Magic Bayes-Ball

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Magic Bayes-Ball

• Livia Predoiu

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Overview

• Motivation
• Preliminaries
• Bayesian Networks
• Bayesian Datalog Programs
• Algorithms
• The Bayes-Ball algorithm for BN
• The class of the Magic Sets algorithms
• Magic Bayes-Ball

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Motivation

• Why a MBB algorithm ?
• Speedup of the inference in BDPs
• (only the minimal relevant part of the BN that
  is equivalent to the BDP is considered)
• procedural definition of the D-Separation for
  BDPs

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Bayesian Networks

• Compact representation of the full joint
  probability distribution

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BDPs

• BDPs Bayesian Datalog Programs
• Datalog fragment of BLPs Kersting De Raedt,
  2000
• BLP Set of prolog clauses with a instead of a
  ?.
• To each clause a cpd is associated
• p(I(head)I(body1), , I(bodyn))
• An atom has an assignment I from a domain that is
  associated to the relation symbol
• A combination rule for each recursive relation
  symbol
• compact representation of a BN

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BDPs an example
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Inference in BDPs

• Querying a BDP
• ?- Q1, , Qn E1e1, , Emem.
• Inference
• 1. construction of the relevant part of the
  equivalent BN
• SLD trees ? support networks ? Union of all
• support networks
• 2. BN inference in the constructed BN

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Inference in BDPs example

• Example query ?- bt(dorothy) pc(dorothy) a
• 1. construction of the SLD tree

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Inference in BDPs example

• example query ?- bt(dorothy) pc(dorothy) a
• 2. Transformation of the SLD tree to the support
  network

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Inference in BDPs

• Disadvantages of this kind of inference
• BN can be to big
• Several proves of the same query or evidence
  variable can occur
• Idea generate the minimal relevant part of the
  BN with a logic Bayes-Ball algorithm

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Bayes-Ball algorithm for BN
(Shachter, 98)

• procedural definition of D-Separation for BN
• A ball is thrown through the net (D-Separation) ?
  Determination of the minimal set of relevant
  nodes for the query

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BB example
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BB example
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Magic Sets algorithms

• Transformation algorithms for Datalog programs
• Forward-Chaining on MQ is goal-oriented
• Supplementary Magic Sets (Sacca Zaniolo, 1987)
  is method of choice (additional supplementary
  predicates rules
  
• ? Acceleration of the inference)

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Magic Sets algorithms

• requirements
• program P must be rectified
• binding pattern of the recursive predicates
  ? Rule/Goal-Graph
• unique binding property ( each recursive
  predicate has 1 binding pattern) ? can be
  created
• Supplementary Magic Sets
• Semi-naive Forward-Chaining

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Supplementary Magic Sets

• Supplementary Magic Sets
• M all facts of P
• M M ? magic fact
• M M ? magic rules ? supplementary rules

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Magic Sets algorithms example

• binding pattern bt b mc b pc b

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Magic Bayes-Ball algorithm

• Logic Bayes-Ball algorithm
• Advantage
• Derivation of a minimal BN that is relevant to
  the query
• realization
• R Set of all relevant atoms
• For all query variables Magic Backward-Chaining
  ? Throw the ball to the parents
• For all evidence variables Magic
  Forward-Chaining ? Throw the ball to the children
• Throw the ball to the parents of a derived
  relevant evidence atom (Backward-Chaining)

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Magic Bayes-Ball Magic Backward-Chaining

• Ball is thrown to the parents
• An evidence atom needs only one proof. ? It
  doesnt throw the ball to the parents
• Realization
• S Set of nodes with selected suspicious atom
• when the answer substitution is known, then a
  connected part of the path in the SLD tree can
  possibly be deleted
• At the node k ? S that is nearest to the root and
  whose selected atom is an evidence atom do
• replace r by the evidence atom at the edge and
• add the first node below that doesnt contain an
  atom from body(r)? or its expansions,
  respectively

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Magic Bayes-Ball Magic Forward-Chaining

• Foward-Chaining on MEi with EDB EDB ? R the
  atoms from R are marked
• If a rule is used that has a marked atom, then
  the derived atom has to be marked, too.
• Derivation of a marked evidence atom ? Ball is
  thrown to the parents predecessors (these get a
  secend mark and are added to R) no usage of the
  evidence atom anymore (d-Separation)

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Magic Bayes-Ball Magic Forward-Chaining

• Construction of the SNs
• atoms with second mark become nodes of the SN.
  For each rule a directed edge from each body atom
  to the head atom is added.
• HB(P) ? HB(MEi) ? additional atoms from MEi have
  to be deleted atoms with
• magic predicates or supplementary predicates (are
  deleted, edges are redirected)
• Binding pattern predicates (replacement with
  original predicate)

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Magic Bayes-Ball Ablauf

• For all query atoms Do magic Backward-Chaining
• While R ? Rnew Do
• For all evidence atomts Do magic Forward-Chaining

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Magic Bayes-Ball

• Only one pass through the Forward-Chaining
• phase is not enough

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Magic Bayes-Ball

• Only one pass through the Forward-Chaining
• phase is not enough

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Summary

• BB determines for a query the minimal relevant
  part of the BN
• MBB is a logic equivalent to the BB
• MBB is a procedural definition of the
  D-Separation at BDPs
• Goal-oriented Forward-Chaining by means of
  Supplementary Magic Sets (Ball is thrown to the
  children)
• Iteration of the Magic Forward-Chaining essential
  for completeness