bayesian Networks

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Probabilistic Horn abduction and Bayesian Networks

• David Poole
• presented by Hrishikesh Goradia

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Introduction

• Logic-based systems for diagnostic problems
• Too many logical possibilities to handle
• Many of the diagnoses not worth considering
• Bayesian networks
• Probabilistic analysis
• Probabilistic Horn Abduction
• Framework for logic-based abduction that
  incorporates probabilities with assumptions
• Extends pure Prolog in a simple way to include
  probabilities

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Motivating Example
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Motivating Example
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Probabilistic Horn Abduction Theory
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Probabilistic Horn Abduction Theory
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Assumptions and Constraints

• Identical hypotheses cannot appear in multiple
  disjoint declarations.
• All atoms in disjoint declarations share the same
  variables.
• Hypotheses cannot form the head of rules.
• No cycles in the knowledge base.
• Knowledge base is both covering and disjoint.

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Bayesian Networks to Probabilistic Horn
Abduction Theory

• A discrete Bayesian network is represented by
  Probabilistic Horn abduction rules that relates
  a random variable ai with its parents ai1, ,
  ain
• The conditional probabilities for the random
  variable are translated into assertions

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Bayesian Networks to Probabilistic Horn
Abduction Theory
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Bayesian Networks to Probabilistic Horn
Abduction Theory
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Probabilistic Horn Abduction Theory to Bayesian
Networks

• Each disjoint declaration maps to a random
  variable.
• Each atom defined by rules also corresponds to a
  random variable.
• Arcs go from the body RV(s) to the head RV in
  each rule.
• Probabilities in the disjoint declarations map
  directly to the conditional probabilities for the
  RVs
• Additional optimizations possible.

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Discussion Independence and Dependence

• Can the world be represented such that all of the
  hypotheses are independent?

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Discussion Independence and Dependence

• Can the world be represented such that all of the
  hypotheses are independent?
• Author claims that it is possible.
• Reichenbachs principle of the common cause If
  coincidences of two events A and B occur more
  frequently than their independent occurrence,
  then there exists a common cause for these events
  

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Discussion Abduction and Prediction

• Is abducing to causes and making assumptions as
  to what to predict from those assumptions the
  right logical analogue of the independence in
  Bayesian networks?

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Discussion Abduction and Prediction

• Is abducing to causes and making assumptions as
  to what to predict from those assumptions the
  right logical analogue of the independence in
  Bayesian networks?
• Author claims that it is true.
• Approach is analogous to Pearls network
  propagation scheme for computing conditional
  probabilities.

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Discussion Causation

• Common problem associated with logical
  formulation of causation If c1is a cause for a
  and c2 is a cause for a, then from c1 we can
  infer c2. Does the probabilistic Horn abduction
  theory overcome this?

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Discussion Causation

• Common problem associated with logical
  formulation of causation If c1is a cause for a
  and c2 is a cause for a, then from c1 we can
  infer c2. Does the probabilistic Horn abduction
  theory overcome this?
• Author claims that it does.
• The Bayesian network represented by the theory
  will have c1 and c2 as disjoint RVs.

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Summary

• Presents a simple framework for Horn clause
  abduction, with probabilities associated with
  hypotheses.
• Finds a relationship between logical and
  probabilistic notions of evidential reasoning.
• Presents a useful representation language that
  provides a compromise between heuristic and
  epistemic adequacy.